Let k and n be integers such that 1 k n. Find the dimension of the vector space of all k-forms on . A) B) n C) k D) n - k E) n(n - 1)(n - 2).....(n - k +1)

The answer of Let k and n be integers such...

Let φ and ψ be two 1-forms on say φ= d + d , ψ= d + d . Show that φ∧ψ = 0 if and only if φ = c ψ for some constant real number c.

The answer of Let φ and ψ be two 1-forms...

If is a k -form, are l-forms on , then = implies = .

The answer of If is a k -form,...

L = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that . A) = t dx + (25 - 2t) dy, t R B) = t dx + (29 - 2t) dy, - < t < C) = t dx + (29 + 2t) dy, - < t < D) = (25 + 9t) dx + t dy, t R E) = (29 - 9t) dx + t dy, - < t <

The answer of L = 9dx - 2dy and...

Simplify a dy dz + b dy + c dx + (a + b +c) dy. A) (a - b + c) dz B) (a +b + c) dz C) (a - b - c) dz D) (a + b -c) dz E) (a + b +c) dy

The answer of Simplify a dy ...

Let e i , i = 1, 2, 3, 4 be the standard basis vectors in R 4 and let A) 32 B) -56 C) -40 D) -14 E) 40

The answer of Let e i , i = 1,...

Expand and simplify: (2a dx + (b + 2a)dy + c dz)∧(a dx + 2b dy + c dz) - (2a 2 - 2ab) ( a - b) dy∧dx . Express your answer in terms of the basis vectors dy∧dz, dz∧dx, and dx∧dy of

The answer of Expand and simplify: (2a dx + (b...

Let η = F 1 dy∧dz - F 2 dx∧dz+ F 3 dx∧ j + b 3 k be vectors in . If η is identified by the vector w = F 1 i + F 2 j + F 3 k then η (u, v) is equal to A) w . (u + v) B) w . (u × v) C) w. (v - u) D) w . (v × u) E) w × (u × v)

The answer of Let η = F 1 dy∧dz -...

Let be a differential k-form and be a differential -form on a domain D and letd( ) (D). Express m in terms of k and l. A) m = k + B) m = k C) m = k + + 2 D) m = k + + 1 E) m = k + 1

The answer of Let be a differential k-form...

Let = xdx + vdv, = dw, = du be differential forms in a domain D .Find d( ). A) 12ut dw B) 6xut dw C) 3x dw D) 3x E) -3x dw

The answer of Let = xdx + vdv,...

Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ($\bigtriangledown$g(x)) D) $\bigtriangledown$ ×$\bigtriangledown$(g(x)) E) $\bigtriangledown$(g(x)) g(x)

The answer of Let g(x) be a differential 0-form on...

Find a differential form such that = 3 dzNote: answer is not unique. A) dz + dx - dy B) xdx + ydy + zdz C) dz + dx + dy D) dz + dx -4 dx dy E) (x + y + z ) dz

The answer of Find a differential form ...

Which of the following is an antiderivative of (3x + 4xy) dy? A) (9xdx -16 ydy) B) (x + y) dx?dy C) (xdx - ydy) D) (x + y) E) - (xdx + ydy)

The answer of Which of the following is an antiderivative...

Let the differential 1-form = dz + dx + dy be defined in a star-like domain .(a) Is closed?(b) Is exact on D? If so, find a differential 1-form such that = . A) is neither closed nor exact B) is both closed and exact, = (xz - + )dx + (yz + sin(z)) dz C) is both closed and exact, = (xz + + )dx + (yz + sin(x)) dy D) is closed but not exact E) is both closed and exact, = zxdx+ xydy + yzdz

The answer of Let the differential 1-form ...

Let the differential 2-form = (3 + 2xy + 6 ) dy be defined in a star-like domain . (a) Is closed? (b) Is exact on D? If so, find a differential 1-form such that = . A) is closed but not exact B) is both closed and exact, = - (x + 2 )dx + dy C) is neither closed nor exact D) is both closed and exact, = (x + 2 )dx + dy E) is both closed and exact, = dx - (x + 2 )dy

The answer of Let the differential 2-form ...

Let the differential 2-form = dz + dx + (1 - 2z) dy be defined in a star-like domain . (a) Is closed? (b) Is exact on D? If so, find a differential 1-form such that = . A) is neither closed nor exact B) is both closed and exact, = ydx + 2xzdy - xydz C) is closed but not exact D) is exact but not closed, = ydx + 2xzdy + xydz E) is both closed and exact, = -ydx - 2xzdy - xydz

The answer of Let the differential 2-form ...

Find the 2-volume of the 2-parallelogram in spanned by the vectors v 1 = (0, - 1, -2, -1) and v 2 = (1, 3, 7, 1). A) 18 units 2 B) 36 units 2 C) 6 units 2 D) 9 units 2 E) 8 units 2

The answer of Find the 2-volume of the 2-parallelogram in...

Find the 3-volume of the 3-parallelogram in spanned by the vectors v 1 = (2, 3, 1, 0),v 2 = (0, -3, -2, 1), and v 3 = (1, 1, 1, 1). A) 10 units 3 B) 50 units 3 C) 100 units 3 D) 540 units 3 E) 6 units 3

The answer of Find the 3-volume of the 3-parallelogram in...

Explore all topics

Deck 18: Differential Forms and Exterior Calculus

Question

Let , , be 1-forms and be a 2-form on . Which of the following is a 3-form on ?

)

11ee7972_a99a_a1a7_88d3_5bac39f31ff0_TB9661_11
D)11ee7971_d7c9_b5d4_88d3_75d03a1ac3e7_TB9661_11 11ee7973_0b66_7d59_88d3_6bcf237f8512_TB9661_11 11ee7972_5038_6616_88d3_2d4d40837c09_TB9661_11 11ee7973_0b66_7d59_88d3_6bcf237f8512_TB9661_11 11ee7972_a99a_a1a7_88d3_5bac39f31ff0_TB9661_11
E) All of the above

Use Space or

to flip the card.

Question

Let , be differential 1-forms and let be a differential 0-form on a domain D in Which of the following is a differential 2-form on D?

A) 11ee7971_d7c9_b5d4_88d3_75d03a1ac3e7_TB9661_11+ 11ee7972_5038_6616_88d3_2d4d40837c09_TB9661_11
B)11ee7971_d7c9_b5d4_88d3_75d03a1ac3e7_TB9661_11

11ee7972_5038_6616_88d3_2d4d40837c09_TB9661_11
C) 311ee7971_d7c9_b5d4_88d3_75d03a1ac3e7_TB9661_11 - 11ee7972_5038_6616_88d3_2d4d40837c09_TB9661_11
D)

+ 11ee7972_5038_6616_88d3_2d4d40837c09_TB9661_11
E) None of the above

Question

Find the dimension of ( ).

A) 7
B) 840
C) 4
D) 3
E) 35

Question

Let k and n be integers such that 1 k 11ee7973_99f4_ef7a_88d3_478f26d4adc3_TB9661_11 n. Find the dimension of the vector space of all k-forms on .

B) n
C) k
D) n - k
E) n(n - 1)(n - 2).....(n - k +1)

Question

Let = . List all basis vectors of ( ).

Question

is a k-form and

is an l-form on

, then

https://storage.examlex.com/TB9661/

Question

Let φ and ψ be two 1-forms on

say φ=

, ψ=

. Show that φ∧ψ = 0 if and only if φ = c ψ for some constant real number c.

Question

A) u . v
B) u × v
C) u + v
D)

Question

is a k -form,

and

are l-forms on

, then

implies

=https://storage.examlex.com/TB9661/

Question

A) u + v
B) u - v
C) u × v
D) v × u
E) v - u

Question

Let = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that .

A) 11ee7975_2bfd_ecfa_88d3_cb60d904737e_TB9661_11 = t dx +

(25 - 2t) dy, t

R
B) 11ee7975_2bfd_ecfa_88d3_cb60d904737e_TB9661_11 = t dx +

(29 - 2t) dy, -

< t < 11ee7975_081b_3819_88d3_837de92b3d96_TB9661_11
C) 11ee7975_2bfd_ecfa_88d3_cb60d904737e_TB9661_11 = t dx +

(29 + 2t) dy, -11ee7975_081b_3819_88d3_837de92b3d96_TB9661_11 < t < 11ee7975_081b_3819_88d3_837de92b3d96_TB9661_11
D) 11ee7975_2bfd_ecfa_88d3_cb60d904737e_TB9661_11 =

(25 + 9t) dx + t dy, t 11ee7974_eec1_32c8_88d3_178ace6e171f_TB9661_11 R
E) 11ee7975_2bfd_ecfa_88d3_cb60d904737e_TB9661_11 =

(29 - 9t) dx + t dy, -11ee7975_081b_3819_88d3_837de92b3d96_TB9661_11 < t < 11ee7975_081b_3819_88d3_837de92b3d96_TB9661_11

Question

= 0 for every k-form

Question

Simplify a dx dy 11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11 dz + b dxdy + c dy11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11dz11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11dx + (a + b +c) dydy.

A) (a - b + c) dx

dz
B) (a +b + c) dx11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11dy11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11dz
C) (a - b - c) dx11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11dy11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11dz
D) (a + b -c) dx11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11dy11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11dz
E) (a + b +c) dx11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11dz11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11dy

Question

Let $Let = 2dx + 5dy, = -dx + 7dy and \theta = -3dx + c dy be 1-forms on . Find the real number c such that = \theta . A) 12 B) 2 C) 17 D) -3 E) 0 <div style=padding-top: 35px>$ = 2dx + 5dy, $Let = 2dx + 5dy, = -dx + 7dy and \theta = -3dx + c dy be 1-forms on . Find the real number c such that = \theta . A) 12 B) 2 C) 17 D) -3 E) 0 <div style=padding-top: 35px>$ = -dx + 7dy and $\theta$ = -3dx + c dy be 1-forms on $Let = 2dx + 5dy, = -dx + 7dy and \theta = -3dx + c dy be 1-forms on . Find the real number c such that = \theta . A) 12 B) 2 C) 17 D) -3 E) 0 <div style=padding-top: 35px>$ . Find the real number c such that 11ee7bbb_18dc_373c_ae82_e55d2b6d55c1_TB9661_11 $Let = 2dx + 5dy, = -dx + 7dy and \theta = -3dx + c dy be 1-forms on . Find the real number c such that = \theta . A) 12 B) 2 C) 17 D) -3 E) 0 <div style=padding-top: 35px>$ 11ee7bbb_3770_208d_ae82_59bc8eff822f_TB9661_11 = 11ee7bbb_18dc_373c_ae82_e55d2b6d55c1_TB9661_11 11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11 $\theta$ .

A) 12
B) 2
C) 17
D) -3
E) 0

Question

Let $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ , $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ , $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ be m-forms, $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ be a k-form, and $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ be an l-form on $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ . Which of the following properties of the wedge product is not always true?

A) ( $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ + $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ ) $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 = $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 + $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11
B) (11ee7bbc_2f42_a9a0_ae82_03df16557ec6_TB9661_11 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 )11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbc_854e_b551_ae82_a1ccb48ff30b_TB9661_11 = 11ee7bbc_2f42_a9a0_ae82_03df16557ec6_TB9661_11 ( $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ 11ee7bbc_854e_b551_ae82_a1ccb48ff30b_TB9661_11 )
C) 11ee7bbc_2f42_a9a0_ae82_03df16557ec6_TB9661_11 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 = - 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbc_2f42_a9a0_ae82_03df16557ec6_TB9661_11
D) (a 11ee7bbc_2f42_a9a0_ae82_03df16557ec6_TB9661_11 )11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 = a ( $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 ) for a $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0. <div style=padding-top: 35px>$ R
E) for every k

\ge

1, there exists a zero k-form such that 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 + 0 = 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11, and 0 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbc_854e_b551_ae82_a1ccb48ff30b_TB9661_11 = 0.

Question

Let e_i , i = 1, 2, 3, 4 be the standard basis vectors in R⁴ and let

A) 32
B) -56
C) -40
D) -14
E) 40

Question

Let ψ = 12dy∧dz -10 dx∧dz + 8 dx∧dy be a 2-form on

. Express ψ as a product of two 1-forms.Hint: Add a suitable multiple of dz∧dz = 0 to ψ.

Question

Let σ = d

∧d

+ d

∧d

be a 2-form on

. Express σ as a product of two 1-forms.

Question

Show that if φ is a k-form on

, then φ∧φ = 0 if k is odd.

Question

Let

Let be the permutation that maps [1, 2, 3, 4, 5] to [3, 4, 5, 2, 1], then sgn( ) = -1.<div style=padding-top: 35px>

be the permutation that maps [1, 2, 3, 4, 5] to [3, 4, 5, 2, 1], then sgn(11ee7bbc_f5df_3c93_ae82_ade676104027_TB9661_11 ) = -1.

Question

Expand and simplify: (2a dx + (b + 2a)dy + c dz)∧(a dx + 2b dy + c dz) - (2a² - 2ab) ( a - b) dy∧dx . Express your answer in terms of the basis vectors dy∧dz, dz∧dx, and dx∧dy of

Expand and simplify: (2a dx + (b + 2a)dy + c dz)∧(a dx + 2b dy + c dz) - (2a<sup>2</sup> - 2ab) ( a - b) dy∧dx . Express your answer in terms of the basis vectors dy∧dz, dz∧dx, and dx∧dy of <div style=padding-top: 35px>

Question

Let ( ) be the vector space of all 3-forms on and ( ) be the vector space of all 5-forms on . If ( ) and ( ) have the same dimension, find n.

A) 2
B) 4
C) 8
D) 15
E) 8 or -1

Question

Let η = F₁ dy∧dz - F₂ dx∧dz+ F₃ dx∧dy j + b₃k
be vectors in . If η is identified by the vector w = F₁ i + F₂ j + F₃ k then η (u, v) is equal to

A) w . (u + v)
B) w . (u × v)
C) w. (v - u)
D) w . (v × u)
E) w × (u × v)

Question

Let $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$ ( $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$ ), 1 $\le$ k $\le$ n be the vector space of all k-forms on $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$ and let $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$ be the dimension of $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$ . Find $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$ .

A) $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$
B) $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$
C) $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$
D) $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$
E) $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1 <div style=padding-top: 35px>$ - 1

Question

Let (x) = (x) d + (x) d + (x) d be a differential 1-form on a domain D in and let . The value of 11ee7bbd_2d92_9334_ae82_4706b097eca0_TB9661_11 (x) on a vector v is equal to

A) 11ee7bbd_2d92_9334_ae82_4706b097eca0_TB9661_11 (x)

v
B) a(x) . v
C) a(x) × v
D) a(x) + v
E) v × a(x)

Question

Let be a differential k-form and be a differential -form on a domain D and letd(11ee7bbd_9e7e_0bb6_ae82_bb619a089092_TB9661_11 11ee7bbd_b7c7_e467_ae82_c37bb85a20c0_TB9661_11 ) (D). Express m in terms of k and l.

A) m = k + 11ee7bbe_0158_713a_ae82_97ce6608d721_TB9661_11
B) m = k 11ee7bbe_0158_713a_ae82_97ce6608d721_TB9661_11
C) m = k + 11ee7bbe_0158_713a_ae82_97ce6608d721_TB9661_11 + 2
D) m = k + 11ee7bbe_0158_713a_ae82_97ce6608d721_TB9661_11 + 1
E) m = k 11ee7bbe_0158_713a_ae82_97ce6608d721_TB9661_11 + 1

Question

Let = 4xz (y) dydz + z(2y + sin(2y)) dz11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11dx + (yz - 2 ) dx11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11dy. Find d11ee7bbe_8ef5_094b_ae82_6b47a89a60fc_TB9661_11 .

A) (y - 8z) dx

dz
B) (8z - y) dx11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11dy11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11dz
C) (y - 4zcos(2y)) dx11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11dy11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11dz
D) y
E) ydx11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11dy11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11dz

Question

Let = xdx + vdv, = dy dw, = dz11ee7bbf_c537_273d_ae82_09ae839f29a0_TB9661_11 dv11ee7bbf_c537_273d_ae82_09ae839f29a0_TB9661_11 du be differential forms in a domain D .Find d(11ee7bbf_ebe2_23df_ae82_0565ecd758af_TB9661_11 11ee7bc0_041d_24a0_ae82_87f021afe142_TB9661_11 11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 11ee7bc0_3408_e781_ae82_7de18ff0ad45_TB9661_11 ).

A) 12ut dx11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dy11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dz11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dt11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 du11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dv11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dw
B) 6xut dx11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dy11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dz11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dt11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 du11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dv11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dw
C) 3x

<strong>Let = xdx + vdv, = dy dw, = dz dv du be differential forms in a domain D .Find d( ).</strong> A) 12ut dx dy dz dt du dv dw B) 6xut dx dy dz dt du dv dw C) 3x dx dy dz dt du dv dw D) 3x E) -3x dx du dv dw <div style=padding-top: 35px>

E) -3x

du11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dv11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dw

Question

Let be a differential k-form and be a differential l-form on a domain D . State without proof a product rule for the exterior derivative of the wedge product 11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 https://storage.examlex.com/TB9661/11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11.

A) d11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11 +

<strong>Let be a differential k-form and be a differential l-form on a domain D . State without proof a product rule for the exterior derivative of the wedge product https://storage.examlex.com/TB9661/ .</strong> A) d + d B) d + d C) d + d D) d + d E) d + d <div style=padding-top: 35px>

11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 d11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11

Question

Let be a differential k-form, be a differential l-form, and be an m-form on a domain D . Find an expression for a product rule for the exterior derivative of the wedge product 11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 .

A) d(11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 ) = (d11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 )11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 + 11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 d(11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 )11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 + 11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11

<strong>Let be a differential k-form, be a differential l-form, and be an m-form on a domain D . Find an expression for a product rule for the exterior derivative of the wedge product .</strong> A) d( ) = (d ) + d( ) + d( ) B) d( ) = (d ) (d ) (d ) C) d( ) = (d ) + d( ) + d( ) D) d( ) = (d ) + d( ) + d( ) E) d( ) = (d ) + d( ) + d( ) <div style=padding-top: 35px>

Question

Let g(x) be a differential 0-form on a domain D in $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x) <div style=padding-top: 35px>$ . If dg(x) = $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x) <div style=padding-top: 35px>$ (x) dx+ $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x) <div style=padding-top: 35px>$ (x) dy + $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x) <div style=padding-top: 35px>$ (x) dz, then the vector field $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x) <div style=padding-top: 35px>$ (x) i + $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x) <div style=padding-top: 35px>$ (x) j + $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x) <div style=padding-top: 35px>$ (x) k is equal to

A) grad (g(x))
B) div(g(x))
C) curl (

\bigtriangledown

g(x))
D)

\bigtriangledown

\bigtriangledown

(g(x))
E)

\bigtriangledown

(g(x)) $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x) <div style=padding-top: 35px>$ g(x)

Question

$ A) div( \bigtriangledown F) B) div(F) C) grad(F) D) curl(F) E) F <div style=padding-top: 35px>$

A) div(

\bigtriangledown

F)
B) div(F)
C) grad(F)
D) curl(F)
E) $ A) div( \bigtriangledown F) B) div(F) C) grad(F) D) curl(F) E) F <div style=padding-top: 35px>$ F

Question

$ A) F . \bigtriangledown B) div(F) C) F . F D) (F × \bigtriangledown ) . F E) <div style=padding-top: 35px>$

A) F .

\bigtriangledown

B) div(F)
C) F . F
D) (F ×

\bigtriangledown

) . F
E) $ A) F . \bigtriangledown B) div(F) C) F . F D) (F × \bigtriangledown ) . F E) <div style=padding-top: 35px>$

Question

Let be a differential k-form and be a differential l-form on a domain D . Then if and only if

A) both k and l are even
B) both k and l are odd
C) k is even and l is odd
D) k is odd and l is even
E) k + l is even

Question

\bigtriangledown

(divF) = 0
B) curl(F) = 0
C) div(curl F)= 0
D) $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.Let = F dx + G dy + H dz be a differential 1-form on a domain D and let be the vector field corresponding to . Using this set up, find the vector differential identity corresponding to the fact = . A) \bigtriangledown (divF) = 0 B) curl(F) = 0 C) div(curl F)= 0 D) F = 0 E) curl(curl F) = 0 <div style=padding-top: 35px>$ F = 0
E) curl(curl F) = 0

Question

You probably know by now that a differential k-form k $\ge$ 1 on a domain D $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ dx + $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ dy + $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ dz and the vector field F = $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ i + $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ j + $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ k. Using this setup, find the vector differential identity corresponding to the fact $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ for any differential 0-form g on a domain D in $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ .

\bigtriangledown

. (

\bigtriangledown

g) = 0
B)

\bigtriangledown

× (

\bigtriangledown

g) = 0
C) $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0 <div style=padding-top: 35px>$ (g) = 0
D)

\bigtriangledown

g = 0
E) (

\bigtriangledown

g).

\bigtriangledown

) = 0

Question

Let F(x, y) and G(x, y) be differential 0-forms on a domain D in

. Prove that(dF)∧(dG) =

dx∧dy.

Question

If g is a differential 0-form and

is a differential k-form on domain D

, then

Question

Find a differential form such that d11ee7bc7_4a83_9687_ae82_51ccf66be046_TB9661_11 = 3 dx dy11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dzNote: answer is not unique.

A) 4xdy11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11dz + ydz11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11dx -2z

<strong>Find a differential form such that d = 3 dx dy dzNote: answer is not unique.</strong> A) 4xdy dz + ydz dx -2z dy B) xdx + ydy + zdz C) zdy dz + xdz dx + y dy D) 2ydy dz + 5zdz dx -4 x dx dy E) (x + y + z ) dx dy dz <div style=padding-top: 35px>

dy
B) xdx + ydy + zdz
C) zdy11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dz + xdz11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dx + y11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11dx11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dy
D) 2ydy11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dz + 5zdz11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dx -4 x11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11dx 11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dy
E) (x + y + z ) dx11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dy11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dz

Question

Prove that every smooth exterior derivative is a closed differential form.

Question

Which of the following is an antiderivative of (3x + 4xy) dx dy?

(9xdx -16

ydy)
B)

(x + y) dx?dy
C)

(xdx - ydy)
D)

(x + y)
E) -

(xdx + ydy)

Question

Let the differential 1-form = zdy dz + xdz11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dx + ydx11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dy be defined in a star-like domain .(a) Is 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 closed?(b) Is 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 exact on D? If so, find a differential 1-form such that 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 = d11ee7bc8_e3af_834b_ae82_7d811aed33cf_TB9661_11 .

A) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is neither closed nor exact
B) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is both closed and exact, 11ee7bc8_e3af_834b_ae82_7d811aed33cf_TB9661_11 = (xz -

<strong>Let the differential 1-form = zdy dz + xdz dx + ydx dy be defined in a star-like domain .(a) Is closed?(b) Is exact on D? If so, find a differential 1-form such that = d .</strong> A) is neither closed nor exact B) is both closed and exact, = (xz - + )dx + (yz + sin(z)) dz C) is both closed and exact, = (xz + + )dx + (yz + sin(x)) dy D) is closed but not exact E) is both closed and exact, = zxdx+ xydy + yzdz <div style=padding-top: 35px>

)dx + (yz + sin(z)) dz
C) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is both closed and exact,11ee7bc8_e3af_834b_ae82_7d811aed33cf_TB9661_11 = (xz +

)dx + (yz + sin(x)) dy
D) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is closed but not exact
E) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is both closed and exact, 11ee7bc8_e3af_834b_ae82_7d811aed33cf_TB9661_11= zxdx+ xydy + yzdz

Question

Let the differential 2-form = (3 + 2xy + 6 )dx dy be defined in a star-like domain .
(a) Is 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 closed?
(b) Is 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 exact on D? If so, find a differential 1-form such that 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 = d11ee7bc9_84dc_f4fe_ae82_1b41b7d4ea2f_TB9661_11 .

<strong>Let the differential 2-form = (3 + 2xy + 6 )dx dy be defined in a star-like domain . (a) Is closed? (b) Is exact on D? If so, find a differential 1-form such that = d .</strong> A) is closed but not exact B) is both closed and exact, = - (x + 2 )dx + dy C) is neither closed nor exact D) is both closed and exact, = (x + 2 )dx + dy E) is both closed and exact, = dx - (x + 2 )dy <div style=padding-top: 35px>

+ 2

)dx +

+ 2

)dx +

dy
E) 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 is both closed and exact, 11ee7bc9_84dc_f4fe_ae82_1b41b7d4ea2f_TB9661_11 =

dx - (x

+ 2

)dy

Question

Let the differential 2-form = xdy dz + ydz11ee7bc9_e7ae_ceff_ae82_cb591a35a817_TB9661_11 dx + (1 - 2z)dx11ee7bc9_e7ae_ceff_ae82_cb591a35a817_TB9661_11 dy be defined in a star-like domain .
(a) Is 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 closed?
(b) Is 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 exact on D? If so, find a differential 1-form such that 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 = dhttps://storage.examlex.com/TB9661/11ee7bca_376e_ea71_ae82_a7b6e43e019b_TB9661_11.

A) 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 is neither closed nor exact
B) 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 is both closed and exact, 11ee7bca_376e_ea71_ae82_a7b6e43e019b_TB9661_11 = ydx + 2xzdy - xydz
C) 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 is closed but not exact
D) 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 is exact but not closed, 11ee7bca_376e_ea71_ae82_a7b6e43e019b_TB9661_11 = ydx + 2xzdy + xydz
E) 11ee7bca_0dd7_5530_ae82_83ba26f1ab04_TB9661_11 is both closed and exact, 11ee7bca_376e_ea71_ae82_a7b6e43e019b_TB9661_11 = -ydx - 2xzdy - xydz

Question

Every (smooth) exact differential k-form on a domain D

is closed.

Question

If $If is a differential k-form, where k \ge 1 is an even integer, then d( d ) = .<div style=padding-top: 35px>$ is a differential k-form, where k

\ge

1 is an even integer, then d(11ee7bca_9ca8_75a3_ae82_5bb03b7ccf94_TB9661_11 $If is a differential k-form, where k \ge 1 is an even integer, then d( d ) = .<div style=padding-top: 35px>$ d11ee7bca_9ca8_75a3_ae82_5bb03b7ccf94_TB9661_11 ) = $If is a differential k-form, where k \ge 1 is an even integer, then d( d ) = .<div style=padding-top: 35px>$ .

Question

Let [r, θ, z] be the cylindrical coordinates of a point in 3-space. Prove that rdr∧dθ∧dz = dx∧dy∧dz.

Question

Let Φ = (2xy -

) dx + (2yz +

) dy + (

- 2zx) dz be a differential 1-form defined on a star-like domain D in

.
(a) Show that Φ is exact on D.
(b) Find a differential 0-form Ψsuch that Φ = dΨ on D.

Question

Find the 2-volume of the 2-parallelogram in spanned by the vectors v₁ = (0, - 1, -2, -1) and v₂ = (1, 3, 7, 1).

A) 18 units²
B) 36 units²
C) 6 units²
D) 9 units²
E) 8

<strong>Find the 2-volume of the 2-parallelogram in spanned by the vectors v<sub>1</sub> = (0, - 1, -2, -1) and v<sub>2</sub> = (1, 3, 7, 1).</strong> A) 18 units<sup>2</sup> B) 36 units<sup>2</sup> C) 6 units<sup>2</sup> D) 9 units<sup>2</sup> E) 8 units<sup>2</sup> <div style=padding-top: 35px>

units²

Question

Find the 3-volume of the 3-parallelogram in spanned by the vectors v₁ = (2, 3, 1, 0),v₂ = (0, -3, -2, 1), and v₃ = (1, 1, 1, 1).

A) 10 units³
B) 50 units³
C) 100 units³
D) 540 units³
E) 6

<strong>Find the 3-volume of the 3-parallelogram in spanned by the vectors v<sub>1</sub> = (2, 3, 1, 0),v<sub>2</sub> = (0, -3, -2, 1), and v<sub>3</sub> = (1, 1, 1, 1).</strong> A) 10 units<sup>3</sup> B) 50 units<sup>3</sup> C) 100 units<sup>3</sup> D) 540 units<sup>3</sup> E) 6 units<sup>3</sup> <div style=padding-top: 35px>

units³

Question

Find the 4-volume of the 4-parallelogram in spanned by the vectors = ( , , , ), , = (1, 0, 0, 0), and = (- , , - , ).

Question

Find all values of the real number $\alpha$ such that the 2-volume of the 2-parallelogram in $Find all values of the real number \alpha such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, \alpha , 8, 3) is equal to 22 . A) \alpha = ± 10 B) \alpha = ± 2 C) \alpha = ± 2 D) \alpha = ± E) \alpha = ± <div style=padding-top: 35px>$ spanned by the vectors $Find all values of the real number \alpha such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, \alpha , 8, 3) is equal to 22 . A) \alpha = ± 10 B) \alpha = ± 2 C) \alpha = ± 2 D) \alpha = ± E) \alpha = ± <div style=padding-top: 35px>$ = (2, 0, 0, 1) and $Find all values of the real number \alpha such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, \alpha , 8, 3) is equal to 22 . A) \alpha = ± 10 B) \alpha = ± 2 C) \alpha = ± 2 D) \alpha = ± E) \alpha = ± <div style=padding-top: 35px>$ = (-2, $\alpha$ , 8, 3) is equal to 22 $Find all values of the real number \alpha such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, \alpha , 8, 3) is equal to 22 . A) \alpha = ± 10 B) \alpha = ± 2 C) \alpha = ± 2 D) \alpha = ± E) \alpha = ± <div style=padding-top: 35px>$ .

\alpha

= ± 10
B)

\alpha

= ± 2 $Find all values of the real number \alpha such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, \alpha , 8, 3) is equal to 22 . A) \alpha = ± 10 B) \alpha = ± 2 C) \alpha = ± 2 D) \alpha = ± E) \alpha = ± <div style=padding-top: 35px>$
C)

\alpha

\alpha

= ± $Find all values of the real number \alpha such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, \alpha , 8, 3) is equal to 22 . A) \alpha = ± 10 B) \alpha = ± 2 C) \alpha = ± 2 D) \alpha = ± E) \alpha = ± <div style=padding-top: 35px>$
E)

\alpha

Question

Find

(x), where M is the 2-manifold in

given parametrically by

sin(2

), 3

) for 0 ≤

≤ 1, 0 ≤

≤ 1.

Question

Find the 2-volume of the 2-manifold in $Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E) <div style=padding-top: 35px>$ given parametrically by $Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E) <div style=padding-top: 35px>$ for 0 $\le$ u $\le$ 1, 0 $\le$ v $\le$ $\pi$ .

A) $Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E) <div style=padding-top: 35px>$ sinh(2)
B) $Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E) <div style=padding-top: 35px>$ sinh(2) +

\pi

\pi

$Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E) <div style=padding-top: 35px>$ (1)
D) 0
E) $Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E) <div style=padding-top: 35px>$

Question

Let f be a smooth real value function over a domain D in , then the graph x_n+1 = f( , ,....., ) is

A) a smooth (n - 1)-manifold in

B) a smooth n-manifold in

C) a smooth (n + 1)-manifold in

D) a smooth (n + 1)-manifold in

E) a smooth n-manifold in

Question

The two equations z =

and z =

define a smooth manifold of dimension two in

Question

One way to describe a smooth k-manifold M in

is to require its points

satisfy a set of (n - k) independent equations in (

,.....,

Question

Find

(x), where M is the 2-manifold in

given parametrically by

for 0 ≤ u ≤ 1, 0 ≤ v ≤ 1.

Question

The k-volume of a k-parallelogram in

spanned by the k vectors,

,......,

is given by det(A), where A is the k × k matrix whose columns are the components of the vectors.

Question

Let x = ( , , ) = ( , , ), J(u) = . Find det( J(u)).

B) 1
C) d

Question

Calculate $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ dx $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ dz, where M is the surface given by z = $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ , 0 $\le$ z $\le$ 1, using the following parametrizations: (i) (x, y, z) = p( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ , $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ ) = ( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ cos( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ ), $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ sin( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ ) $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ )
(ii) (x, y, z) = p( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ , $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ )= ( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ , $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ , $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ )

A) (i) $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ (ii) - $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$
B) (i) $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ (ii) 0
C) (i) $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ (ii) - $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$
D) (i) 6

\pi

(ii) - $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$
E) (i) - $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$ (ii) $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii) <div style=padding-top: 35px>$

Question

The 2-manifold M in R⁴given by the equations 0 < x₄ < 1,
0 < < 1 has normals It is oriented by the 2-form
ω(x)( v₁ , v₂ ) = det( n₁ n₂ v₁ v₂ ). Let be a parametrization for M. Which of the following statements is true?

A) P is orientation preserving for M.
B) P does not determine an orientation for M.
C) P is orientation reversing for M, but no orientation preserving parametrization for M exists.
D) P is orientation reversing for M, but q(u) = (

<strong>The 2-manifold M in R<sup>4 </sup>given by the equations 0 < x<sub>4</sub> < 1, 0 < < 1 has normals It is oriented by the 2-form ω(x)( v<sub>1</sub> , v<sub>2</sub> ) = det( n<sub>1</sub> n<sub>2</sub> v<sub>1</sub> v<sub>2</sub> ). Let be a parametrization for M. Which of the following statements is true? </strong> A) P is orientation preserving for M. B) P does not determine an orientation for M. C) P is orientation reversing for M, but no orientation preserving parametrization for M exists. D) P is orientation reversing for M, but q(u) = ( , 5 , , -2 ) would be orientation preserving parametrization for M. E) P is orientation reversing for M, but q(u) = (- , 5 , , 2 ) would be an orientation-preserving parametrization for M. <div style=padding-top: 35px>

, 5

, -2

) would be orientation preserving parametrization for M.
E) P is orientation reversing for M, but q(u) = (-

, 5

, 2

) would be an orientation-preserving parametrization for M.

Question

Find , where M is the 2-manifold in given parametrically by for 0 < < 1, 0 < < 2.

A) -130
B) 135
C) -

D) 130
E) -13

Question

Evaluate the integral of $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ = $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ + $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ 11ee7bcb_75db_a3b7_ae82_1108d245f507_TB9661_11 d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ + $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ 11ee7bcb_75db_a3b7_ae82_1108d245f507_TB9661_11 d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ over the upper hemispherical surface $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ + $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ + $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ = 16, $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi <div style=padding-top: 35px>$ $\ge$ 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing.

A) ±12

\pi

B) ± 4

\pi

C) ± 48

\pi

D) ± 6

\pi

E) ± 24

\pi

Question

If M is the part of the surface z = g(x, y) in

that lies above a closed region D in the

, then the integral of the differential 2-form

= f(x, y) dx

dy over M is independent of the function g.

Question

Let M be the smooth 2-manifold $Let M be the smooth 2-manifold , x = p( \theta , ) = (cos( \theta )sin( ), sin( \theta )sin( ), cos( ),0 \le\theta \le 2 \pi , and let be a parametrization for M. If M is oriented by the differential 2-form = zdx dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.<div style=padding-top: 35px>$ , x = p(

\theta

, $Let M be the smooth 2-manifold , x = p( \theta , ) = (cos( \theta )sin( ), sin( \theta )sin( ), cos( ),0 \le\theta \le 2 \pi , and let be a parametrization for M. If M is oriented by the differential 2-form = zdx dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.<div style=padding-top: 35px>$ ) = (cos(

\theta

)sin(11ee7bce_0cad_3050_ae82_0fc996929baa_TB9661_11 ), sin(

\theta

)sin(11ee7bce_0cad_3050_ae82_0fc996929baa_TB9661_11 ), cos(11ee7bce_0cad_3050_ae82_0fc996929baa_TB9661_11 ),0

\le

\theta

\le

\pi

, and let $Let M be the smooth 2-manifold , x = p( \theta , ) = (cos( \theta )sin( ), sin( \theta )sin( ), cos( ),0 \le\theta \le 2 \pi , and let be a parametrization for M. If M is oriented by the differential 2-form = zdx dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.<div style=padding-top: 35px>$ be a parametrization for M. If M is oriented by the differential 2-form $Let M be the smooth 2-manifold , x = p( \theta , ) = (cos( \theta )sin( ), sin( \theta )sin( ), cos( ),0 \le\theta \le 2 \pi , and let be a parametrization for M. If M is oriented by the differential 2-form = zdx dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.<div style=padding-top: 35px>$ = zdx $Let M be the smooth 2-manifold , x = p( \theta , ) = (cos( \theta )sin( ), sin( \theta )sin( ), cos( ),0 \le\theta \le 2 \pi , and let be a parametrization for M. If M is oriented by the differential 2-form = zdx dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.<div style=padding-top: 35px>$ dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.

Question

Consider the unit cube Q =

with the standard orientation given by

.Express the orientations of the bottom and the front faces of Q as differential 1-forms evaluated at the cross product of vectors u, v in

Question

Let S be a piece with boundary of a smooth 3-manifold in R⁴ (hypersurface) given by the equation = g( , , ) and let= d d 11ee7bcb_c6fc_6099_ae82_719973faadb8_TB9661_11 d . Apart from sign due to orientation of S, is equal to

A) the 3-volume of the projection of S onto the hyperplane

<strong>Let S be a piece with boundary of a smooth 3-manifold in R<sup>4</sup> (hypersurface) given by the equation = g( , , ) and let = d d d . Apart from sign due to orientation of S, is equal to</strong> A) the 3-volume of the projection of S onto the hyperplane = 0 B) the 4-volume of the projection of S onto the hyperplane = 0 C) D) E) None of the above <div style=padding-top: 35px>

= 0
B) the 4-volume of the projection of S onto the hyperplane

= 0
C)

E) None of the above

Question

Use the generalized Stokes's Theorem to find where and D is the oriented boundary of the domain .

A) 5
B) 8
C) 240
D) 3
E) 37

Question

Use the generalized Stokes's Theorem to find $Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz dx - 9z dx dy and D is the oriented boundary of the conical domain D = . A) 8 \pi B) 32 \pi C) 24 \pi D) 12 \pi E) 4 \pi <div style=padding-top: 35px>$ where $Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz dx - 9z dx dy and D is the oriented boundary of the conical domain D = . A) 8 \pi B) 32 \pi C) 24 \pi D) 12 \pi E) 4 \pi <div style=padding-top: 35px>$ = 7x dy $Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz dx - 9z dx dy and D is the oriented boundary of the conical domain D = . A) 8 \pi B) 32 \pi C) 24 \pi D) 12 \pi E) 4 \pi <div style=padding-top: 35px>$ dz + (3y + 2z) dz11ee7bcc_0cc6_216b_ae82_3f9133291a84_TB9661_11 dx - 9z dx11ee7bcc_0cc6_216b_ae82_3f9133291a84_TB9661_11 dy and $Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz dx - 9z dx dy and D is the oriented boundary of the conical domain D = . A) 8 \pi B) 32 \pi C) 24 \pi D) 12 \pi E) 4 \pi <div style=padding-top: 35px>$ D is the oriented boundary of the conical domain D = $Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz dx - 9z dx dy and D is the oriented boundary of the conical domain D = . A) 8 \pi B) 32 \pi C) 24 \pi D) 12 \pi E) 4 \pi <div style=padding-top: 35px>$ .

A) 8

\pi

B) 32

\pi

C) 24

\pi

D) 12

\pi

E) 4

\pi

Question

Integrate the differential 3-form = dx dy11ee7bcc_8068_caae_ae82_1bce1a65fd1b_TB9661_11dz over the boundary of the 4-dimensional tetrahedron T = .

A) -648
B) -

C) -36
D) 648
E) 36

Question

Let C be the curve of intersection of the cylinder $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3 <div style=padding-top: 35px>$ + $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3 <div style=padding-top: 35px>$ = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3 <div style=padding-top: 35px>$ , where $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3 <div style=padding-top: 35px>$ = -3 $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3 <div style=padding-top: 35px>$ z dx + sin(y) dy + (3x $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3 <div style=padding-top: 35px>$ + x +7) dz.

A) 16

\pi

B) 28

\pi

C) 14

\pi

D) 20

\pi

E) 3

Question

State the Divergence Theorem and Stokes's Theorem in 3-space, and Green's Theorem in 2-space in terms of differential forms.

Question

Let Ω be the differential 3-form $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$ be the 4-dimensional ball of radius α in R⁴ ; that is
$ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$
(a) Use the generalized Stokes's Theorem to evaluate $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$
(b) Use part (a) to find the 4-volume of the ball.
Hint: You may use symmetry and the transformation , .
$ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$

A) 4

\pi

$ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$
B) $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$ $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$ $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$
C) $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$ $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$ $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$
D) 2 $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$ $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$
E) $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi <div style=padding-top: 35px>$

\pi

Question

Let D be a closed bounded domain in

and lot Ψ = xdy∧dz + ydz∧dx + zdx∧dy. Show that the volume V of D is given by

Question

Integrate the differential (n -1)-form over the boundary of the n-dimensional cube
Note: The hat ∧ is used to indicate a missing component.

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1/76

Deck 18: Differential Forms and Exterior Calculus

Let , , be 1-forms and be a 2-form on . Which of the following is a 3-form on ?

A) 11ee7971_d7c9_b5d4_88d3_75d03a1ac3e7_TB9661_11 + 11ee7972_1f9d_0245_88d3_430fabc730ac_TB9661_11 + 11ee7972_a99a_a1a7_88d3_5bac39f31ff0_TB9661_11
B) 11ee7971_d7c9_b5d4_88d3_75d03a1ac3e7_TB9661_11 + 11ee7972_e728_fe48_88d3_fd489bade70a_TB9661_11
C) (11ee7971_d7c9_b5d4_88d3_75d03a1ac3e7_TB9661_11 + Let , , be 1-forms and be a 2-form on . Which of the following is a 3-form on ? A) + + B) + C) ( + ) D) E) All of the above

)

11ee7973_0b66_7d59_88d3_6bcf237f8512_TB9661_11

Let , be differential 1-forms and let be a differential 0-form on a domain D in Which of the following is a differential 2-form on D?

A) 11ee7971_d7c9_b5d4_88d3_75d03a1ac3e7_TB9661_11+ 11ee7972_5038_6616_88d3_2d4d40837c09_TB9661_11
B)11ee7971_d7c9_b5d4_88d3_75d03a1ac3e7_TB9661_11 Let , be differential 1-forms and let be a differential 0-form on a domain D in Which of the following is a differential 2-form on D? A) + B) C) 3 - D) + E) None of the above

11ee7972_5038_6616_88d3_2d4d40837c09_TB9661_11
C) 311ee7971_d7c9_b5d4_88d3_75d03a1ac3e7_TB9661_11 - 11ee7972_5038_6616_88d3_2d4d40837c09_TB9661_11
D) Let , be differential 1-forms and let be a differential 0-form on a domain D in Which of the following is a differential 2-form on D? A) + B) C) 3 - D) + E) None of the above

+ 11ee7972_5038_6616_88d3_2d4d40837c09_TB9661_11
E) None of the above

Find the dimension of ( ).

A) 7
B) 840
C) 4
D) 3
E) 35

Let k and n be integers such that 1 k 11ee7973_99f4_ef7a_88d3_478f26d4adc3_TB9661_11 n. Find the dimension of the vector space of all k-forms on .

B) n
C) k
D) n - k
E) n(n - 1)(n - 2).....(n - k +1)

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k this deck

Let = . List all basis vectors of ( ).

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k this deck

is a k-form and If is a k-form and is an l-form on , then = https://storage.examlex.com/TB9661/ .

is an l-form on If is a k-form and is an l-form on , then = https://storage.examlex.com/TB9661/ .

, then

https://storage.examlex.com/TB9661/ If is a k-form and is an l-form on , then = https://storage.examlex.com/TB9661/ .

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k this deck

Let φ and ψ be two 1-forms on

say φ=

, ψ=

. Show that φ∧ψ = 0 if and only if φ = c ψ for some constant real number c.

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k this deck

A) u . v
B) u × v
C) u + v
D) A) u . v B) u × v C) u + v D) + E)

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k this deck

is a k -form, If is a k -form, and are l-forms on , then = implies =https://storage.examlex.com/TB9661/ .

and

are l-forms on If is a k -form, and are l-forms on , then = implies =https://storage.examlex.com/TB9661/ .

, then

implies

=https://storage.examlex.com/TB9661/ If is a k -form, and are l-forms on , then = implies =https://storage.examlex.com/TB9661/ .

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k this deck

A) u + v
B) u - v
C) u × v
D) v × u
E) v - u

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k this deck

Let = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that .

A) 11ee7975_2bfd_ecfa_88d3_cb60d904737e_TB9661_11 = t dx + Let = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that . A) = t dx + (25 - 2t) dy, t R B) = t dx + (29 - 2t) dy, - < t < C) = t dx + (29 + 2t) dy, - < t < D) = (25 + 9t) dx + t dy, t R E) = (29 - 9t) dx + t dy, - < t <

(25 - 2t) dy, t Let = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that . A) = t dx + (25 - 2t) dy, t R B) = t dx + (29 - 2t) dy, - < t < C) = t dx + (29 + 2t) dy, - < t < D) = (25 + 9t) dx + t dy, t R E) = (29 - 9t) dx + t dy, - < t <

R
B) 11ee7975_2bfd_ecfa_88d3_cb60d904737e_TB9661_11 = t dx + Let = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that . A) = t dx + (25 - 2t) dy, t R B) = t dx + (29 - 2t) dy, - < t < C) = t dx + (29 + 2t) dy, - < t < D) = (25 + 9t) dx + t dy, t R E) = (29 - 9t) dx + t dy, - < t <

(29 - 2t) dy, - Let = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that . A) = t dx + (25 - 2t) dy, t R B) = t dx + (29 - 2t) dy, - < t < C) = t dx + (29 + 2t) dy, - < t < D) = (25 + 9t) dx + t dy, t R E) = (29 - 9t) dx + t dy, - < t <

< t < 11ee7975_081b_3819_88d3_837de92b3d96_TB9661_11
C) 11ee7975_2bfd_ecfa_88d3_cb60d904737e_TB9661_11 = t dx + Let = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that . A) = t dx + (25 - 2t) dy, t R B) = t dx + (29 - 2t) dy, - < t < C) = t dx + (29 + 2t) dy, - < t < D) = (25 + 9t) dx + t dy, t R E) = (29 - 9t) dx + t dy, - < t <

(29 + 2t) dy, -11ee7975_081b_3819_88d3_837de92b3d96_TB9661_11 < t < 11ee7975_081b_3819_88d3_837de92b3d96_TB9661_11
D) 11ee7975_2bfd_ecfa_88d3_cb60d904737e_TB9661_11 = Let = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that . A) = t dx + (25 - 2t) dy, t R B) = t dx + (29 - 2t) dy, - < t < C) = t dx + (29 + 2t) dy, - < t < D) = (25 + 9t) dx + t dy, t R E) = (29 - 9t) dx + t dy, - < t <

(25 + 9t) dx + t dy, t 11ee7974_eec1_32c8_88d3_178ace6e171f_TB9661_11 R
E) 11ee7975_2bfd_ecfa_88d3_cb60d904737e_TB9661_11 = Let = 9dx - 2dy and = -dx+ 3dy be 1 -forms on . Find all 1-forms on such that . A) = t dx + (25 - 2t) dy, t R B) = t dx + (29 - 2t) dy, - < t < C) = t dx + (29 + 2t) dy, - < t < D) = (25 + 9t) dx + t dy, t R E) = (29 - 9t) dx + t dy, - < t <

(29 - 9t) dx + t dy, -11ee7975_081b_3819_88d3_837de92b3d96_TB9661_11 < t < 11ee7975_081b_3819_88d3_837de92b3d96_TB9661_11

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k this deck

= 0 for every k-form = 0 for every k-form on .

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A) (a - b + c) dx Simplify a dx dy dz + b dx dy + c dy dx + (a + b +c) dy dy. A) (a - b + c) dx dz B) (a +b + c) dx dz C) (a - b - c) dx dz D) (a + b -c) dx dz E) (a + b +c) dx dy

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Let $Let = 2dx + 5dy, = -dx + 7dy and \theta = -3dx + c dy be 1-forms on . Find the real number c such that = \theta . A) 12 B) 2 C) 17 D) -3 E) 0$ = 2dx + 5dy, $Let = 2dx + 5dy, = -dx + 7dy and \theta = -3dx + c dy be 1-forms on . Find the real number c such that = \theta . A) 12 B) 2 C) 17 D) -3 E) 0$ = -dx + 7dy and $\theta$ = -3dx + c dy be 1-forms on $Let = 2dx + 5dy, = -dx + 7dy and \theta = -3dx + c dy be 1-forms on . Find the real number c such that = \theta . A) 12 B) 2 C) 17 D) -3 E) 0$ . Find the real number c such that 11ee7bbb_18dc_373c_ae82_e55d2b6d55c1_TB9661_11 $Let = 2dx + 5dy, = -dx + 7dy and \theta = -3dx + c dy be 1-forms on . Find the real number c such that = \theta . A) 12 B) 2 C) 17 D) -3 E) 0$ 11ee7bbb_3770_208d_ae82_59bc8eff822f_TB9661_11 = 11ee7bbb_18dc_373c_ae82_e55d2b6d55c1_TB9661_11 11ee7bba_a357_125b_ae82_afb9ee65da13_TB9661_11 $\theta$ .

A) 12
B) 2
C) 17
D) -3
E) 0

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Let $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ , $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ , $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ be m-forms, $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ be a k-form, and $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ be an l-form on $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ . Which of the following properties of the wedge product is not always true?

A) ( $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ + $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ ) $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 = $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 + $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11
B) (11ee7bbc_2f42_a9a0_ae82_03df16557ec6_TB9661_11 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 )11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbc_854e_b551_ae82_a1ccb48ff30b_TB9661_11 = 11ee7bbc_2f42_a9a0_ae82_03df16557ec6_TB9661_11 ( $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ 11ee7bbc_854e_b551_ae82_a1ccb48ff30b_TB9661_11 )
C) 11ee7bbc_2f42_a9a0_ae82_03df16557ec6_TB9661_11 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 = - 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbc_2f42_a9a0_ae82_03df16557ec6_TB9661_11
D) (a 11ee7bbc_2f42_a9a0_ae82_03df16557ec6_TB9661_11 )11ee7bbb_c9fe_971f_ae82_3b4fcccd2417_TB9661_11 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 = a ( $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ 11ee7bbb_54a1_6c9e_ae82_5397d9492c20_TB9661_11 ) for a $Let , , be m-forms, be a k-form, and be an l-form on . Which of the following properties of the wedge product is not always true? A) ( + ) = + B) ( ) = ( ) C) = - D) (a ) = a ( ) for a R E) for every k \ge 1, there exists a zero k-form such that + 0 = , and 0 = 0.$ R
E) for every k

\ge

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Let e_i , i = 1, 2, 3, 4 be the standard basis vectors in R⁴ and let

A) 32
B) -56
C) -40
D) -14
E) 40

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Let ψ = 12dy∧dz -10 dx∧dz + 8 dx∧dy be a 2-form on

. Express ψ as a product of two 1-forms.Hint: Add a suitable multiple of dz∧dz = 0 to ψ.

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Let σ = d

∧d

+ d

∧d

be a 2-form on Let σ = d ∧d + d ∧d be a 2-form on . Express σ as a product of two 1-forms.

. Express σ as a product of two 1-forms.

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Show that if φ is a k-form on

, then φ∧φ = 0 if k is odd.

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Let

Let be the permutation that maps [1, 2, 3, 4, 5] to [3, 4, 5, 2, 1], then sgn( ) = -1.

be the permutation that maps [1, 2, 3, 4, 5] to [3, 4, 5, 2, 1], then sgn(11ee7bbc_f5df_3c93_ae82_ade676104027_TB9661_11 ) = -1.

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Expand and simplify: (2a dx + (b + 2a)dy + c dz)∧(a dx + 2b dy + c dz) - (2a² - 2ab) ( a - b) dy∧dx . Express your answer in terms of the basis vectors dy∧dz, dz∧dx, and dx∧dy of

Expand and simplify: (2a dx + (b + 2a)dy + c dz)∧(a dx + 2b dy + c dz) - (2a<sup>2</sup> - 2ab) ( a - b) dy∧dx . Express your answer in terms of the basis vectors dy∧dz, dz∧dx, and dx∧dy of

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Let ( ) be the vector space of all 3-forms on and ( ) be the vector space of all 5-forms on . If ( ) and ( ) have the same dimension, find n.

A) 2
B) 4
C) 8
D) 15
E) 8 or -1

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Let η = F₁ dy∧dz - F₂ dx∧dz+ F₃ dx∧dy j + b₃k
be vectors in . If η is identified by the vector w = F₁ i + F₂ j + F₃ k then η (u, v) is equal to

A) w . (u + v)
B) w . (u × v)
C) w. (v - u)
D) w . (v × u)
E) w × (u × v)

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Let $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$ ( $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$ ), 1 $\le$ k $\le$ n be the vector space of all k-forms on $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$ and let $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$ be the dimension of $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$ . Find $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$ .

A) $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$
B) $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$
C) $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$
D) $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$
E) $Let ( ), 1 \le k \le n be the vector space of all k-forms on and let be the dimension of . Find . A) B) C) D) E) - 1$ - 1

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Let (x) = (x) d + (x) d + (x) d be a differential 1-form on a domain D in and let . The value of 11ee7bbd_2d92_9334_ae82_4706b097eca0_TB9661_11 (x) on a vector v is equal to

A) 11ee7bbd_2d92_9334_ae82_4706b097eca0_TB9661_11 (x) Let (x) = (x) d + (x) d + (x) d be a differential 1-form on a domain D in and let . The value of (x) on a vector v is equal to A) (x) v B) a(x) . v C) a(x) × v D) a(x) + v E) v × a(x)

v
B) a(x) . v
C) a(x) × v
D) a(x) + v
E) v × a(x)

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A) (y - 8z) dx Let = 4xz (y) dy dz + z(2y + sin(2y)) dz dx + (yz - 2 ) dx dy. Find d . A) (y - 8z) dx dz B) (8z - y) dx dz C) (y - 4zcos(2y)) dx dz D) y E) ydx dz

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dx11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dy11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dz11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dt11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 du11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dv11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dw
D) 3x Let = xdx + vdv, = dy dw, = dz dv du be differential forms in a domain D .Find d( ). A) 12ut dx dy dz dt du dv dw B) 6xut dx dy dz dt du dv dw C) 3x dx dy dz dt du dv dw D) 3x E) -3x dx du dv dw

E) -3x

du11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dv11ee7bbe_ab46_077c_ae82_9ffd70004b08_TB9661_11 dw

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A) d11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11 + Let be a differential k-form and be a differential l-form on a domain D . State without proof a product rule for the exterior derivative of the wedge product https://storage.examlex.com/TB9661/ . A) d + d B) d + d C) d + d D) d + d E) d + d

11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 d11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11
B) d11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11 + 11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 d11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11
C) d11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11 + Let be a differential k-form and be a differential l-form on a domain D . State without proof a product rule for the exterior derivative of the wedge product https://storage.examlex.com/TB9661/ . A) d + d B) d + d C) d + d D) d + d E) d + d

11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 d11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11
D) d11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11 + Let be a differential k-form and be a differential l-form on a domain D . State without proof a product rule for the exterior derivative of the wedge product https://storage.examlex.com/TB9661/ . A) d + d B) d + d C) d + d D) d + d E) d + d

11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 d11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11
E) d11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11 + Let be a differential k-form and be a differential l-form on a domain D . State without proof a product rule for the exterior derivative of the wedge product https://storage.examlex.com/TB9661/ . A) d + d B) d + d C) d + d D) d + d E) d + d

11ee7bc0_83e8_98e2_ae82_39e0b41feeac_TB9661_11 11ee7bc1_6b1c_3825_ae82_afc27e21c96d_TB9661_11 d11ee7bc1_2249_2ef3_ae82_ad8c0cfa69a8_TB9661_11

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d(11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 )
B) d(11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 ) = (d11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 )11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 (d11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 )11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 (d11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 )
C) d(11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 ) = Let be a differential k-form, be a differential l-form, and be an m-form on a domain D . Find an expression for a product rule for the exterior derivative of the wedge product . A) d( ) = (d ) + d( ) + d( ) B) d( ) = (d ) (d ) (d ) C) d( ) = (d ) + d( ) + d( ) D) d( ) = (d ) + d( ) + d( ) E) d( ) = (d ) + d( ) + d( )

(d11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 )11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 + Let be a differential k-form, be a differential l-form, and be an m-form on a domain D . Find an expression for a product rule for the exterior derivative of the wedge product . A) d( ) = (d ) + d( ) + d( ) B) d( ) = (d ) (d ) (d ) C) d( ) = (d ) + d( ) + d( ) D) d( ) = (d ) + d( ) + d( ) E) d( ) = (d ) + d( ) + d( )

11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 d(11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 )11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 + Let be a differential k-form, be a differential l-form, and be an m-form on a domain D . Find an expression for a product rule for the exterior derivative of the wedge product . A) d( ) = (d ) + d( ) + d( ) B) d( ) = (d ) (d ) (d ) C) d( ) = (d ) + d( ) + d( ) D) d( ) = (d ) + d( ) + d( ) E) d( ) = (d ) + d( ) + d( )

11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 d( 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 )
D) d(11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 ) = (d11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 )11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 + Let be a differential k-form, be a differential l-form, and be an m-form on a domain D . Find an expression for a product rule for the exterior derivative of the wedge product . A) d( ) = (d ) + d( ) + d( ) B) d( ) = (d ) (d ) (d ) C) d( ) = (d ) + d( ) + d( ) D) d( ) = (d ) + d( ) + d( ) E) d( ) = (d ) + d( ) + d( )

11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 d(11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 )
E) d(11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 ) = (d11ee7bc3_71c2_6b7a_ae82_2ff308d77e4f_TB9661_11 )11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc3_ff64_cfeb_ae82_d9001ab9380d_TB9661_11 11ee7bc5_64fc_b24d_ae82_a1e274b00064_TB9661_11 11ee7bc4_a745_09fc_ae82_03df54d144ea_TB9661_11 + Let be a differential k-form, be a differential l-form, and be an m-form on a domain D . Find an expression for a product rule for the exterior derivative of the wedge product . A) d( ) = (d ) + d( ) + d( ) B) d( ) = (d ) (d ) (d ) C) d( ) = (d ) + d( ) + d( ) D) d( ) = (d ) + d( ) + d( ) E) d( ) = (d ) + d( ) + d( )

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Let g(x) be a differential 0-form on a domain D in $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x)$ . If dg(x) = $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x)$ (x) dx+ $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x)$ (x) dy + $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x)$ (x) dz, then the vector field $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x)$ (x) i + $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x)$ (x) j + $Let g(x) be a differential 0-form on a domain D in . If dg(x) = (x) dx+ (x) dy + (x) dz, then the vector field (x) i + (x) j + (x) k is equal to A) grad (g(x)) B) div(g(x)) C) curl ( \bigtriangledown g(x)) D) \bigtriangledown × \bigtriangledown (g(x)) E) \bigtriangledown (g(x)) g(x)$ (x) k is equal to

A) grad (g(x))
B) div(g(x))
C) curl (

\bigtriangledown

g(x))
D)

\bigtriangledown

\bigtriangledown

(g(x))
E)

\bigtriangledown

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$ A) div( \bigtriangledown F) B) div(F) C) grad(F) D) curl(F) E) F$

A) div(

\bigtriangledown

F)
B) div(F)
C) grad(F)
D) curl(F)
E) $ A) div( \bigtriangledown F) B) div(F) C) grad(F) D) curl(F) E) F$ F

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$ A) F . \bigtriangledown B) div(F) C) F . F D) (F × \bigtriangledown ) . F E)$

A) F .

\bigtriangledown

B) div(F)
C) F . F
D) (F ×

\bigtriangledown

) . F
E) $ A) F . \bigtriangledown B) div(F) C) F . F D) (F × \bigtriangledown ) . F E)$

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Let be a differential k-form and be a differential l-form on a domain D . Then if and only if

A) both k and l are even
B) both k and l are odd
C) k is even and l is odd
D) k is odd and l is even
E) k + l is even

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\bigtriangledown

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You probably know by now that a differential k-form k $\ge$ 1 on a domain D $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0$ $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0$ is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0$ dx + $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0$ dy + $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0$ dz and the vector field F = $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0$ i + $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0$ j + $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0$ k. Using this setup, find the vector differential identity corresponding to the fact $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0$ for any differential 0-form g on a domain D in $You probably know by now that a differential k-form k \ge 1 on a domain D is very similar to a vector field on D, and hence a correspondence between the two may be established.For instance, we may set a correspondence between the 1-form dx + dy + dz and the vector field F = i + j + k. Using this setup, find the vector differential identity corresponding to the fact for any differential 0-form g on a domain D in . A) \bigtriangledown . ( \bigtriangledown g) = 0 B) \bigtriangledown × ( \bigtriangledown g) = 0 C) (g) = 0 D) \bigtriangledown g = 0 E) ( \bigtriangledown g). \bigtriangledown ) = 0$ .

\bigtriangledown

. (

\bigtriangledown

g) = 0
B)

\bigtriangledown

× (

\bigtriangledown

\bigtriangledown

g = 0
E) (

\bigtriangledown

g).

\bigtriangledown

) = 0

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Let F(x, y) and G(x, y) be differential 0-forms on a domain D in

. Prove that(dF)∧(dG) = Let F(x, y) and G(x, y) be differential 0-forms on a domain D in . Prove that(dF)∧(dG) = dx∧dy.

dx∧dy.

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If g is a differential 0-form and

is a differential k-form on domain D If g is a differential 0-form and is a differential k-form on domain D , then .

, then

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Find a differential form such that d11ee7bc7_4a83_9687_ae82_51ccf66be046_TB9661_11 = 3 dx dy11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11 dzNote: answer is not unique.

A) 4xdy11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11dz + ydz11ee7bc7_61d1_9f38_ae82_4b21c548d46f_TB9661_11dx -2z Find a differential form such that d = 3 dx dy dzNote: answer is not unique. A) 4xdy dz + ydz dx -2z dy B) xdx + ydy + zdz C) zdy dz + xdz dx + y dy D) 2ydy dz + 5zdz dx -4 x dx dy E) (x + y + z ) dx dy dz

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Prove that every smooth exterior derivative is a closed differential form.

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Which of the following is an antiderivative of (3x + 4xy) dx dy?

(9xdx -16

ydy)
B)

(x + y) dx?dy
C) Which of the following is an antiderivative of (3x + 4xy) dx dy? A) (9xdx -16 ydy) B) (x + y) dx?dy C) (xdx - ydy) D) (x + y) E) - (xdx + ydy)

(xdx - ydy)
D) Which of the following is an antiderivative of (3x + 4xy) dx dy? A) (9xdx -16 ydy) B) (x + y) dx?dy C) (xdx - ydy) D) (x + y) E) - (xdx + ydy)

(x + y)
E) - Which of the following is an antiderivative of (3x + 4xy) dx dy? A) (9xdx -16 ydy) B) (x + y) dx?dy C) (xdx - ydy) D) (x + y) E) - (xdx + ydy)

(xdx + ydy)

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)dx + (yz + sin(z)) dz
C) 11ee7bc8_9b9f_54e9_ae82_77ffde740b6c_TB9661_11 is both closed and exact,11ee7bc8_e3af_834b_ae82_7d811aed33cf_TB9661_11 = (xz +

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A) 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 is closed but not exact
B) 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 is both closed and exact, 11ee7bc9_84dc_f4fe_ae82_1b41b7d4ea2f_TB9661_11 = - (x Let the differential 2-form = (3 + 2xy + 6 )dx dy be defined in a star-like domain . (a) Is closed? (b) Is exact on D? If so, find a differential 1-form such that = d . A) is closed but not exact B) is both closed and exact, = - (x + 2 )dx + dy C) is neither closed nor exact D) is both closed and exact, = (x + 2 )dx + dy E) is both closed and exact, = dx - (x + 2 )dy

+ 2

)dx +

dy
C) 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 is neither closed nor exact
D)11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 is both closed and exact, 11ee7bc9_84dc_f4fe_ae82_1b41b7d4ea2f_TB9661_11 = (x Let the differential 2-form = (3 + 2xy + 6 )dx dy be defined in a star-like domain . (a) Is closed? (b) Is exact on D? If so, find a differential 1-form such that = d . A) is closed but not exact B) is both closed and exact, = - (x + 2 )dx + dy C) is neither closed nor exact D) is both closed and exact, = (x + 2 )dx + dy E) is both closed and exact, = dx - (x + 2 )dy

+ 2

)dx +

dy
E) 11ee7bc9_5715_4a9c_ae82_31f72d013c41_TB9661_11 is both closed and exact, 11ee7bc9_84dc_f4fe_ae82_1b41b7d4ea2f_TB9661_11 = Let the differential 2-form = (3 + 2xy + 6 )dx dy be defined in a star-like domain . (a) Is closed? (b) Is exact on D? If so, find a differential 1-form such that = d . A) is closed but not exact B) is both closed and exact, = - (x + 2 )dx + dy C) is neither closed nor exact D) is both closed and exact, = (x + 2 )dx + dy E) is both closed and exact, = dx - (x + 2 )dy

dx - (x

+ 2

)dy

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Every (smooth) exact differential k-form on a domain D

is closed.

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If $If is a differential k-form, where k \ge 1 is an even integer, then d( d ) = .$ is a differential k-form, where k

\ge

1 is an even integer, then d(11ee7bca_9ca8_75a3_ae82_5bb03b7ccf94_TB9661_11 $If is a differential k-form, where k \ge 1 is an even integer, then d( d ) = .$ d11ee7bca_9ca8_75a3_ae82_5bb03b7ccf94_TB9661_11 ) = $If is a differential k-form, where k \ge 1 is an even integer, then d( d ) = .$ .

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Let [r, θ, z] be the cylindrical coordinates of a point in 3-space. Prove that rdr∧dθ∧dz = dx∧dy∧dz.

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Let Φ = (2xy - Let Φ = (2xy - ) dx + (2yz + ) dy + ( - 2zx) dz be a differential 1-form defined on a star-like domain D in . (a) Show that Φ is exact on D. (b) Find a differential 0-form Ψsuch that Φ = dΨ on D.

) dx + (2yz + Let Φ = (2xy - ) dx + (2yz + ) dy + ( - 2zx) dz be a differential 1-form defined on a star-like domain D in . (a) Show that Φ is exact on D. (b) Find a differential 0-form Ψsuch that Φ = dΨ on D.

) dy + (

- 2zx) dz be a differential 1-form defined on a star-like domain D in Let Φ = (2xy - ) dx + (2yz + ) dy + ( - 2zx) dz be a differential 1-form defined on a star-like domain D in . (a) Show that Φ is exact on D. (b) Find a differential 0-form Ψsuch that Φ = dΨ on D.

.
(a) Show that Φ is exact on D.
(b) Find a differential 0-form Ψsuch that Φ = dΨ on D.

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Find the 2-volume of the 2-parallelogram in spanned by the vectors v₁ = (0, - 1, -2, -1) and v₂ = (1, 3, 7, 1).

A) 18 units²
B) 36 units²
C) 6 units²
D) 9 units²
E) 8 Find the 2-volume of the 2-parallelogram in spanned by the vectors v1 = (0, - 1, -2, -1) and v2 = (1, 3, 7, 1). A) 18 units2 B) 36 units2 C) 6 units2 D) 9 units2 E) 8 units2

units²

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Find the 3-volume of the 3-parallelogram in spanned by the vectors v₁ = (2, 3, 1, 0),v₂ = (0, -3, -2, 1), and v₃ = (1, 1, 1, 1).

A) 10 units³
B) 50 units³
C) 100 units³
D) 540 units³
E) 6 Find the 3-volume of the 3-parallelogram in spanned by the vectors v1 = (2, 3, 1, 0),v2 = (0, -3, -2, 1), and v3 = (1, 1, 1, 1). A) 10 units3 B) 50 units3 C) 100 units3 D) 540 units3 E) 6 units3

units³

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Find the 4-volume of the 4-parallelogram in spanned by the vectors = ( , , , ), , = (1, 0, 0, 0), and = (- , , - , ).

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Find all values of the real number $\alpha$ such that the 2-volume of the 2-parallelogram in $Find all values of the real number \alpha such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, \alpha , 8, 3) is equal to 22 . A) \alpha = ± 10 B) \alpha = ± 2 C) \alpha = ± 2 D) \alpha = ± E) \alpha = ±$ spanned by the vectors $Find all values of the real number \alpha such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, \alpha , 8, 3) is equal to 22 . A) \alpha = ± 10 B) \alpha = ± 2 C) \alpha = ± 2 D) \alpha = ± E) \alpha = ±$ = (2, 0, 0, 1) and $Find all values of the real number \alpha such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, \alpha , 8, 3) is equal to 22 . A) \alpha = ± 10 B) \alpha = ± 2 C) \alpha = ± 2 D) \alpha = ± E) \alpha = ±$ = (-2, $\alpha$ , 8, 3) is equal to 22 $Find all values of the real number \alpha such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, \alpha , 8, 3) is equal to 22 . A) \alpha = ± 10 B) \alpha = ± 2 C) \alpha = ± 2 D) \alpha = ± E) \alpha = ±$ .

\alpha

= ± 10
B)

\alpha

\alpha

\alpha

\alpha

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Find

(x), where M is the 2-manifold in Find d (x), where M is the 2-manifold in given parametrically by , sin(2 ), 3 ) for 0 ≤ ≤ 1, 0 ≤ ≤ 1.

given parametrically by Find d (x), where M is the 2-manifold in given parametrically by , sin(2 ), 3 ) for 0 ≤ ≤ 1, 0 ≤ ≤ 1.

sin(2

), 3

) for 0 ≤

≤ 1, 0 ≤

≤ 1.

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Find the 2-volume of the 2-manifold in $Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E)$ given parametrically by $Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E)$ for 0 $\le$ u $\le$ 1, 0 $\le$ v $\le$ $\pi$ .

A) $Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E)$ sinh(2)
B) $Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E)$ sinh(2) +

\pi

\pi

$Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E)$ (1)
D) 0
E) $Find the 2-volume of the 2-manifold in given parametrically by for 0 \le u \le 1, 0 \le v \le \pi . A) sinh(2) B) sinh(2) + \pi C) \pi (1) D) 0 E)$

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Let f be a smooth real value function over a domain D in , then the graph x_n+1 = f( , ,....., ) is

A) a smooth (n - 1)-manifold in Let f be a smooth real value function over a domain D in , then the graph xn+1 = f( , ,....., ) is A) a smooth (n - 1)-manifold in B) a smooth n-manifold in C) a smooth (n + 1)-manifold in D) a smooth (n + 1)-manifold in E) a smooth n-manifold in

B) a smooth n-manifold in Let f be a smooth real value function over a domain D in , then the graph xn+1 = f( , ,....., ) is A) a smooth (n - 1)-manifold in B) a smooth n-manifold in C) a smooth (n + 1)-manifold in D) a smooth (n + 1)-manifold in E) a smooth n-manifold in

C) a smooth (n + 1)-manifold in Let f be a smooth real value function over a domain D in , then the graph xn+1 = f( , ,....., ) is A) a smooth (n - 1)-manifold in B) a smooth n-manifold in C) a smooth (n + 1)-manifold in D) a smooth (n + 1)-manifold in E) a smooth n-manifold in

D) a smooth (n + 1)-manifold in Let f be a smooth real value function over a domain D in , then the graph xn+1 = f( , ,....., ) is A) a smooth (n - 1)-manifold in B) a smooth n-manifold in C) a smooth (n + 1)-manifold in D) a smooth (n + 1)-manifold in E) a smooth n-manifold in

E) a smooth n-manifold in Let f be a smooth real value function over a domain D in , then the graph xn+1 = f( , ,....., ) is A) a smooth (n - 1)-manifold in B) a smooth n-manifold in C) a smooth (n + 1)-manifold in D) a smooth (n + 1)-manifold in E) a smooth n-manifold in

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The two equations z = The two equations z = and z = in define a smooth manifold of dimension two in .

and z =

define a smooth manifold of dimension two in The two equations z = and z = in define a smooth manifold of dimension two in .

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One way to describe a smooth k-manifold M in

is to require its points One way to describe a smooth k-manifold M in is to require its points satisfy a set of (n - k) independent equations in ( , ,....., ).

satisfy a set of (n - k) independent equations in ( One way to describe a smooth k-manifold M in is to require its points satisfy a set of (n - k) independent equations in ( , ,....., ).

,.....,

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Find

(x), where M is the 2-manifold in Find d (x), where M is the 2-manifold in given parametrically by for 0 ≤ u ≤ 1, 0 ≤ v ≤ 1.

given parametrically by Find d (x), where M is the 2-manifold in given parametrically by for 0 ≤ u ≤ 1, 0 ≤ v ≤ 1.

for 0 ≤ u ≤ 1, 0 ≤ v ≤ 1.

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The k-volume of a k-parallelogram in

spanned by the k vectors, The k-volume of a k-parallelogram in spanned by the k vectors, ,......, is given by det(A), where A is the k × k matrix whose columns are the components of the vectors.

,......,

is given by det(A), where A is the k × k matrix whose columns are the components of the vectors. The k-volume of a k-parallelogram in spanned by the k vectors, ,......, is given by det(A), where A is the k × k matrix whose columns are the components of the vectors.

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Let x = ( , , ) = ( , , ), J(u) = . Find det( J(u)).

B) 1
C) d

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Calculate $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ dx $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ dz, where M is the surface given by z = $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ , 0 $\le$ z $\le$ 1, using the following parametrizations: (i) (x, y, z) = p( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ , $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ ) = ( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ cos( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ ), $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ sin( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ ) $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ )
(ii) (x, y, z) = p( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ , $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ )= ( $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ , $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ , $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ )

A) (i) $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ (ii) - $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$
B) (i) $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ (ii) 0
C) (i) $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ (ii) - $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$
D) (i) 6

\pi

(ii) - $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$
E) (i) - $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$ (ii) $Calculate dx dz, where M is the surface given by z = , 0 \le z \le 1, using the following parametrizations: (i) (x, y, z) = p( , ) = ( cos( ), sin( ) ) (ii) (x, y, z) = p( , )= ( , , ) A) (i) (ii) - B) (i) (ii) 0 C) (i) (ii) - D) (i) 6 \pi (ii) - E) (i) - (ii)$

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, 5

, -2

) would be orientation preserving parametrization for M.
E) P is orientation reversing for M, but q(u) = (-

, 5

, 2

) would be an orientation-preserving parametrization for M.

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Find , where M is the 2-manifold in given parametrically by for 0 < < 1, 0 < < 2.

A) -130
B) 135
C) - Find , where M is the 2-manifold in given parametrically by for 0 < < 1, 0 < < 2. A) -130 B) 135 C) - D) 130 E) -13

D) 130
E) -13

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Evaluate the integral of $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ = $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ + $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ 11ee7bcb_75db_a3b7_ae82_1108d245f507_TB9661_11 d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ + $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ 11ee7bcb_75db_a3b7_ae82_1108d245f507_TB9661_11 d $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ over the upper hemispherical surface $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ + $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ + $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ = 16, $Evaluate the integral of = d d + d d + d d over the upper hemispherical surface + + = 16, \ge 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing. A) ±12 \pi B) ± 4 \pi C) ± 48 \pi D) ± 6 \pi E) ± 24 \pi$ $\ge$ 0.Note: There are two possible answers, depending on whether the parametrization used is orientation preserving or orientation reversing.

A) ±12

\pi

B) ± 4

\pi

C) ± 48

\pi

D) ± 6

\pi

E) ± 24

\pi

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If M is the part of the surface z = g(x, y) in

that lies above a closed region D in the If M is the part of the surface z = g(x, y) in that lies above a closed region D in the , then the integral of the differential 2-form = f(x, y) dx dy over M is independent of the function g.

, then the integral of the differential 2-form If M is the part of the surface z = g(x, y) in that lies above a closed region D in the , then the integral of the differential 2-form = f(x, y) dx dy over M is independent of the function g.

= f(x, y) dx If M is the part of the surface z = g(x, y) in that lies above a closed region D in the , then the integral of the differential 2-form = f(x, y) dx dy over M is independent of the function g.

dy over M is independent of the function g.

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\theta

\theta

)sin(11ee7bce_0cad_3050_ae82_0fc996929baa_TB9661_11 ), sin(

\theta

)sin(11ee7bce_0cad_3050_ae82_0fc996929baa_TB9661_11 ), cos(11ee7bce_0cad_3050_ae82_0fc996929baa_TB9661_11 ),0

\le

\theta

\le

\pi

, and let $Let M be the smooth 2-manifold , x = p( \theta , ) = (cos( \theta )sin( ), sin( \theta )sin( ), cos( ),0 \le\theta \le 2 \pi , and let be a parametrization for M. If M is oriented by the differential 2-form = zdx dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.$ be a parametrization for M. If M is oriented by the differential 2-form $Let M be the smooth 2-manifold , x = p( \theta , ) = (cos( \theta )sin( ), sin( \theta )sin( ), cos( ),0 \le\theta \le 2 \pi , and let be a parametrization for M. If M is oriented by the differential 2-form = zdx dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.$ = zdx $Let M be the smooth 2-manifold , x = p( \theta , ) = (cos( \theta )sin( ), sin( \theta )sin( ), cos( ),0 \le\theta \le 2 \pi , and let be a parametrization for M. If M is oriented by the differential 2-form = zdx dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.$ dy, determine whether the parametrization p is orientation preserving or orientation reversing for M.

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Consider the unit cube Q = Consider the unit cube Q = in with the standard orientation given by .Express the orientations of the bottom and the front faces of Q as differential 1-forms evaluated at the cross product of vectors u, v in .

with the standard orientation given by Consider the unit cube Q = in with the standard orientation given by .Express the orientations of the bottom and the front faces of Q as differential 1-forms evaluated at the cross product of vectors u, v in .

.Express the orientations of the bottom and the front faces of Q as differential 1-forms evaluated at the cross product of vectors u, v in Consider the unit cube Q = in with the standard orientation given by .Express the orientations of the bottom and the front faces of Q as differential 1-forms evaluated at the cross product of vectors u, v in .

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A) the 3-volume of the projection of S onto the hyperplane Let S be a piece with boundary of a smooth 3-manifold in R4 (hypersurface) given by the equation = g( , , ) and let = d d d . Apart from sign due to orientation of S, is equal to A) the 3-volume of the projection of S onto the hyperplane = 0 B) the 4-volume of the projection of S onto the hyperplane = 0 C) D) E) None of the above

= 0
B) the 4-volume of the projection of S onto the hyperplane Let S be a piece with boundary of a smooth 3-manifold in R4 (hypersurface) given by the equation = g( , , ) and let = d d d . Apart from sign due to orientation of S, is equal to A) the 3-volume of the projection of S onto the hyperplane = 0 B) the 4-volume of the projection of S onto the hyperplane = 0 C) D) E) None of the above

= 0
C)

E) None of the above

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Use the generalized Stokes's Theorem to find where and D is the oriented boundary of the domain .

A) 5
B) 8
C) 240
D) 3
E) 37

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Use the generalized Stokes's Theorem to find $Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz dx - 9z dx dy and D is the oriented boundary of the conical domain D = . A) 8 \pi B) 32 \pi C) 24 \pi D) 12 \pi E) 4 \pi$ where $Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz dx - 9z dx dy and D is the oriented boundary of the conical domain D = . A) 8 \pi B) 32 \pi C) 24 \pi D) 12 \pi E) 4 \pi$ = 7x dy $Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz dx - 9z dx dy and D is the oriented boundary of the conical domain D = . A) 8 \pi B) 32 \pi C) 24 \pi D) 12 \pi E) 4 \pi$ dz + (3y + 2z) dz11ee7bcc_0cc6_216b_ae82_3f9133291a84_TB9661_11 dx - 9z dx11ee7bcc_0cc6_216b_ae82_3f9133291a84_TB9661_11 dy and $Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz dx - 9z dx dy and D is the oriented boundary of the conical domain D = . A) 8 \pi B) 32 \pi C) 24 \pi D) 12 \pi E) 4 \pi$ D is the oriented boundary of the conical domain D = $Use the generalized Stokes's Theorem to find where = 7x dy dz + (3y + 2z) dz dx - 9z dx dy and D is the oriented boundary of the conical domain D = . A) 8 \pi B) 32 \pi C) 24 \pi D) 12 \pi E) 4 \pi$ .

A) 8

\pi

B) 32

\pi

C) 24

\pi

D) 12

\pi

E) 4

\pi

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Integrate the differential 3-form = dx dy11ee7bcc_8068_caae_ae82_1bce1a65fd1b_TB9661_11dz over the boundary of the 4-dimensional tetrahedron T = .

A) -648
B) - Integrate the differential 3-form = dx dy dz over the boundary of the 4-dimensional tetrahedron T = . A) -648 B) - C) -36 D) 648 E) 36

C) -36
D) 648
E) 36

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Let C be the curve of intersection of the cylinder $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3$ + $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3$ = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3$ , where $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3$ = -3 $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3$ z dx + sin(y) dy + (3x $Let C be the curve of intersection of the cylinder + = 4 and the plane y = x + 1. Use the generalized Stokes's Theorem to evaluate , where = -3 z dx + sin(y) dy + (3x + x +7) dz. A) 16 \pi B) 28 \pi C) 14 \pi D) 20 \pi E) 3$ + x +7) dz.

A) 16

\pi

B) 28

\pi

C) 14

\pi

D) 20

\pi

E) 3

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State the Divergence Theorem and Stokes's Theorem in 3-space, and Green's Theorem in 2-space in terms of differential forms.

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Let Ω be the differential 3-form $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$ be the 4-dimensional ball of radius α in R⁴ ; that is
$ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$
(a) Use the generalized Stokes's Theorem to evaluate $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$
(b) Use part (a) to find the 4-volume of the ball.
Hint: You may use symmetry and the transformation , .
$ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$

A) 4

\pi

$ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$
B) $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$ $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$ $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$
C) $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$ $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$ $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$
D) 2 $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$ $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$
E) $ Let Ω be the differential 3-form be the 4-dimensional ball of radius α in R4 ; that is (a) Use the generalized Stokes's Theorem to evaluate (b) Use part (a) to find the 4-volume of the ball. Hint: You may use symmetry and the transformation , . A) 4 \pi B) C) D) 2 E) \pi$

\pi

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Let D be a closed bounded domain in

and lot Ψ = xdy∧dz + ydz∧dx + zdx∧dy. Show that the volume V of D is given by Let D be a closed bounded domain in and lot Ψ = xdy∧dz + ydz∧dx + zdx∧dy. Show that the volume V of D is given by

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Integrate the differential (n -1)-form over the boundary of the n-dimensional cube
Note: The hat ∧ is used to indicate a missing component.

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