Question 1

Accepted Answer

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)

Question 2

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

Accepted Answer

A)  -648 
B)  -   
C)  -36 
D)  648 
E)  36 
A)  -648 
B)  -   
C)  -36 
D)  648 
E)  36

Question 3

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.

Accepted Answer

ψ = (2 dx

Question 4

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

Accepted Answer

A)   dz +  dx -  dy 
B)  xdx + ydy + zdz 
C)    dz +   dx +   dy 
D)    dz +   dx -4  dx   dy 
E)  (x + y + z )     dz 
A)   dz +  dx -  dy 
B)  xdx + ydy + zdz 
C)    dz +   dx +   dy 
D)    dz +   dx -4  dx   dy 
E)  (x + y + z )     dz

Question 5

If   is a k-form and   is an l-form on   , then   =    .

Accepted Answer

A) True 
 B)False

Question 6

Accepted Answer

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

Question 7

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

Accepted Answer

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 
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 8

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.

Accepted Answer

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

Question 9

Let η = FS1U1B11S1U1B0  dy∧dz - FS1U1B12S1U1B0 dx∧dz+ FS1U1B13S1U1B0   dx∧   j + bS1U1B13S1U1B0k
be vectors in  . If η is identified by the vector  w = FS1U1B11S1U1B0  i +  FS1U1B12S1U1B0 j + FS1U1B13S1U1B0   k  then  η (u, v) is equal to

Accepted Answer

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

Question 10

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

Accepted Answer

A) True 
 B)False

Question 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

A)      
B)      
C)      
D)      
E)      
A)      
B)      
C)      
D)      
E)

Question 13

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

Accepted Answer

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 <   
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 <

Question 14

Let   =   . List all basis vectors of   (   ).

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 15

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?

Accepted Answer

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. 
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.

Question 16

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

Accepted Answer

A possible answer is

Question 17

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   .

Accepted Answer

A)  $\alpha$= ± 10 
B)  $\alpha$  = ± 2   
C)  $\alpha$ = ± 2   
D) $\alpha$ = ±   
E) $\alpha$ = ±   
A)  $\alpha$= ± 10 
B)  $\alpha$  = ± 2   
C)  $\alpha$ = ± 2   
D) $\alpha$ = ±   
E) $\alpha$ = ±

Question 18

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 19

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

Accepted Answer

A) True 
 B)False

Question 20

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

Accepted Answer

Let Φ be a differential k-form on a doma

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

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.

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.

If is a k-form and is an l-form on , then = .

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

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.

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

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

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

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

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

Let = . List all basis vectors of ( ).

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?

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

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

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

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

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Ordinary Differential Equations

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Exam 18: Differential Forms and Exterior Calculus

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

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.

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.

If is a k-form and is an l-form on , then = .

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

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.

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

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

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

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

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

Let = . List all basis vectors of ( ).

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?

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

Find all values of the real number α\alphaα such that the 2-volume of the 2-parallelogram in spanned by the vectors = (2, 0, 0, 1) and = (-2, α\alphaα , 8, 3) is equal to 22 .

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

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

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

Preliminaries

Limits and Continuity

Differentiation

Transcendental Functions

More Applications of Differentiation

Integration

Techniques of Integration

Applications of Integration

Conics, Parametric Curves, and Polar Curves

Sequences, Series, and Power Series

Vectors and Coordinate Geometry in 3-Space

Vector Functions and Curves

Partial Differentiation

Applications of Partial Derivatives

Multiple Integration

Vector Fields

Vector Calculus

Ordinary Differential Equations

Filters

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