Exam 15: Vector Fields

Evaluate $\iint_{S} f ( x , y ) d S$ , where $f ( x , y ) = y + 3$ $S : \mathbf { r } ( u , v ) = u \mathbf { i } + v \mathbf { j } + \frac { v } { 10 } \mathbf { k } , 0 \leq \mathrm { u } \leq 1,0 \leq v \leq 10$

$\text { Evaluate } \iint _ { S } ( x , y , z ) d S \text {, where } f ( x , y , z ) = \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } \text { and } S \text { is given by }$ $z = \sqrt { x ^ { 2 } + y ^ { 2 } } , x ^ { 2 } + y ^ { 2 } \leq 121$

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. $\int \frac { 2 x } { C \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } } d x + \frac { 2 y } { \left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } } d y$ $C$ : circle $( x - 2 ) ^ { 2 } + ( y - 10 ) ^ { 2 } = 9$ clockwise from $( 5,10 )$ to $( - 1,10 )$

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function $\mathbf { r } ( u , v ) = 9 \cos v \cos u \mathbf { i } + 9 \cos v \sin u \mathbf { j } + 6 \sin v \mathbf { k }$ .

Calculate the line integral along $\int \mathbf { F } \cdot d \mathbf { r } \text { for } \mathbf { F } ( x , y ) = 3 y \mathbf { i } + 5 x ^ { 2 } \mathbf { j } \text { and } C \text { is any path }$ starting at the point $( 3,5 )$ and ending at $( 8,0 )$ .

Evaluate the integral $\int_{c}(4 x-y) d x+(x+6 y) d y$ along the path $C$ , defined as $y$ -axis from $y = 0$ to $y = 4$ .

Match the following vector-valued function with its graph. $\mathbf { r } ( u , v ) = u \cos v \mathbf { i } + u \sin v \mathbf { j } + u \mathbf { k }$

Evaluate the integral along the path C defined as $y = 9 - x ^ { 2 } \text { from } ( 0,3 ) \text { to } ( 3,0 ) \text {. }$

Find the gradient vector for the scalar function. (That is, find the conservative vector field for the potential function.) $f ( x , y ) = \sin 4 x \cos 6 y$

Use Green's Theorem to evaluate the integral $\int _ { C } ( y - x ) d x + ( 2 y - y ) d y \text { for the }$ path $C$ defined as $x = 6 \cos \theta , y = 7 \sin \theta$ .

Evaluate $\iint _ { S } ( x - 5 y + z ) d S$ , where $S$ is $z = 4 , x ^ { 2 } + y ^ { 2 } \leq 4$ .

Green's Theorem to evaluate the integral $\int _ { C } \left( x ^ { 2 } - y ^ { 2 } \right) d x + 10 x y d y$ for the path $C : x ^ { 2 } + y ^ { 2 } = 64$ .

Find the value of the line integral $\int_{c}(4 x-3 y+1) d x-(3 x+y-5) d y$ . (Hint: If $\mathbf { F } ( x , y ) = ( 4 x - 3 y + 1 ) \mathbf { i } - ( 3 x + y - 5 ) \mathbf { j }$ is conservative, the integration may be easier on an alternate path.)

Match the following vector -valued function with its graph. $\mathbf { r } ( u , v ) = u \mathbf { i } + v \mathbf { j } + u v \mathbf { k }$

Find a vector-valued function whose graph is the plane $x + y + z = 4 \text {. }$

Use Stokes's Theorem to evaluate $\int _ { C } \mathbf { F } \cdot d r$ where $\mathbf { F } ( x , y , z ) = y z \mathbf { i } + ( 2 - 3 y ) \mathbf { j } + \left( x ^ { 2 } + y ^ { 2 } \right) \mathbf { k } , x ^ { 2 } + y ^ { 2 } \leq 400$ and $S$ is the first-octant portion of $x ^ { 2 } + y ^ { 2 } \leq 400$ over $x ^ { 2 } + y ^ { 2 } = 400$ . Use a computer algebra system to verify your result.

Use a computer algebra system and the result "The centroid of the region having area $A$ bounded by the simple closed path $C$ is $\bar { x } = \frac { 1 } { 2 A } \int _ { C } x ^ { 2 } d y , \bar { y } = - \frac { 1 } { 2 A } \int _ { C } y ^ { 2 } d x ^ { \prime \prime }$ to find the centroid of the region bounded by the graphs of $y = \sqrt { 196 - x ^ { 2 } }$ and $y = 0$ .

Find the maximum value of $\int _ { C _ { 1 } } y ^ { 3 } d x + \left( 225 x - x ^ { 3 } \right) d y , \text { where } C \text { is any closed curve }$ in the xy-plane, oriented counterclockwise.

$\text { Use Stokes's Theorem to evaluate } \int _ { C } \mathbf { F } \cdot d r \text { where } \mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i } + z ^ { 2 } \mathbf { j } - x y z \mathbf { k }$ and $S$ is $z = \sqrt { 25 - x ^ { 2 } - y ^ { 2 } }$ .Use a computer algebra system to verify your result.

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field $\mathbf { F } ( x , y , z ) = 5 \mathbf { i } + 6 \mathbf { j } - 7 \mathbf { k }$ . Find $\iint _ { S } ( \operatorname { curl } \mathbf { F } ) \cdot \mathbf { N } d S$ , where $S$ is the upper surface of the cylindrical container.