Question 1

Find   , where F(x, y, z) = x   y i +   ln x j - z   y k, S is the sphere of radius 3 centred at the origin, and   is the unit outward normal field on S.

Accepted Answer

A)  24 $\pi$ 
B)  12 $\pi$ 
C)  36 $\pi$ 
D)  72 $\pi$ 
E)  54 $\pi$ 
A)  24 $\pi$ 
B)  12 $\pi$ 
C)  36 $\pi$ 
D)  72 $\pi$ 
E)  54 $\pi$

Question 2

Calculate the surface integral   where G = (x + y) i + (y + z) j + (z + x) k and S is the sphere   with outward normal.

Accepted Answer

A)  32 $\pi$ 
B)  16 $\pi$ 
C)  8 $\pi$ 
D)  64 $\pi$ 
E)  256 $\pi$ 
A)  32 $\pi$ 
B)  16 $\pi$ 
C)  8 $\pi$ 
D)  64 $\pi$ 
E)  256 $\pi$

Question 3

Use Green's Theorem to evaluate the line integral   counterclockwise around the square with vertices (0, 3), (3, 0), (-3, 0), and (0, -3).

Accepted Answer

A)  18 
B)  180 
C)  -36 
D)  0 
E)  36 
A)  18 
B)  180 
C)  -36 
D)  0 
E)  36

Question 4

Let F = -y i + x j + z k. Use Stokes's Theorem to find the flux of $\textbf{          curl F}$ upward through the paraboloid   where u    [0, 1] and v    [0, 2 $\pi$].

Accepted Answer

A)  2 $\pi$ 
B)  3 $\pi$ 
C)  4 $\pi$ 
D)  6 $\pi$ 
E)   $\pi$ 
A)  2 $\pi$ 
B)  3 $\pi$ 
C)  4 $\pi$ 
D)  6 $\pi$ 
E)   $\pi$

Question 5

Find grad f(1, 0, -1) if f(x, y, z) = xy + yz.

Accepted Answer

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

Question 6

Find the acute angle (to the nearest degree) between the normals of the paraboloid z = x2 + y2 - 6 and the sphere x2 + y2 + z2 = 26 at the point (-3, 1, 4) on both surfaces.

Accepted Answer

Question 7

Find  .F if F (x, y, z) =   xy   ,   yz, -xyz   .

Accepted Answer

A)  2y   +   yz - xy 
B)  y   + 2xyz - xy 
C)  y   +   z - xy 
D)  y   +   z + 2 xy 
E)  2y   +   z + xy 
A)  2y   +   yz - xy 
B)  y   + 2xyz - xy 
C)  y   +   z - xy 
D)  y   +   z + 2 xy 
E)  2y   +   z + xy

Question 8

The curl of a vector field F is defined by

Accepted Answer

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

Question 9

Evaluate the line integral   where C is the circle given by the parametric equations   for

Accepted Answer

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

Question 10

Find the outward flux of F = ln(   +   ) i -   j + z   k across the boundary of the region

Accepted Answer

A)  4    $\pi$ - 3 $\pi$ln 2 + 2 $\pi$ 
B)  4    $\pi$ - 3 $\pi$ln 2 - 2 $\pi$ 
C)  4    $\pi$ + 3 $\pi$ln 2 - 2 $\pi$ 
D)  4    $\pi$ + 3 $\pi$ln 2 + 2 $\pi$ 
E)  4    $\pi$ +    $\pi$ ln 2 - 2 $\pi$ 
A)  4    $\pi$ - 3 $\pi$ln 2 + 2 $\pi$ 
B)  4    $\pi$ - 3 $\pi$ln 2 - 2 $\pi$ 
C)  4    $\pi$ + 3 $\pi$ln 2 - 2 $\pi$ 
D)  4    $\pi$ + 3 $\pi$ln 2 + 2 $\pi$ 
E)  4    $\pi$ +    $\pi$ ln 2 - 2 $\pi$

Question 11

Evaluate   F = x   y i + xz j + z   y k and S is the sphere of radius 3 with centre at the origin and unit outward normal field   .

Accepted Answer

A)  32$\pi$ 
B)  34$\pi$ 
C)  36$\pi$ 
D)  38$\pi$ 
E)  72$\pi$ 
A)  32$\pi$ 
B)  34$\pi$ 
C)  36$\pi$ 
D)  38$\pi$ 
E)  72$\pi$

Question 12

If f(x, y, z) =   z + cos(z   ), find   f.

Accepted Answer

A)  2zx sin(z   ) i + 2yz j + (   +   sin(z   )) k 
B)  2x sin(z   ) i + 2yz j + (   -   sin(z   )) k 
C)  -2x sin(z   ) i + 2yz j + (   -   sin(z   )) k 
D)  -2zx sin(z   ) i + 2yz j + (   -   sin(z   )) k 
E)  -2zx cos(z   ) i + 2yz j + (   -   cos(z   )) k 
A)  2zx sin(z   ) i + 2yz j + (   +   sin(z   )) k 
B)  2x sin(z   ) i + 2yz j + (   -   sin(z   )) k 
C)  -2x sin(z   ) i + 2yz j + (   -   sin(z   )) k 
D)  -2zx sin(z   ) i + 2yz j + (   -   sin(z   )) k 
E)  -2zx cos(z   ) i + 2yz j + (   -   cos(z   )) k

Question 13

Use Green's theorem in the plane to show that the area A of a regular plane region R enclosed by a positively oriented, piecewise smooth, simple closed curve C is given by A =     dx + x dy).

Accepted Answer

The answer of Use Green's theorem in the plane to...

Question 14

If the vector field H = f(r) r, r $\neq$ 0 is solenoidal, find an expression for f(r).

Accepted Answer

A)  f(r) = c   , where c is an arbitrary constant 
B)  f(r) = c   , where c is an arbitrary constant 
C)  f(r) = c   , where c is an arbitrary constant 
D)  f(r) = c   , where c is an arbitrary constant 
E)  f(r) = c   , where c is an arbitrary constant 
A)  f(r) = c   , where c is an arbitrary constant 
B)  f(r) = c   , where c is an arbitrary constant 
C)  f(r) = c   , where c is an arbitrary constant 
D)  f(r) = c   , where c is an arbitrary constant 
E)  f(r) = c   , where c is an arbitrary constant

Question 15

Let F = f(x, y, z) i + g(x, y, z) j + h(x, y, z) k be a vector field in 3-space whose components f, g, and h have continuous second partial derivatives. Calculate div curl F in terms of those partials.

Accepted Answer

A)    +   +   
B)  -   -   -   
C)    +   +   
D)  0 
E)    -   +   
A)    +   +   
B)  -   -   -   
C)    +   +   
D)  0 
E)    -   +

Question 16

Let C be a cone whose base is an arbitrarily shaped region in the plane z = h > 0 having area A, and whose vertex is at the origin. By calculating the flux of   out of C through its entire surface both directly and by using the Divergence Theorem, find the volume of C.

Accepted Answer

A)    Ah 
B)    Ah 
C)    Ah 
D)    Ah 
E)  3 Ah 
A)    Ah 
B)    Ah 
C)    Ah 
D)    Ah 
E)  3 Ah

Question 17

The divergence of a vector field F is defined by

Accepted Answer

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

Question 18

Compute the divergence for the vector field F = (xy + xz) i + (yz + yx) j + (zx + zy) k.

Accepted Answer

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

Question 19

Show that there does not exist a twice continuously differentiable vector field G such that $\textbf{      curl G   }$  = x i + y j + z k.

Accepted Answer

Assume on the contrary that su

Question 20

Let  = arctan(x) - arctan(z) and   =   . Find a simplified expression for $\textbf{         grad}$(  ) × $\textbf{         grad}$(  ) .

Accepted Answer

A)    j 
B)  0 (zero vector field) 
C)    j 
D)  0 (zero scalar field) 
E)  -   j 
A)    j 
B)  0 (zero vector field) 
C)    j 
D)  0 (zero scalar field) 
E)  -   j

Find , where F(x, y, z) = x y i + ln x j - z y k, S is the sphere of radius 3 centred at the origin, and is the unit outward normal field on S.

Calculate the surface integral where G = (x + y) i + (y + z) j + (z + x) k and S is the sphere with outward normal.

Use Green's Theorem to evaluate the line integral counterclockwise around the square with vertices (0, 3), (3, 0), (-3, 0), and (0, -3).

Find grad f(1, 0, -1) if f(x, y, z) = xy + yz.

Find the acute angle (to the nearest degree) between the normals of the paraboloid z = x² + y² - 6 and the sphere x² + y² + z² = 26 at the point (-3, 1, 4) on both surfaces.

Find .F if F (x, y, z) = xy , yz, -xyz .

The curl of a vector field F is defined by

Evaluate the line integral where C is the circle given by the parametric equations for

Find the outward flux of F = ln( + ) i - j + z k across the boundary of the region

Evaluate F = x y i + xz j + z y k and S is the sphere of radius 3 with centre at the origin and unit outward normal field .

If f(x, y, z) = z + cos(z ), find f.

Use Green's theorem in the plane to show that the area A of a regular plane region R enclosed by a positively oriented, piecewise smooth, simple closed curve C is given by A = dx + x dy).

If the vector field H = f(r) r, r $\neq$ 0 is solenoidal, find an expression for f(r).

Let F = f(x, y, z) i + g(x, y, z) j + h(x, y, z) k be a vector field in 3-space whose components f, g, and h have continuous second partial derivatives. Calculate div curl F in terms of those partials.

Let C be a cone whose base is an arbitrarily shaped region in the plane z = h > 0 having area A, and whose vertex is at the origin. By calculating the flux of out of C through its entire surface both directly and by using the Divergence Theorem, find the volume of C.

The divergence of a vector field F is defined by

Compute the divergence for the vector field F = (xy + xz) i + (yz + yx) j + (zx + zy) k.

Show that there does not exist a twice continuously differentiable vector field G such that $\textbf{ curl G }$ = x i + y j + z k.

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Limits and Continuity

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Exam 17: Vector Calculus

Find , where F(x, y, z) = x y i + ln x j - z y k, S is the sphere of radius 3 centred at the origin, and is the unit outward normal field on S.

Calculate the surface integral where G = (x + y) i + (y + z) j + (z + x) k and S is the sphere with outward normal.

Use Green's Theorem to evaluate the line integral counterclockwise around the square with vertices (0, 3), (3, 0), (-3, 0), and (0, -3).

Let F = -y i + x j + z k. Use Stokes's Theorem to find the flux of curl F\textbf{ curl F} curl F upward through the paraboloid where u [0, 1] and v 11ee7b17_3372_5854_ae82_d19d2ea0c252_TB9661_11 [0, 2 π\piπ ].

Find grad f(1, 0, -1) if f(x, y, z) = xy + yz.

Find the acute angle (to the nearest degree) between the normals of the paraboloid z = x2 + y2 - 6 and the sphere x2 + y2 + z2 = 26 at the point (-3, 1, 4) on both surfaces.

Find .F if F (x, y, z) = xy , yz, -xyz .

The curl of a vector field F is defined by

Evaluate the line integral where C is the circle given by the parametric equations for

Find the outward flux of F = ln( + ) i - j + z k across the boundary of the region

Evaluate F = x y i + xz j + z y k and S is the sphere of radius 3 with centre at the origin and unit outward normal field .

If f(x, y, z) = z + cos(z ), find f.

Use Green's theorem in the plane to show that the area A of a regular plane region R enclosed by a positively oriented, piecewise smooth, simple closed curve C is given by A = dx + x dy).

If the vector field H = f(r) r, r ≠\neq= 0 is solenoidal, find an expression for f(r).

Let F = f(x, y, z) i + g(x, y, z) j + h(x, y, z) k be a vector field in 3-space whose components f, g, and h have continuous second partial derivatives. Calculate div curl F in terms of those partials.

Let C be a cone whose base is an arbitrarily shaped region in the plane z = h > 0 having area A, and whose vertex is at the origin. By calculating the flux of out of C through its entire surface both directly and by using the Divergence Theorem, find the volume of C.

The divergence of a vector field F is defined by

Compute the divergence for the vector field F = (xy + xz) i + (yz + yx) j + (zx + zy) k.

Show that there does not exist a twice continuously differentiable vector field G such that curl G \textbf{ curl G } curl G = x i + y j + z k.

Let = arctan(x) - arctan(z) and = . Find a simplified expression for grad\textbf{ grad} grad (11ee7bac_4a3a_9aaa_ae82_759e3f104991_TB9661_11 ) × grad\textbf{ grad} grad (11ee7bac_77f1_7c2b_ae82_019616e1397c_TB9661_11 ) .

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

Find the acute angle (to the nearest degree) between the normals of the paraboloid z = x² + y² - 6 and the sphere x² + y² + z² = 26 at the point (-3, 1, 4) on both surfaces.

If the vector field H = f(r) r, r $\neq$ 0 is solenoidal, find an expression for f(r).

Show that there does not exist a twice continuously differentiable vector field G such that $\textbf{ curl G }$ = x i + y j + z k.