Question 1

For r = x i + y j + z k, evaluate and simplify   .   .

Accepted Answer

A)    
B)    
C)    
D)  |r| 
E)  0 
A)    
B)    
C)    
D)  |r| 
E)  0

Question 2

Let C be a non-self-intersecting closed curve in the xy-plane oriented counterclockwise and bounding a region R having area A and centroid   . In terms of these quantities, evaluate the line integral   .

Accepted Answer

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

Question 3

Find the flux of F = x i +   j +   k out of the cube bounded by the coordinate planes and the planes   and

Accepted Answer

A)  0 
B)  1 
C)    
D)  3 
E)    
A)  0 
B)  1 
C)    
D)  3 
E)

Question 4

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

Accepted Answer

A)     sin(     ) +   cos(    ) -   cot(1   ) 
B)      sin(   ) +   cos(    ) -    cot(   ) 
C)     sin(    ) +   cos(   ) -    cot(  ) 
D)     sin(   ) +   cos(   ) -    cot(   ) 
E)  4   sin(   ) -   cos(   ) -    $\theta$ cot(   ) 
A)     sin(     ) +   cos(    ) -   cot(1   ) 
B)      sin(   ) +   cos(    ) -    cot(   ) 
C)     sin(    ) +   cos(   ) -    cot(  ) 
D)     sin(   ) +   cos(   ) -    cot(   ) 
E)  4   sin(   ) -   cos(   ) -    $\theta$ cot(   )

Question 5

Given F = 4y i + x j + 2z k, find   over the hemisphere   with outward normal   .

Accepted Answer

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

Question 6

If r = x i + y j + z k and r = |r|, evaluate and simplify div   .

Accepted Answer

A)  0 
B)    
C)    
D)    
E)    
A)  0 
B)    
C)    
D)    
E)

Question 7

Evaluate the surface integral   where   is the unit inner normal to the surface S of the region lying between the two paraboloids

Accepted Answer

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

Question 8

Compute curl F for F = (x - z) i + (y - x) j + (z - y) k.

Accepted Answer

A)  - i + j - k 
B)  i + j + k 
C)  - i + j + k 
D)  i - j 
E)  0 
A)  - i + j - k 
B)  i + j + k 
C)  - i + j + k 
D)  i - j 
E)  0

Question 9

Compute the divergence and the curl of the vector field r = x i + y j + z k.

Accepted Answer

A)    . r = 2,   × r = 0 
B)   . r = 3,  × r = 0 
C)    . r = 3,   × r = r 
D)    . r = 1,   × r = 0 
E)    . r = 2,   × r = r 
A)    . r = 2,   × r = 0 
B)   . r = 3,  × r = 0 
C)    . r = 3,   × r = r 
D)    . r = 1,   × r = 0 
E)    . r = 2,   × r = r

Question 10

Calculate the divergence of the vector field F =   y i +   x j + xyz k.

Accepted Answer

A)  5xy 
B)  4xy + yz 
C)  6xy 
D)  2xy + 2yz + xz 
E)  4xy + xz 
A)  5xy 
B)  4xy + yz 
C)  6xy 
D)  2xy + 2yz + xz 
E)  4xy + xz

Question 11

Let w be a function of x, y, and z having continuous second partial derivatives.Calculate curl grad w in terms of those partials.

Accepted Answer

A)    +   +   
B)    i +   j +   k 
C)    i +   j +   k 
D)  0 
E)    +   +   
A)    +   +   
B)    i +   j +   k 
C)    i +   j +   k 
D)  0 
E)    +   +

Question 12

Find the flux of r = x i + y j + z k out of the cone with base   +   $\le$ 16, z = 0, and vertex at (0, 0, 3).

Accepted Answer

A)  46$\pi$ 
B)  48$\pi$ 
C)  50$\pi$ 
D)  52$\pi$ 
E)  16$\pi$ 
A)  46$\pi$ 
B)  48$\pi$ 
C)  50$\pi$ 
D)  52$\pi$ 
E)  16$\pi$

For r = x i + y j + z k, evaluate and simplify . .

Let C be a non-self-intersecting closed curve in the xy-plane oriented counterclockwise and bounding a region R having area A and centroid . In terms of these quantities, evaluate the line integral .

Find the flux of F = x i + j + k out of the cube bounded by the coordinate planes and the planes and

Given F = 4y i + x j + 2z k, find over the hemisphere with outward normal .

If r = x i + y j + z k and r = |r|, evaluate and simplify div .

Evaluate the surface integral where is the unit inner normal to the surface S of the region lying between the two paraboloids

Compute curl F for F = (x - z) i + (y - x) j + (z - y) k.

Compute the divergence and the curl of the vector field r = x i + y j + z k.

Calculate the divergence of the vector field F = y i + x j + xyz k.

Let w be a function of x, y, and z having continuous second partial derivatives.Calculate curl grad w in terms of those partials.

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Exam 17: Vector Calculus

For r = x i + y j + z k, evaluate and simplify . .

Let C be a non-self-intersecting closed curve in the xy-plane oriented counterclockwise and bounding a region R having area A and centroid . In terms of these quantities, evaluate the line integral .

Find the flux of F = x i + j + k out of the cube bounded by the coordinate planes and the planes and

Using spherical polar coordinates, find . F for F = sin( θ\thetaθ ) + sin( ) + cos( 11ee7bb1_1f6d_7966_ae82_e9760b18acae_TB9661_11 ) .

Given F = 4y i + x j + 2z k, find over the hemisphere with outward normal .

If r = x i + y j + z k and r = |r|, evaluate and simplify div .

Evaluate the surface integral where is the unit inner normal to the surface S of the region lying between the two paraboloids

Compute curl F for F = (x - z) i + (y - x) j + (z - y) k.

Compute the divergence and the curl of the vector field r = x i + y j + z k.

Calculate the divergence of the vector field F = y i + x j + xyz k.

Let w be a function of x, y, and z having continuous second partial derivatives.Calculate curl grad w in terms of those partials.

Find the flux of r = x i + y j + z k out of the cone with base + ≤\le≤ 16, z = 0, and vertex at (0, 0, 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