Exam 17: Integrals and Vector Fields

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. - $\mathbf { F } = x ^ { 2 } \mathbf { i } + y ^ { 2 } \mathbf { j } + z \mathbf { k } \text {; D: the solid cube cut by the coordinate planes and the planes } x = 3 , y = 3 \text {, and } z = 3$

Calculate the circulation of the field F around the closed curve C. - $\mathbf { F } = \frac { 7 } { 10 } \mathrm { x } ^ { 2 } \mathrm { yi } + \frac { 7 } { 10 } \mathrm { xy } ^ { 2 } \mathrm { j } ;$ curve $\mathrm { C }$ is $\mathbf { r } ( \mathrm { t } ) = 3 \cos \mathrm { ti } + 3 \sin \mathrm { tj } , 0 \leq \mathrm { t } \leq 2 \pi$

Find the divergence of the field F. -F $= 8 x ^ { 6 } \mathbf { i } - 8 x y j - 3 x z k$

Find the flux of the vector field F across the surface S in the indicated direction. - $\mathbf { F } = 2 \mathrm { x } ^ { 2 } \mathrm { j } - \mathrm { z } ^ { 3 } \mathrm { k } ; \mathrm { S }$ is the portion of the parabolic cylinder $\mathrm { y } = 2 \mathrm { x } ^ { 2 }$ for which $0 \leq \mathrm { z } \leq 1$ and $- 1 \leq \mathrm { x } \leq 1$ ; direction is outward (away from the $y - z$ plane)

Solve the problem. -Assuming all the necessary derivatives exist, show that if $\int _ { C } \frac { \partial f } { \partial y } d x - \frac { \partial f } { \partial x } d y = 0$ for all closed curves C to which Green's Theorem applies, then $f$ satisfies the Laplace equation $\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } = 0$ for all regions bounded by closed curves $\mathrm { C }$ to which Green's Theorem applies.

Apply Green's Theorem to evaluate the integral. - $\oint _ { C } ( 8 y + x ) d x + ( y + 3 x ) d y$ $C$ : The circle $( x - 6 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 9$

Parametrize the surface S. - $S \text { is the portion of the cylinder } x ^ { 2 } + y ^ { 2 } = 25 \text { that lies between } z = 7 \text { and } z = 8 \text {. }$

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch the vector field in the plane along with its horizontal and vertical components at a representative assortment of points on the circle $x ^ { 2 } + y ^ { 2 } = 4$ . - $\mathbf { F } = \frac { \mathrm { y } } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } } \mathbf { i } - \frac { \mathrm { x } } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } } \mathrm { j }$

Find the gradient field of the function. - $f ( x , y , z ) = e ^ { x ^ { 6 } + y ^ { 5 } + z ^ { 7 } }$

Solve the problem. -The shape and density of a thin shell are indicated below. Find the coordinates of the center of mass. Shell: "nose" of the paraboloid $x ^ { 2 } + y ^ { 2 } = 2 z$ cut by the plane $z = 3$ Density: $\delta = \sqrt { x ^ { 2 } + y ^ { 2 } + 1 }$

Calculate the area of the surface S. - $S$ is the portion of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 100$ between $z = - 5 \sqrt { 2 }$ and $z = 5 \sqrt { 2 }$ .

Solve the problem. -a). For the field $F ( x , y ) = M i + N j$ and the closed counterclockwise plane curve $C$ in the $x y - p l a n e$ , show that $\int _ { C } F \cdot n d s = \int _ { C } M d y - N d x .$ b). How would the equality change if the closed path is followed clockwise?

Evaluate the surface integral of the function g over the surface S. - $G ( x , y , z ) = x ^ { 2 } + y ^ { 2 } + z ^ { 2 } ; S$ is the surface of the cube formed from the coordinate planes and the planes $x = 1$ , $y = 1$ , and $z = 1$

Evaluate the line integral along the curve C. - ds,C is the path given by: :(t)=(2t)+(2t) from (2,0,0) to (0,2,0) :(t)=(2t)+(2t) from (0,2,0) to (0,0,2) :(t)=(2t)+(2t) from (0,0,2) to (2,0,0)

Evaluate the surface integral of G over the surface S. - $\mathrm { S }$ is the portion of the cone $\mathrm { z } = 3 \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } , 0 \leq \mathrm { z } \leq 3 ; \mathrm { G } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { z } - \mathrm { y }$

$\text { Find the center of mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$ - $r ( t ) = ( 5 \cos 3 t ) i + ( 5 \sin 3 t ) j + 15 t k , 0 \leq t \leq 2 \pi ; \delta ( x , y , z ) = 3 ( 1 + \sin 3 t \cos 3 t )$

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. - $\mathbf { F } = \mathrm { e } ^ { \mathrm { yz } } \mathrm { i } + 6 \mathrm { yj } + 8 \mathrm { z } ^ { 2 } \mathbf { k } ;$ D: the solid sphere $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } \leq 64$

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. - $\mathbf { F } = 2 \mathrm { x } ^ { 3 } \mathbf { i } + 2 \mathrm { y } ^ { 3 } \mathrm { j } + 2 \mathrm { z } ^ { 3 } \mathbf { k }$ ; D: the thick sphere $16 \leq \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } \leq 25$

Using Green's Theorem, calculate the area of the indicated region. -The astroid $\mathbf { r } ( \mathrm { t } ) = \left( 5 \cos ^ { 3 } \mathrm { t } \right) \mathbf { i } + \left( 5 \sin ^ { 3 } \mathrm { t } \right) \mathbf { j } , 0 \leq \mathrm { t } \leq 2 \pi$

Test the vector field F to determine if it is conservative. - $\mathbf { F } = 4 x ^ { 3 } y ^ { 4 } z ^ { 4 } i + 4 x ^ { 4 } y ^ { 3 } z ^ { 4 } j + 4 x ^ { 4 } y ^ { 4 } z ^ { 3 } k$