Question 1

Accepted Answer

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)    F D

Question 2

= 0 for every k-form   on   .

Accepted Answer

A) True 
 B)False  False

Question 3

Simplify a   dy   dz + b  dy + c  dx + (a + b +c)  dy.

Accepted Answer

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 
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 A

Question 4

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

Accepted Answer

The answer of Find   d   (x),...

Question 5

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

Accepted Answer

The answer of Let [r, θ, z] be the cylindrical...

Question 6

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)   dy over M is independent of the function g.

Accepted Answer

A) True 
 B)False

Question 7

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           .

Accepted Answer

A)  d(          ) = (  )         +     d(  )     +       d(  ) 
B)  d(          ) = (  )  (  )  (  ) 
C)  d(          ) =   (  )         +       d(  )     +           d(   ) 
D)  d(          ) = (  )         +       d(  )     +           d(  ) 
E)  d(          ) = (  )         +       d(  )     +           d(  ) 
A)  d(          ) = (  )         +     d(  )     +       d(  ) 
B)  d(          ) = (  )  (  )  (  ) 
C)  d(          ) =   (  )         +       d(  )     +           d(   ) 
D)  d(          ) = (  )         +       d(  )     +           d(  ) 
E)  d(          ) = (  )         +       d(  )     +           d(  )

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

The answer of Let D be a closed bounded domain...

Question 10

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

Accepted Answer

A)    
B)    
C)    
D)    
E)    - 1 
A)    
B)    
C)    
D)    
E)    - 1

Question 11

Let   = 4xz   (y)  dz + z(2y + sin(2y))  dx + (yz - 2   )  dy. Find   .

Accepted Answer

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

Question 12

Show that if φ is a k-form on   , then φ∧φ = 0 if k is odd.

Accepted Answer

Recall that if φ is a k-form and ψ is an

Question 13

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).

Accepted Answer

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

Question 14

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

Accepted Answer

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   
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 15

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

Accepted Answer

A)    
B)  1 
C)  d   d   
D)    d   d   
E)    d   d   
A)    
B)  1 
C)  d   d   
D)    d   d   
E)    d   d

Question 16

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

Accepted Answer

A) True 
 B)False

Question 17

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

Accepted Answer

A) True 
 B)False

Question 18

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    =   .

Accepted Answer

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 
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

Question 19

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

Accepted Answer

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

Question 20

Expand and simplify: (2a dx + (b + 2a)dy + c dz)∧(a dx + 2b dy + c dz) - (2aS1U1P12S1S1P0  - 2ab) ( a - b) dy∧dx  . Express your answer in terms of the basis vectors dy∧dz, dz∧dx, and dx∧dy of

Accepted Answer

-3bc + 4ac

= 0 for every k-form on .

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.

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

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

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.

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 .

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 = 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 .

Show that if φ is a k-form on , then φ∧φ = 0 if k is odd.

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).

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

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

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

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

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

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

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Exam 18: Differential Forms and Exterior Calculus

= 0 for every k-form on .

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.

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

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

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.

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 .

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 ( ), 1 ≤\le≤ k ≤\le≤ n be the vector space of all k-forms on and let be the dimension of . Find .

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 .

Show that if φ is a k-form on , then φ∧φ = 0 if k is odd.

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).

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

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

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

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

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

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

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

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).

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

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