Question 1

Evaluate   , where F = y i + zx j +   k and C are the positively oriented boundary of the triangle in which the plane   with upward normal, intersects the first octant of space.

Accepted Answer

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

Question 2

Every conservative vector field is irrotational.

Accepted Answer

A) True 
 B)False  True

Question 3

Find the flux of F = 2   y i +     j out of the rectangle 0 $\le$ x $\le$ ln(3), 0 $\le$ y $\le$2.

Accepted Answer

A)  4 
B)  8 
C)  16 
D)  32 
E)  24 
A)  4 
B)  8 
C)  16 
D)  32 
E)  24 C

Question 4

A vector field F satisfying the equation div F = 0 in domain D is called:

Accepted Answer

A)  irrotational in D 
B)  a scalar potential 
C)  solenoidal in D 
D)  conservative in D 
E)  a vector potential 
A)  irrotational in D 
B)  a scalar potential 
C)  solenoidal in D 
D)  conservative in D 
E)  a vector potential

Question 5

For what value of the constant C is the vector field F =   i + C(xy + yz) j +   k. solenoidal?
 If C has that value, find a vector potential G for F having the form G(x, y, z) =   (x, y, z) i +   y k.

Accepted Answer

A)  C = -2, G = y   i +   y k 
B)  C = -2, G = - y   i +   y k 
C)  C = -2, G = x   i +   y k 
D)  C = -2, G = - x   i +   y k 
E)  C = 2, G = y   i +   y k 
A)  C = -2, G = y   i +   y k 
B)  C = -2, G = - y   i +   y k 
C)  C = -2, G = x   i +   y k 
D)  C = -2, G = - x   i +   y k 
E)  C = 2, G = y   i +   y k

Question 6

. F = F .   for any sufficiently smooth vector field F.

Accepted Answer

A) True 
 B)False

Question 7

Show that div (   r) = (n + 3)   .You may use the following fact: grad (   ) = n   r

Accepted Answer

The answer of Show that div (   r)...

Question 8

Use Green's Theorem to compute the integral   clockwise around the circle of radius 3 centred at the origin.

Accepted Answer

A)  18  $\pi$ 
B)  9  $\pi$ 
C)  127  $\pi$ 
D)  243  $\pi$ 
E)  0 
A)  18  $\pi$ 
B)  9  $\pi$ 
C)  127  $\pi$ 
D)  243  $\pi$ 
E)  0

Question 9

Compute the gradient of the function f(x, y) =   sin y +   cos x.

Accepted Answer

A)  (   cos y - 2y cos x) i + (2x sin y +   sin x) j 
B)  (2x sin y +   sin x) i + (   cos y - 2y cos x) j 
C)  (2x sin y -   sin x) i + (   cos y + 2y cos x) j 
D)  (   cos y + 2y cos x) i + (2x sin y -   sin x) j 
E)  (2x sin y) i + (2y cos x) j 
A)  (   cos y - 2y cos x) i + (2x sin y +   sin x) j 
B)  (2x sin y +   sin x) i + (   cos y - 2y cos x) j 
C)  (2x sin y -   sin x) i + (   cos y + 2y cos x) j 
D)  (   cos y + 2y cos x) i + (2x sin y -   sin x) j 
E)  (2x sin y) i + (2y cos x) j

Question 10

Find the flux of F =   out of (a) the disk   +   $\le$  , (b) an arbitrary plane region not containing the origin in its interior or on its boundary, and (c) an arbitrary plane region containing the origin in its interior.

Accepted Answer

A)  (a) 0 $~~~~~~~~$(b) 0 $~~~~~~~~$(c) 0 
B)  (a) 2 $\pi$ $~~~~~~~~$(b) 0 $~~~~~~~~$(c) 2 $\pi$ 
C)  (a) 2 $\pi$a $~~~~~~~~$(b) 0 $~~~~~~~~$(c) 2 $\pi$ 
D)  (a) 0 $~~~~~~~~$(b) 2 $\pi$ $~~~~~~~~$(c) 0 
E)  None of the above 
A)  (a) 0 $~~~~~~~~$(b) 0 $~~~~~~~~$(c) 0 
B)  (a) 2 $\pi$ $~~~~~~~~$(b) 0 $~~~~~~~~$(c) 2 $\pi$ 
C)  (a) 2 $\pi$a $~~~~~~~~$(b) 0 $~~~~~~~~$(c) 2 $\pi$ 
D)  (a) 0 $~~~~~~~~$(b) 2 $\pi$ $~~~~~~~~$(c) 0 
E)  None of the above

Question 11

If F = x i + y j, calculate the flux of F upward through the part of the surface z = 4 - x2 - y2 that lies above the (x, y) plane by applying the Divergence Theorem to the volume bounded by the surface and the disk that it cuts out of the (x, y) plane.

Accepted Answer

A)  14 $\pi$ 
B)  16 $\pi$ 
C)  18 $\pi$ 
D)  20 $\pi$ 
E)  8 $\pi$ 
A)  14 $\pi$ 
B)  16 $\pi$ 
C)  18 $\pi$ 
D)  20 $\pi$ 
E)  8 $\pi$

Question 12

Verify that the vector field  F  =  (2x  yS1U1P12S1S1P0zS1U1P12S1S1P0 - sin(x)sin(y)) i  +  (2 xS1U1P12S1S1P0y  zS1U1P12S1S1P0+ cos(x)cos(y)) j  +  (2xS1U1P12S1S1P0yS1U1P12S1S1P0  z +  ) k  is conservative and find a scalar potential  f(x, y, z)  for it that satisfies  f(0, 0, 0) = 1.

Accepted Answer

A)    
B)  f(x, y, z) =       + cos(x)sin(y) +   + 1 
C)  f(x, y, z) =       + sin(x)cos(y) +   + 1 
D)  f(x, y, z) =       + cos(x)sin(y) +   
E)  f(x, y, z) = xyz + cos(x)sin(y) +   
A)    
B)  f(x, y, z) =       + cos(x)sin(y) +   + 1 
C)  f(x, y, z) =       + sin(x)cos(y) +   + 1 
D)  f(x, y, z) =       + cos(x)sin(y) +   
E)  f(x, y, z) = xyz + cos(x)sin(y) +

Question 13

Compute div F for F = (2x + yz) i + (   +   ) j + (x sin(z) +   ) k.

Accepted Answer

A)  2 + 2y +   + cos(z) + 3   
B)  2 + 2y + z   - x cos(z) + 3   
C)  2 + 2y + z   + x cos(z) + 3   
D)  2 + 2y + x cos(z) 
E)  2 + 2y +   - cos(z) + 3   
A)  2 + 2y +   + cos(z) + 3   
B)  2 + 2y + z   - x cos(z) + 3   
C)  2 + 2y + z   + x cos(z) + 3   
D)  2 + 2y + x cos(z) 
E)  2 + 2y +   - cos(z) + 3

Question 14

If r = x i + y j + z k and f(u) is any differentiable function of one variable, evaluate and simplify   .

Accepted Answer

A)  0 
B)  r 
C)  2r 
D)  3r 
E)  4r 
A)  0 
B)  r 
C)  2r 
D)  3r 
E)  4r

Question 15

Evaluate      and  S  is the part of the sphere       that lies above the xy-plane with outward normal field.

Accepted Answer

A)  2 $\pi$ 
B)  4 $\pi$ 
C)  6 $\pi$ 
D)  8 $\pi$ 
E)  0 
A)  2 $\pi$ 
B)  4 $\pi$ 
C)  6 $\pi$ 
D)  8 $\pi$ 
E)  0

Question 16

Use Stokes's Theorem to evaluate the line integral   where C is the triangle with vertices (1, 1, 1), (0, 1, 0) and (0, 0, 0) oriented counterclockwise as seen from high on the z-axis.

Accepted Answer

A)  0 
B)  1 
C)  2 
D)  -2 
E)  -1 
A)  0 
B)  1 
C)  2 
D)  -2 
E)  -1

Question 17

Calculate the divergence of the vector field F(x, y, z) = (   - xz) i + (z   -   ) j - xy   k.

Accepted Answer

A)    - z - zx   - 2   - y   
B)    + zx   - 2   + 4xy   
C)    - z + zx   - 2   - 4xy   
D)    - z + zx   - 4xy   
E)    + zx   + 2   + 4xy   
A)    - z - zx   - 2   - y   
B)    + zx   - 2   + 4xy   
C)    - z + zx   - 2   - 4xy   
D)    - z + zx   - 4xy   
E)    + zx   + 2   + 4xy

Question 18

Use a line integral to find the area enclosed by the x-axis and one arch of the cycloid given parametrically by the equations x(t) = 3(t - sin(t)), y(t) =3(1 - cos(t)), 0 $\le$ t $\le$ 2$\pi$.

Accepted Answer

A)  36 $\pi$ 
B)  18 $\pi$ 
C)  27 $\pi$ 
D)  54 $\pi$ 
E)  9 $\pi$ 
A)  36 $\pi$ 
B)  18 $\pi$ 
C)  27 $\pi$ 
D)  54 $\pi$ 
E)  9 $\pi$

Question 19

Use Stokes's Theorem to evaluate the line integral   + (2x - y) dy + (y + z) dz,where C is the triangle cut from the plane P with equation   by the three coordinate planes. C has orientation inherited from the upward normal on P.

Accepted Answer

A)  18 
B)  -6 
C)  6 
D)  24 
E)  -24 
A)  18 
B)  -6 
C)  6 
D)  24 
E)  -24

Question 20

Verify that the vector field F =   i + (1 - xy) j - xz k is solenoidal, and find a vector potential G for it having the form G(x, y, z) =   (x, y, z) i +   y k.

Accepted Answer

A)  (xyz + z) i +   y k 
B)  (   y + z) i +   y k 
C)  (xyz - z) i +   y k 
D)  (   z - z) i +   y k 
E)  xyz i +   y k 
A)  (xyz + z) i +   y k 
B)  (   y + z) i +   y k 
C)  (xyz - z) i +   y k 
D)  (   z - z) i +   y k 
E)  xyz i +   y k

Evaluate , where F = y i + zx j + k and C are the positively oriented boundary of the triangle in which the plane with upward normal, intersects the first octant of space.

Every conservative vector field is irrotational.

A vector field F satisfying the equation div F = 0 in domain D is called:

For what value of the constant C is the vector field F = i + C(xy + yz) j + k. solenoidal? If C has that value, find a vector potential G for F having the form G(x, y, z) = (x, y, z) i + y k.

. F = F . 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 for any sufficiently smooth vector field F.

Show that div ( r) = (n + 3) .You may use the following fact: grad ( ) = n r

Use Green's Theorem to compute the integral clockwise around the circle of radius 3 centred at the origin.

Compute the gradient of the function f(x, y) = sin y + cos x.

If F = x i + y j, calculate the flux of F upward through the part of the surface z = 4 - x² - y² that lies above the (x, y) plane by applying the Divergence Theorem to the volume bounded by the surface and the disk that it cuts out of the (x, y) plane.

Verify that the vector field F = (2x y²z² - sin(x)sin(y)) i + (2 x²y z²+ cos(x)cos(y)) j + (2x²y² z + ) k is conservative and find a scalar potential f(x, y, z) for it that satisfies f(0, 0, 0) = 1.

Compute div F for F = (2x + yz) i + ( + ) j + (x sin(z) + ) k.

If r = x i + y j + z k and f(u) is any differentiable function of one variable, evaluate and simplify .

Evaluate and S is the part of the sphere that lies above the xy-plane with outward normal field.

Use Stokes's Theorem to evaluate the line integral where C is the triangle with vertices (1, 1, 1), (0, 1, 0) and (0, 0, 0) oriented counterclockwise as seen from high on the z-axis.

Calculate the divergence of the vector field F(x, y, z) = ( - xz) i + (z - ) j - xy k.

Use a line integral to find the area enclosed by the x-axis and one arch of the cycloid given parametrically by the equations x(t) = 3(t - sin(t)), y(t) =3(1 - cos(t)), 0 $\le$ t $\le$ 2 $\pi$ .

Use Stokes's Theorem to evaluate the line integral + (2x - y) dy + (y + z) dz,where C is the triangle cut from the plane P with equation by the three coordinate planes. C has orientation inherited from the upward normal on P.

Verify that the vector field F = i + (1 - xy) j - xz k is solenoidal, and find a vector potential G for it having the form G(x, y, z) = (x, y, z) i + y k.

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

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Vector Functions and Curves

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Exam 17: Vector Calculus

Evaluate , where F = y i + zx j + k and C are the positively oriented boundary of the triangle in which the plane with upward normal, intersects the first octant of space.

Every conservative vector field is irrotational.

Find the flux of F = 2 y i + j out of the rectangle 0 ≤\le≤ x ≤\le≤ ln(3), 0 ≤\le≤ y ≤\le≤ 2.

A vector field F satisfying the equation div F = 0 in domain D is called:

For what value of the constant C is the vector field F = i + C(xy + yz) j + k. solenoidal? If C has that value, find a vector potential G for F having the form G(x, y, z) = (x, y, z) i + y k.

. F = F . 11ee7bac_9657_298c_ae82_2b2223aa180c_TB9661_11 for any sufficiently smooth vector field F.

Show that div ( r) = (n + 3) .You may use the following fact: grad ( ) = n r

Use Green's Theorem to compute the integral clockwise around the circle of radius 3 centred at the origin.

Compute the gradient of the function f(x, y) = sin y + cos x.

Find the flux of F = out of (a) the disk + ≤\le≤ , (b) an arbitrary plane region not containing the origin in its interior or on its boundary, and (c) an arbitrary plane region containing the origin in its interior.

If F = x i + y j, calculate the flux of F upward through the part of the surface z = 4 - x2 - y2 that lies above the (x, y) plane by applying the Divergence Theorem to the volume bounded by the surface and the disk that it cuts out of the (x, y) plane.

Verify that the vector field F = (2x y2z2 - sin(x)sin(y)) i + (2 x2y z2+ cos(x)cos(y)) j + (2x2y2 z + ) k is conservative and find a scalar potential f(x, y, z) for it that satisfies f(0, 0, 0) = 1.

Compute div F for F = (2x + yz) i + ( + ) j + (x sin(z) + ) k.

If r = x i + y j + z k and f(u) is any differentiable function of one variable, evaluate and simplify .

Evaluate and S is the part of the sphere that lies above the xy-plane with outward normal field.

Use Stokes's Theorem to evaluate the line integral where C is the triangle with vertices (1, 1, 1), (0, 1, 0) and (0, 0, 0) oriented counterclockwise as seen from high on the z-axis.

Calculate the divergence of the vector field F(x, y, z) = ( - xz) i + (z - ) j - xy k.

Use a line integral to find the area enclosed by the x-axis and one arch of the cycloid given parametrically by the equations x(t) = 3(t - sin(t)), y(t) =3(1 - cos(t)), 0 ≤\le≤ t ≤\le≤ 2 π\piπ .

Use Stokes's Theorem to evaluate the line integral + (2x - y) dy + (y + z) dz,where C is the triangle cut from the plane P with equation by the three coordinate planes. C has orientation inherited from the upward normal on P.

Verify that the vector field F = i + (1 - xy) j - xz k is solenoidal, and find a vector potential G for it having the form G(x, y, z) = (x, y, z) i + y k.

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

Differential Forms and Exterior Calculus

Ordinary Differential Equations

Filters

If F = x i + y j, calculate the flux of F upward through the part of the surface z = 4 - x² - y² that lies above the (x, y) plane by applying the Divergence Theorem to the volume bounded by the surface and the disk that it cuts out of the (x, y) plane.

Verify that the vector field F = (2x y²z² - sin(x)sin(y)) i + (2 x²y z²+ cos(x)cos(y)) j + (2x²y² z + ) k is conservative and find a scalar potential f(x, y, z) for it that satisfies f(0, 0, 0) = 1.

Use a line integral to find the area enclosed by the x-axis and one arch of the cycloid given parametrically by the equations x(t) = 3(t - sin(t)), y(t) =3(1 - cos(t)), 0 $\le$ t $\le$ 2 $\pi$ .