Exam 13: Vector Calculus

Consider the vector field $\mathbf { F } ( x , y ) = - \frac { y } { x ^ { 2 } + y ^ { 2 } } \mathbf { i } + \frac { x } { x ^ { 2 } + y ^ { 2 } } \mathbf { j }$ (a) Compute the curl of F.(b) Compute the divergence of F.

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(Short Answer)

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Question 1

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(a) 0
(b) 0

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ where $\mathbf { F } ( x , y , z ) = ( 2 x y + \ln z ) \mathbf { i } + \left( x ^ { 2 } + 1 + z \cos y z \right) \mathbf { j } + \left( \frac { x } { 2 } + y \cos y z \right) \mathbf { k }$ and C is the curve $x = \sqrt { 3 \cos t + 1 }$ , $y = 2 \sin ^ { 3 } t$ , $z = \sin t + \cos t$ , $0 \leq t \leq \frac { \pi } { 2 }$ .

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Question 2

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$f ( x , y , z ) = x ^ { 2 } y + y + \sin ( y z ) + \left. x \ln z \right| _ { 2,0,1 } ^ { 1,2,1 } = \sin 2 + 4$

Find the mass of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ whose density at each point is proportional to its distance to the $x y -$ plane.

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Question 3

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$16 \pi k$

Find the z-coordinate of the centroid of the upper hemisphere with uniform density whose equation is given by $z = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }$ .

(Short Answer)

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Question 4

Evaluate the line integral $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( y e ^ { x y } + \sin y \right) \mathbf { i } + \left( x e ^ { x y } + x \cos y \right) \mathbf { j }$ and C is the curve $x = \sqrt { 3 \cos t + 1 }$ , $y = 2 \sin ^ { 3 } t$ , $0 \leq t \leq \frac { \pi } { 2 }$ .

(Essay)

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Question 5

For what value of the constant $b$ is there a function $f ( x , y )$ such that $\nabla f = b x ^ { 2 } y ^ { 2 } \mathbf { i } + x ^ { 3 } y \mathbf { j }$ ?

(Multiple Choice)

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Question 6

Find the gradient vector field of $f ( x , y , z ) = x \ln ( y - z )$ .

(Essay)

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Question 7

For what value of the constant b is the vector field $\mathbf { F } = b x y \mathbf { i } + x ^ { 2 } y \mathbf { j }$ irrotational?

(Multiple Choice)

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Question 8

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathrm { F } ( x , y ) = \left( \arctan y , - \frac { x y ^ { 2 } } { 1 + y ^ { 2 } } \right)$ and the curve C is a triangle from $( 0,0 )$ , to $( 2,2 )$ , to $( 1,4 )$ , then back to $( 0,0 )$ .

(Multiple Choice)

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Question 9

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathrm { F } ( x , y ) = \left( \arctan y , - \frac { x y ^ { 2 } } { 1 + y ^ { 2 } } \right)$ and the curve C is given by C : $x = 2 \cos t , y = 2 \sin t , 0 \leq t \leq 2 \pi$ .

(Multiple Choice)

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Question 10

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } ( x , y , z ) = - y \mathbf { i } + x \mathbf { j } + 2 z \mathbf { k }$ and S is the upper half of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25$ , with upward orientation.

(Short Answer)

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Question 11

Evaluate $\int _ { c } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( \frac { - y } { x ^ { 2 } + y ^ { 2 } } , \frac { x } { x ^ { 2 } + y ^ { 2 } } \right)$ and the curve $C$ is given by $C : x = 5 \sin t , y = 5 \cos t , 0 \leq t \leq 4 \pi$ .

(Multiple Choice)

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Question 12

Use Stokes' Theorem to evaluate $\int _ { C } ( 3 z - 2 y ) d x + ( 4 x - 2 y ) d y + ( z + 2 y ) d z$ where C is the circle $x = 3 \cos t , y = 3 \sin t , z = 2,0 \leq t \leq 2 \pi$ .

(Multiple Choice)

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Question 13

Determine whether $\mathrm { F } ( x , y , z ) = \left\langle x ^ { 2 } \cos \left( x ^ { 3 } \right) , y e ^ { y ^ { 2 } } , z \sin z \right\rangle$ is conservative and if so, find a potential function.

(Essay)

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Question 14

Show that $\int _ { C } 2 x d x + 2 y d y + 2 z d z = \mathbf { a } \cdot \mathbf { a }$ , where $\mathbf { a } = a _ { 1 } \mathbf { i } + a _ { 2 } \mathbf { j } + a _ { 3 } \mathbf { k }$ , and C is any path from $( 0,0,0 )$ to $\left( a _ { 1 } , a _ { 2 } , a _ { 3 } \right)$ .

(Essay)

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Question 15

Let $\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - y } , \frac { z } { y - x } , \ln ( x - y ) \right\rangle$ , find $\nabla \cdot ( \nabla \times \mathbf { F } )$ at the point $( 1,3,1 )$ .

(Multiple Choice)

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Question 16

Find a formula for the vector field graphed below. (There are many possible answers.)

(Essay)

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Question 17

Evaluate the line integral $\int _ { C } x d x + y d y$ , where $C$ is the curve $x = t , y = t ^ { 3 } , 1 \leq t \leq 2$ .

(Multiple Choice)

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Question 18

Evaluate the surface integral $\iint _ { S } \left( y ^ { 2 } + z ^ { 2 } \right) d S$ , where S is the part of the paraboloid $x = 4 - y ^ { 2 } - z ^ { 2 }$ that lies in front of the plane $x = 0$ .

(Short Answer)

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Question 19

Evaluate $\int _ { C } \frac { 1 } { 1 + x } d s$ where the curve $C$ is given by $x = t , y = \frac { 2 } { 3 } t ^ { \frac { 3 } { 2 } } , 0 \leq t \leq 3$ .

(Multiple Choice)

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Question 20

Consider the vector field $\mathbf { F } ( x , y ) = - \frac { y } { x ^ { 2 } + y ^ { 2 } } \mathbf { i } + \frac { x } { x ^ { 2 } + y ^ { 2 } } \mathbf { j }$ (a) Compute the curl of F.(b) Compute the divergence of F.

Find the mass of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ whose density at each point is proportional to its distance to the $x y -$ plane.

Find the z-coordinate of the centroid of the upper hemisphere with uniform density whose equation is given by $z = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }$ .

For what value of the constant $b$ is there a function $f ( x , y )$ such that $\nabla f = b x ^ { 2 } y ^ { 2 } \mathbf { i } + x ^ { 3 } y \mathbf { j }$ ?

Find the gradient vector field of $f ( x , y , z ) = x \ln ( y - z )$ .

For what value of the constant b is the vector field $\mathbf { F } = b x y \mathbf { i } + x ^ { 2 } y \mathbf { j }$ irrotational?

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathrm { F } ( x , y ) = \left( \arctan y , - \frac { x y ^ { 2 } } { 1 + y ^ { 2 } } \right)$ and the curve C is a triangle from $( 0,0 )$ , to $( 2,2 )$ , to $( 1,4 )$ , then back to $( 0,0 )$ .

Evaluate $\int _ { C } \mathbf { F } d \mathbf { r }$ where $\mathrm { F } ( x , y ) = \left( \arctan y , - \frac { x y ^ { 2 } } { 1 + y ^ { 2 } } \right)$ and the curve C is given by C : $x = 2 \cos t , y = 2 \sin t , 0 \leq t \leq 2 \pi$ .

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } ( x , y , z ) = - y \mathbf { i } + x \mathbf { j } + 2 z \mathbf { k }$ and S is the upper half of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 25$ , with upward orientation.

Evaluate $\int _ { c } \mathbf { F } d \mathbf { r }$ where $\mathbf { F } ( x , y ) = \left( \frac { - y } { x ^ { 2 } + y ^ { 2 } } , \frac { x } { x ^ { 2 } + y ^ { 2 } } \right)$ and the curve $C$ is given by $C : x = 5 \sin t , y = 5 \cos t , 0 \leq t \leq 4 \pi$ .

Use Stokes' Theorem to evaluate $\int _ { C } ( 3 z - 2 y ) d x + ( 4 x - 2 y ) d y + ( z + 2 y ) d z$ where C is the circle $x = 3 \cos t , y = 3 \sin t , z = 2,0 \leq t \leq 2 \pi$ .

Determine whether $\mathrm { F } ( x , y , z ) = \left\langle x ^ { 2 } \cos \left( x ^ { 3 } \right) , y e ^ { y ^ { 2 } } , z \sin z \right\rangle$ is conservative and if so, find a potential function.

Show that $\int _ { C } 2 x d x + 2 y d y + 2 z d z = \mathbf { a } \cdot \mathbf { a }$ , where $\mathbf { a } = a _ { 1 } \mathbf { i } + a _ { 2 } \mathbf { j } + a _ { 3 } \mathbf { k }$ , and C is any path from $( 0,0,0 )$ to $\left( a _ { 1 } , a _ { 2 } , a _ { 3 } \right)$ .

Let $\mathbf { F } ( x , y , z ) = \left\langle \frac { z } { x - y } , \frac { z } { y - x } , \ln ( x - y ) \right\rangle$ , find $\nabla \cdot ( \nabla \times \mathbf { F } )$ at the point $( 1,3,1 )$ .

Find a formula for the vector field graphed below. (There are many possible answers.)

Evaluate the line integral $\int _ { C } x d x + y d y$ , where $C$ is the curve $x = t , y = t ^ { 3 } , 1 \leq t \leq 2$ .

Evaluate the surface integral $\iint _ { S } \left( y ^ { 2 } + z ^ { 2 } \right) d S$ , where S is the part of the paraboloid $x = 4 - y ^ { 2 } - z ^ { 2 }$ that lies in front of the plane $x = 0$ .

Evaluate $\int _ { C } \frac { 1 } { 1 + x } d s$ where the curve $C$ is given by $x = t , y = \frac { 2 } { 3 } t ^ { \frac { 3 } { 2 } } , 0 \leq t \leq 3$ .

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Exam 13: Vector Calculus

Consider the vector field F(x,y)=−yx2+y2i+xx2+y2j\mathbf { F } ( x , y ) = - \frac { y } { x ^ { 2 } + y ^ { 2 } } \mathbf { i } + \frac { x } { x ^ { 2 } + y ^ { 2 } } \mathbf { j }F(x,y)=−x2+y2y​i+x2+y2x​j (a) Compute the curl of F.(b) Compute the divergence of F.

Find the mass of the sphere x2+y2+z2=4x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4x2+y2+z2=4 whose density at each point is proportional to its distance to the xy−x y -xy− plane.

Find the z-coordinate of the centroid of the upper hemisphere with uniform density whose equation is given by z=25−x2−y2z = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }z=25−x2−y2​ .

For what value of the constant bbb is there a function f(x,y)f ( x , y )f(x,y) such that ∇f=bx2y2i+x3yj\nabla f = b x ^ { 2 } y ^ { 2 } \mathbf { i } + x ^ { 3 } y \mathbf { j }∇f=bx2y2i+x3yj ?

Find the gradient vector field of f(x,y,z)=xln⁡(y−z)f ( x , y , z ) = x \ln ( y - z )f(x,y,z)=xln(y−z) .

For what value of the constant b is the vector field F=bxyi+x2yj\mathbf { F } = b x y \mathbf { i } + x ^ { 2 } y \mathbf { j }F=bxyi+x2yj irrotational?

Determine whether F(x,y,z)=⟨x2cos⁡(x3),yey2,zsin⁡z⟩\mathrm { F } ( x , y , z ) = \left\langle x ^ { 2 } \cos \left( x ^ { 3 } \right) , y e ^ { y ^ { 2 } } , z \sin z \right\rangleF(x,y,z)=⟨x2cos(x3),yey2,zsinz⟩ is conservative and if so, find a potential function.

Find a formula for the vector field graphed below. (There are many possible answers.)

Evaluate the line integral ∫Cxdx+ydy\int _ { C } x d x + y d y∫C​xdx+ydy , where CCC is the curve x=t,y=t3,1≤t≤2x = t , y = t ^ { 3 } , 1 \leq t \leq 2x=t,y=t3,1≤t≤2 .

Evaluate the surface integral ∬S(y2+z2)dS\iint _ { S } \left( y ^ { 2 } + z ^ { 2 } \right) d S∬S​(y2+z2)dS , where S is the part of the paraboloid x=4−y2−z2x = 4 - y ^ { 2 } - z ^ { 2 }x=4−y2−z2 that lies in front of the plane x=0x = 0x=0 .

Evaluate ∫C11+xds\int _ { C } \frac { 1 } { 1 + x } d s∫C​1+x1​ds where the curve CCC is given by x=t,y=23t32,0≤t≤3x = t , y = \frac { 2 } { 3 } t ^ { \frac { 3 } { 2 } } , 0 \leq t \leq 3x=t,y=32​t23​,0≤t≤3 .

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Integrals

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Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

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Consider the vector field $\mathbf { F } ( x , y ) = - \frac { y } { x ^ { 2 } + y ^ { 2 } } \mathbf { i } + \frac { x } { x ^ { 2 } + y ^ { 2 } } \mathbf { j }$ (a) Compute the curl of F.(b) Compute the divergence of F.

Find the mass of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4$ whose density at each point is proportional to its distance to the $x y -$ plane.

Find the z-coordinate of the centroid of the upper hemisphere with uniform density whose equation is given by $z = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }$ .

For what value of the constant $b$ is there a function $f ( x , y )$ such that $\nabla f = b x ^ { 2 } y ^ { 2 } \mathbf { i } + x ^ { 3 } y \mathbf { j }$ ?

Find the gradient vector field of $f ( x , y , z ) = x \ln ( y - z )$ .

For what value of the constant b is the vector field $\mathbf { F } = b x y \mathbf { i } + x ^ { 2 } y \mathbf { j }$ irrotational?

Determine whether $\mathrm { F } ( x , y , z ) = \left\langle x ^ { 2 } \cos \left( x ^ { 3 } \right) , y e ^ { y ^ { 2 } } , z \sin z \right\rangle$ is conservative and if so, find a potential function.

Evaluate the line integral $\int _ { C } x d x + y d y$ , where $C$ is the curve $x = t , y = t ^ { 3 } , 1 \leq t \leq 2$ .

Evaluate the surface integral $\iint _ { S } \left( y ^ { 2 } + z ^ { 2 } \right) d S$ , where S is the part of the paraboloid $x = 4 - y ^ { 2 } - z ^ { 2 }$ that lies in front of the plane $x = 0$ .

Evaluate $\int _ { C } \frac { 1 } { 1 + x } d s$ where the curve $C$ is given by $x = t , y = \frac { 2 } { 3 } t ^ { \frac { 3 } { 2 } } , 0 \leq t \leq 3$ .