Exam 15: Vector Fields

Find the area of the surface over the given region. Use a computer algebra system to verify your results. The sphere, $\overrightarrow { \mathbf { r } } ( u , v ) = 10 \sin u \cos v \hat { \mathbf { i } } + 10 \sin u \sin v \hat { \mathbf { j } } + 10 \cos u \hat { \mathbf { k } } , \quad 0 \leq u \leq \pi , 0 \leq v \leq 2 \pi$

Find the flux of $\vec { F }$ over the closed surface let $\vec{N}$ be the outward unit normal vector of the surface). $\overrightarrow { \mathbf { F } } ( x , y , z ) = 49 x y \hat { \mathbf { i } } + z ^ { 2 } \tilde { \mathbf { j } } + y z \tilde { \mathbf { k } }$ $S :$ cube bounded by $x = 0 , x = 2 , y = 0 , y = 2 , z = 0 , z = 2$

Find the area of the surface of revolution $\mathbf { r } ( u , v ) = 11 \sin u \cos v \mathbf { i } + 11 u \mathbf { j } + 11 \sin u \sin v \mathbf { k }$ , where $0 \leq u \leq \pi$ and $0 \leq v \leq 2 \pi$ .

The motion of a liquid in a cylindrical container of radius 1 is described by the velocity field $\mathbf { F } ( x , y , z ) = - 8 z \mathbf { i } + 2 y \mathbf { k }$ . Find $\iint _ { S } ( \operatorname { curl } \mathbf { F } ) \cdot \mathbf { N } d S$ , where $S$ is the upper surface of the cylindrical container.

A stone weighing 2 pounds is attached to the end of a four-foot string and is whirled horizontally with one end held fixed. It makes 1 revolution per second. Find the work done by the Force F that keeps the stone moving in a circular path. [Hint: Use Force = (mass)(centripetal Acceleration).] Round your answer to two decimal places, if required.

Verify Green's Theorem by evaluating both integrals $\int _ { c } 14 y ^ { 2 } d x + 14 x ^ { 2 } d y = \int _ { R } \int \left( \frac { \partial N } { \partial x } - \frac { \partial M } { \partial y } \right) \partial A$ for the path $C$ defined as the boundary of the region lying between the graphs of $y = x$ and $y = x ^ { 2 }$ .

Use Green's Theorem to evaluate the line integral $\int _ { C } e ^ { x } \cos ( 13 y ) d x - 13 e ^ { x } \sin ( 13 y ) d y$ where $C$ is $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$

$\text { Let } \mathbf { F } ( x , y , z ) = x z \mathbf { i } + z y \mathbf { j } + 2 z ^ { 2 } \mathbf { k } \text { and let } S \text { be the surface bounded by }$ $z = 17 - x ^ { 2 } - y ^ { 2 }$ and $z = 0$ . Verify the Divergence Theorem by evaluating $\iint_{S} \mathbf{F} \cdot \mathbf{N} d s$ as a surface integral and as a triple integral. Round your answer to two decimal places.

$\text { Find } \operatorname { div } ( \overrightarrow { \mathrm { F } } \times \overrightarrow { \mathrm { G } } ) \text {. }$ (x,y,z)=8+9x+10y (x,y,z)=8x-8y+8z

Find $\operatorname { div } ( \operatorname { curl } \mathbf { F } ) = \nabla \cdot ( \nabla \times \mathbf { F } ) \text { for the vector field given by }$ $\mathbf { F } ( x , y , z ) = x ^ { 8 } z \mathbf { i } - 2 x z \mathbf { j } + y z \mathbf { k }$

Evaluate $\iint_{S} f(x, y) d S$ , where $f(x, y)=x+y$ and $S$ is given by $\mathbf { r } ( u , v ) = 8 \cos u \mathbf { i } + 8 \sin u \mathbf { j } + v \mathbf { k } , 0 \leq \mathrm { u } \leq \frac { \pi } { 2 } , 0 \leq v \leq 4$

$\text { Find the flux of } \mathbf { F } ( x , y , z ) = x \mathbf { i } + y \mathbf { j } - 2 z \mathbf { k } \text { through }$ $S : z = \sqrt { 121 - x ^ { 2 } - y ^ { 2 } } , \iint _ { S } \mathbf { F } \cdot \mathbf { N } d S$ where $\mathbf { N }$ is the upward unit normal vector to $S$ .

Find the value of the line integral $\int_{c} \mathbf { F } \text {. } d \mathbf { r } \text {, where } \mathbf { F } ( x , y ) = e ^ { y } ( z \mathbf { i } + x z \mathbf { j } + x \mathbf { k } ) \text { and }$ $\mathbf { r } ( t ) = 3 \cos t \mathbf { i } + 3 \sin t \mathbf { j } + 8 \mathbf { k } , 0 \leq t \leq \pi$ (Hint: If $\mathbf { F }$ is conservative, the integration may be easier on an alternate path.)

$\text { Find the value of the line integral }$$\int_{C} y^{3} d x+3 x y^{2} d y$ (Hint: If $\mathbf { F } ( x , y ) = y ^ { 3 } \mathbf { i } + 3 x y ^ { 2 } \mathbf { j }$ is conservative, the integration may be easier on an alternate path.)

$\text { Let } \mathbf { F } ( x , y , z ) = ( 2 x - y ) \mathbf { i } - ( 2 y - z ) \mathbf { j } + z \mathbf { k } \text { and let } S \text { be the surface bounded by the }$ plane $2 x + 4 y + 2 z = 12$ and the coordinates planes. Verify the Divergence Theorem by evaluating $\iint_{S} \mathbf{F} \cdot \mathbf{N} d s$ as a surface integral and as a triple integral.

$\text { Find the value of the line integral } \int _ { C } \mathbf { F } \cdot d \mathbf { r } \text {, where } \mathbf { F } ( x , y ) = 4 \mathbf { i } + 2 z \mathbf { j } + 2 y \mathbf { k } \text { and }$$\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } + t ^ { 2 } \mathbf { k } , 0 \leq t \leq \pi$ (Hint: If $\mathbf { F }$ is conservative, the integration may be easier on an alternate path.)

Find the flux $\vec { F }$ of through $S$ , $\iint _ { S } \vec { F } \cdot \vec { N } d S$ , where $\vec { N }$ is the upward unit normal vector to S. $\overrightarrow { \mathbf { F } } ( x , y , z ) = 2 z \hat { \mathbf { i } } - 8 \hat { \mathbf { j } } + y \hat { \mathbf { k } }$ $S : x + y + z = 9$ , first octant

Evaluate the line integral $\int \sin x \sin y d x - \cos x \cos y d y \text { using the Fundamental }$ Theorem of Line Integrals, where $C$ is the line segment from $( 0 , - \pi )$ to $\left( \frac { 3 \pi } { 2 } , \frac { \pi } { 2 } \right)$ .

The surface of the dome on a new museum is given by $\mathbf { r } = ( u , v ) = 24 \sin u \cos v \mathbf { i } + 24 \sin u \sin v \mathbf { j } + 24 \cos u \mathbf { k }$ , where $0 \leq u \leq \frac { \pi } { 3 }$ and $0 \leq v \leq 2 \pi$ and $\mathbf { r }$ is in meters. Find the surface area of the dome.

Find the curl for the vector field at the given point. $\overrightarrow { \mathbf { F } } ( x , y , z ) = 7 x y z \hat { \mathbf { i } } + 7 y \hat { \mathbf { j } } + 7 z \hat { \mathbf { k } } , \quad ( 7,8,7 )$