Question 1

Define the curl of a vector field F.

Accepted Answer

A)  F ×   
B)    F 
C)    × F 
D)    . F 
E)    F 
A)  F ×   
B)    F 
C)    × F 
D)    . F 
E)    F

Question 2

If r = x i + y j + z k and k is a constant vector field in R3, then

Accepted Answer

A)  div ( k × r) = 0 
B)  div ( k × r) = 0. 
C)  grad ( k . r) = 2k 
D)  curl ( k × r) = 0 
E)  curl ( k × r) = 0. 
A)  div ( k × r) = 0 
B)  div ( k × r) = 0. 
C)  grad ( k . r) = 2k 
D)  curl ( k × r) = 0 
E)  curl ( k × r) = 0.

Question 3

Evaluate   where S is the first-octant part of the sphere of radius a centred at the origin. (Hint: Even though S is not a closed surface, it is still easiest to use the Divergence Theorem because the integrand in the surface integral is zero on the coordinate planes.)

Accepted Answer

A)     $\pi$    
B)   $\pi$   
C)     $\pi$  
D)  2 $\pi$   
E)     $\pi$   
A)     $\pi$    
B)   $\pi$   
C)     $\pi$  
D)  2 $\pi$   
E)     $\pi$

Question 4

Let  and F be sufficiently smooth scalar and vector fields, respectively.Express the well-known identity  . (  F ) = (    ) . F +   ( . F) using the notations grad , div or curl.

Accepted Answer

A)  curl (  F) = grad (  ) . F +     div (F) 
B)  div (  F) = curl (  ) . F +    grad (F) 
C)  div (  F) = grad (  ) . F +    div (F) 
D)  grad (  F) = div (  ) . F +    curl (F) 
E)  curl (  F) = div (  ) . F +     grad (F) 
A)  curl (  F) = grad (  ) . F +     div (F) 
B)  div (  F) = curl (  ) . F +    grad (F) 
C)  div (  F) = grad (  ) . F +    div (F) 
D)  grad (  F) = div (  ) . F +    curl (F) 
E)  curl (  F) = div (  ) . F +     grad (F)

Question 5

Let F = -   i +   j and let C be the boundary of circle   +   = 9 oriented counterclockwise. Use Green's Theorem to evaluate

Accepted Answer

A)  9 $\pi$ 
B)  0 
C)  -2 $\pi$ 
D)  2 $\pi$ 
E)  3 $\pi$ 
A)  9 $\pi$ 
B)  0 
C)  -2 $\pi$ 
D)  2 $\pi$ 
E)  3 $\pi$

Question 6

In cylindrical coordinates,   f for f =   sin($\theta$) +   .

Accepted Answer

A)  2r sin($\theta$)   + r cos($\theta$)   + 2z k 
B)  2r sin($\theta$)   +   cos($\theta$)   + 2z k 
C)  2r cos($\theta$)   + r sin($\theta$)   + 2z k 
D)  2r cos($\theta$)   +   sin($\theta$)   + 2z k 
E)  r sin($\theta$)   +   cos($\theta$)   + 2z k 
A)  2r sin($\theta$)   + r cos($\theta$)   + 2z k 
B)  2r sin($\theta$)   +   cos($\theta$)   + 2z k 
C)  2r cos($\theta$)   + r sin($\theta$)   + 2z k 
D)  2r cos($\theta$)   +   sin($\theta$)   + 2z k 
E)  r sin($\theta$)   +   cos($\theta$)   + 2z k

Question 7

In cylindrical coordinates, find   . F for F =     + rz cos($\theta$)   + rz sin($\theta$) k.

Accepted Answer

A)  4   - z sin($\theta$) + sin($\theta$) 
B)  4   - z sin($\theta$) + r sin($\theta$) 
C)  3   - sin($\theta$) + r sin($\theta$) 
D)  3   - z sin($\theta$) - r sin($\theta$) 
E)  3   - z sin($\theta$) + sin($\theta$) 
A)  4   - z sin($\theta$) + sin($\theta$) 
B)  4   - z sin($\theta$) + r sin($\theta$) 
C)  3   - sin($\theta$) + r sin($\theta$) 
D)  3   - z sin($\theta$) - r sin($\theta$) 
E)  3   - z sin($\theta$) + sin($\theta$)

Question 8

Use Stokes's Theorem to evaluate the integral    where C is the curve of intersection of the sphere       and the plane       oriented counterclockwise as seen from high on the  z-axis.

Accepted Answer

A)  2 $\pi$ 
B)  3 $\pi$ 
C)  4 $\pi$ 
D)  5 $\pi$ 
E)  0 
A)  2 $\pi$ 
B)  3 $\pi$ 
C)  4 $\pi$ 
D)  5 $\pi$ 
E)  0

Question 9

Let F be a smooth vector field in 3-space satisfying the condition   Find the flux of curl F upward through the part of the   lying above the xy-plane.

Accepted Answer

A)  81 $\pi$ 
B)  72 $\pi$ 
C)  27 $\pi$ 
D)  18 $\pi$ 
E)  None of the above 
A)  81 $\pi$ 
B)  72 $\pi$ 
C)  27 $\pi$ 
D)  18 $\pi$ 
E)  None of the above

Question 10

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

Accepted Answer

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

Question 11

Use Green's theorem in the plane to find the x-coordinate of the centroid of a regular plane region R (with areaA) enclosed by a positively oriented, piecewise smooth, simple closed curve C .

Accepted Answer

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

Question 12

A vector field F is called $\textbf{      solenoidal    }$ in a domain D if

Accepted Answer

A)    F = 0 in D 
B)  curl(F) = 0 in D 
C)  F =     in D for some scalar field   
D)  div(F) = 0 in D 
E)  grad(F) = 0 in D 
A)    F = 0 in D 
B)  curl(F) = 0 in D 
C)  F =     in D for some scalar field   
D)  div(F) = 0 in D 
E)  grad(F) = 0 in D

Question 13

In spherical coordinates, find   f for f =   sin($\theta$) cos($\theta$).

Accepted Answer

A)  3   sin($\theta$) cos($\theta$)   +   cos($\theta$) cos($\theta$)   -   sin($\theta$)   
B)  3   sin($\theta$) cos($\theta$)   +   cos($\theta$) cos($\theta$)   -   sin($\theta$) sin($\theta$)   
C)    
D)  3   sin($\theta$) cos($\theta$)   -   cos($\theta$) cos($\theta$)   +   sin($\theta$)   
E)  3   sin($\theta$) cos($\theta$)   -   cos($\theta$) cos($\theta$)   -   sin($\theta$)   
A)  3   sin($\theta$) cos($\theta$)   +   cos($\theta$) cos($\theta$)   -   sin($\theta$)   
B)  3   sin($\theta$) cos($\theta$)   +   cos($\theta$) cos($\theta$)   -   sin($\theta$) sin($\theta$)   
C)    
D)  3   sin($\theta$) cos($\theta$)   -   cos($\theta$) cos($\theta$)   +   sin($\theta$)   
E)  3   sin($\theta$) cos($\theta$)   -   cos($\theta$) cos($\theta$)   -   sin($\theta$)

Question 14

If C is the positively oriented boundary of a plane region R having area 3 units and centroid at the point (12, 6), evaluate (i)   (ii)   dx + 3xy dy

Accepted Answer

A)  (i) 36 (ii) 15 
B)  (i) -36 (ii) 18 
C)  (i) -18 (ii) 36 
D)  (i) -4 (ii) 2 
E)  (i) 432 (ii) 1080 
A)  (i) 36 (ii) 15 
B)  (i) -36 (ii) 18 
C)  (i) -18 (ii) 36 
D)  (i) -4 (ii) 2 
E)  (i) 432 (ii) 1080

Question 15

Find all values of the nonzero constant real numbers a, b, and c so that the vector field F = a cos(   x + 2y )cosh (c z) i + b cos (   x + 2y)cosh (c z) j + c sin(   x + 2y)sinh(c z) k is both $\textbf{     irrotational    }$ and $\textbf{     solenoidal    }$ .

Accepted Answer

A)  a = -   , b = -2, c = 3 
B)  a =   , b = 2, c = 2 
C)  a = -   , b = -2, c =   ± 2 
D)  a =   , b = 2, c = ± 3 
E)  a =   , b = -2, c = 9 
A)  a = -   , b = -2, c = 3 
B)  a =   , b = 2, c = 2 
C)  a = -   , b = -2, c =   ± 2 
D)  a =   , b = 2, c = ± 3 
E)  a =   , b = -2, c = 9

Question 16

Using spherical polar coordinates, find   × F for F =   sin($\theta$)   + sin($\theta$)   +   cos($\theta$)   .

Accepted Answer

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

Question 17

Evaluate the integral   (   ) - 2y) dx + (3x - ysin(   )) dy counterclockwise around the triangle in the xy-plane having vertices (0, 0), (2, 2), and (2, 0).

Accepted Answer

A)  5 
B)  20 
C)  0 
D)  10 
E)  2 
A)  5 
B)  20 
C)  0 
D)  10 
E)  2

Question 18

Find the flux of   i - xy j +3z k out of the solid region bounded by the parabolic cylinder   and the planes   , and

Accepted Answer

A)  208 
B)  112 
C)  64 
D)  48 
E)  176 
A)  208 
B)  112 
C)  64 
D)  48 
E)  176

Question 19

Use Green's Theorem to compute the integral   where C is the triangle formed by the lines y = -x + 1, x = 0 and y = 0, oriented clockwise.

Accepted Answer

A)  3 
B)  2 
C)  1 
D)  0 
E)   $\pi$ 
A)  3 
B)  2 
C)  1 
D)  0 
E)   $\pi$

Question 20

Evaluate   clockwise around the triangle with vertices (0, 0), (3, 0), and (3, 3).

Accepted Answer

A)  27 
B)  9 
C)  -9 
D)  -27 
E)  0 
A)  27 
B)  9 
C)  -9 
D)  -27 
E)  0

Define the curl of a vector field F.

If r = x i + y j + z k and k is a constant vector field in R³, then

Evaluate where S is the first-octant part of the sphere of radius a centred at the origin. (Hint: Even though S is not a closed surface, it is still easiest to use the Divergence Theorem because the integrand in the surface integral is zero on the coordinate planes.)

Let F = - i + j and let C be the boundary of circle + = 9 oriented counterclockwise. Use Green's Theorem to evaluate

In cylindrical coordinates, find $In cylindrical coordinates, find f for f = sin( \theta ) + .$ f for f = $In cylindrical coordinates, find f for f = sin( \theta ) + .$ sin( $\theta$ ) + $In cylindrical coordinates, find f for f = sin( \theta ) + .$ .

Use Stokes's Theorem to evaluate the integral where C is the curve of intersection of the sphere and the plane oriented counterclockwise as seen from high on the z-axis.

Let F be a smooth vector field in 3-space satisfying the condition Find the flux of curl F upward through the part of the lying above the xy-plane.

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

Use Green's theorem in the plane to find the x-coordinate of the centroid of a regular plane region R (with areaA) enclosed by a positively oriented, piecewise smooth, simple closed curve C .

A vector field F is called $\textbf{ solenoidal }$ in a domain D if

In spherical coordinates, find $In spherical coordinates, find f for f = sin( \theta ) cos( \theta ).$ f for f = $In spherical coordinates, find f for f = sin( \theta ) cos( \theta ).$ sin( $\theta$ ) cos( $\theta$ ).

If C is the positively oriented boundary of a plane region R having area 3 units and centroid at the point (12, 6), evaluate (i) (ii) dx + 3xy dy

Evaluate the integral ( ) - 2y) dx + (3x - ysin( )) dy counterclockwise around the triangle in the xy-plane having vertices (0, 0), (2, 2), and (2, 0).

Find the flux of i - xy j +3z k out of the solid region bounded by the parabolic cylinder and the planes , and

Use Green's Theorem to compute the integral where C is the triangle formed by the lines y = -x + 1, x = 0 and y = 0, oriented clockwise.

Evaluate clockwise around the triangle with vertices (0, 0), (3, 0), and (3, 3).

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Exam 17: Vector Calculus

Define the curl of a vector field F.

If r = x i + y j + z k and k is a constant vector field in R3, then

Evaluate where S is the first-octant part of the sphere of radius a centred at the origin. (Hint: Even though S is not a closed surface, it is still easiest to use the Divergence Theorem because the integrand in the surface integral is zero on the coordinate planes.)

Let F = - i + j and let C be the boundary of circle + = 9 oriented counterclockwise. Use Green's Theorem to evaluate

In cylindrical coordinates, find f for f = sin( θ\thetaθ ) + .

In cylindrical coordinates, find . F for F = + rz cos( θ\thetaθ ) + rz sin( θ\thetaθ ) k.

Use Stokes's Theorem to evaluate the integral where C is the curve of intersection of the sphere and the plane oriented counterclockwise as seen from high on the z-axis.

Let F be a smooth vector field in 3-space satisfying the condition Find the flux of curl F upward through the part of the lying above the xy-plane.

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

Use Green's theorem in the plane to find the x-coordinate of the centroid of a regular plane region R (with areaA) enclosed by a positively oriented, piecewise smooth, simple closed curve C .

A vector field F is called solenoidal \textbf{ solenoidal } solenoidal in a domain D if

In spherical coordinates, find f for f = sin( θ\thetaθ ) cos( θ\thetaθ ).

If C is the positively oriented boundary of a plane region R having area 3 units and centroid at the point (12, 6), evaluate (i) (ii) dx + 3xy dy

Find all values of the nonzero constant real numbers a, b, and c so that the vector field F = a cos( x + 2y )cosh (c z) i + b cos ( x + 2y)cosh (c z) j + c sin( x + 2y)sinh(c z) k is both irrotational \textbf{ irrotational } irrotational and solenoidal \textbf{ solenoidal } solenoidal .

Using spherical polar coordinates, find × F for F = sin( θ\thetaθ ) + sin( θ\thetaθ ) + cos( θ\thetaθ ) .

Evaluate the integral ( ) - 2y) dx + (3x - ysin( )) dy counterclockwise around the triangle in the xy-plane having vertices (0, 0), (2, 2), and (2, 0).

Find the flux of i - xy j +3z k out of the solid region bounded by the parabolic cylinder and the planes , and

Use Green's Theorem to compute the integral where C is the triangle formed by the lines y = -x + 1, x = 0 and y = 0, oriented clockwise.

Evaluate clockwise around the triangle with vertices (0, 0), (3, 0), and (3, 3).

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 r = x i + y j + z k and k is a constant vector field in R³, then

In cylindrical coordinates, find $In cylindrical coordinates, find f for f = sin( \theta ) + .$ f for f = $In cylindrical coordinates, find f for f = sin( \theta ) + .$ sin( $\theta$ ) + $In cylindrical coordinates, find f for f = sin( \theta ) + .$ .

A vector field F is called $\textbf{ solenoidal }$ in a domain D if

In spherical coordinates, find $In spherical coordinates, find f for f = sin( \theta ) cos( \theta ).$ f for f = $In spherical coordinates, find f for f = sin( \theta ) cos( \theta ).$ sin( $\theta$ ) cos( $\theta$ ).