Exam 17: Integrals and Vector Fields

Using Green's Theorem, find the outward flux of F across the closed curve C. - $\mathbf { F } = \sin 10 \mathrm { yi } + \cos 6 \mathrm { x } \mathbf { j } ; \mathrm { C }$ is the rectangle with vertices at $( 0,0 ) , \left( \frac { \pi } { 10 } , 0 \right) , \left( \frac { \pi } { 10 } , \frac { \pi } { 6 } \right)$ , and $\left( 0 , \frac { \pi } { 6 } \right)$

Find the potential function f for the field F. - $\mathbf { F } = - \frac { 1 } { x } \mathbf { i } + \frac { 1 } { \mathrm { y } } \mathrm { j } - \frac { 1 } { \mathrm { z } } \mathbf { k }$

Evaluate. The differential is exact. - $\int _ { ( 0,0,0 ) } ^ { ( 4,6,2 ) } \left( 2 x y ^ { 2 } - 2 x z ^ { 2 } \right) d x + 2 x ^ { 2 } y d y - 2 x ^ { 2 } z d z$

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. - $\mathbf { F } = x \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } \mathrm { i } + \mathrm { y } \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } \mathrm { j } + \mathrm { z } \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } \mathbf { k } ;$ D: the thick cylinder $5 \leq \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } \leq 6,3 \leq \mathrm { z } \leq 4$

Find the flux of the curl of field F through the shell S. - $\mathbf { F } = 7 \mathrm { yi } - 5 \mathrm { z } \mathbf { j } - 5 \mathrm { x } \mathbf { k } ; \mathrm { S } : \mathbf { r } ( \mathrm { r } , \theta ) = \mathrm { r } \cos \theta \mathbf { i } + \mathrm { r } \sin \theta \mathbf { j } + \left( 36 - \mathrm { r } ^ { 2 } \right) \mathbf { k } , 0 \leq \mathrm { r } \leq 6$ and $0 \leq \theta \leq 2 \pi$

Evaluate. The differential is exact. - $\int _ { ( 0,0,0 ) } ^ { ( 1,1,1 ) } 9 x ^ { 8 } y ^ { 8 } z ^ { 5 } d x + 8 x ^ { 9 } y ^ { 7 } z ^ { 5 } d y + 5 x ^ { 9 } y ^ { 8 } z ^ { 4 } d z$

Find the flux of the vector field F across the surface S in the indicated direction. - $\mathbf { F } = \frac { z ^ { 2 } } { 9 } \mathbf { k } ; \mathrm { S }$ is the upper hemisphere of $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } = 9$ ; direction is outward

Solve the problem. -The base of the closed cubelike surface is the unit square in the xy-plane. The four sides lie in the planes x = 0, x = 1, y = 0, and y = 1. The top is an arbitrary smooth surface whose identity is unknown. Let F = xi - 4yj + (z + 11)k and suppose the outward flux through the side parallel to the yz-plane is 2 and through The side parallel to the xz-plane is -5. What is the outward flux through the top?

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. - $\mathbf { F } = ( \mathrm { y } - \mathrm { x } ) \mathbf { i } + ( \mathrm { z } - \mathrm { y } ) \mathbf { j } + ( \mathrm { z } - \mathrm { x } ) \mathbf { k } ; \mathrm { D }$ ; the region cut from the solid cylinder $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } \leq 9$ by the planes $\mathrm { z } = 0$ and $\mathrm { z } = 5$

Find the flux of the curl of field F through the shell S. - $\mathbf{F}=\mathrm{e}^{\mathrm{x}} \mathbf{i}+\mathrm{e}^{\mathrm{y}} \mathbf{j}+6 \mathrm{xy} \mathbf{k} ; \mathrm{S} \text { is the portion of the paraboloid } 2-\mathrm{x}^{2}-\mathrm{y}^{2}=\mathrm{z} \text { that lies above the } x y-\text { plane }$

Parametrize the surface S. - $S \text { is the lower portion of the sphere } x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 49 \text { cut by the cone } z = \sqrt { x ^ { 2 } + y ^ { 2 } } \text {. }$

Evaluate the line integral of f(x,y) along the curve C. - $f ( x , y ) = y ^ { 2 } + x ^ { 2 } , C$ : the perimeter of the circle $x ^ { 2 } + y ^ { 2 } = 16$

Solve the problem. -For the surface $z = f ( x , y )$ , show that the surface integral $\iint g ( x , y , z ) d \sigma =$ $\iint \mathrm { g } ( \mathrm { x } , \mathrm { y } , \mathrm { f } ( \mathrm { x } , \mathrm { y } ) ) \sqrt { \left[ \mathrm { f } _ { \mathrm { x } } ( \mathrm { x } , \mathrm { y } ) \right] ^ { 2 } + \left[ \mathrm { f } _ { \mathrm { y } } ( \mathrm { x } , \mathrm { y } ) \right] ^ { 2 } + 1 } \mathrm { dx } d y$

Find the potential function f for the field F. - $\mathbf { F } = \frac { 1 } { \mathrm { z } } \mathbf { i } - 2 \mathbf { j } - \frac { \mathrm { x } } { \mathrm { z } ^ { 2 } } \mathbf { k }$

Solve the problem. -Assuming $C$ is a simple closed path, what is special about the integral $\left. \left. \int _ { C } ( 4 x + 5 \sin 8 x \cos 8 y ) \right) d x + ( 3 x + 5 \cos 8 x \sin 8 y ) \right) d y$ ? Give reasons for your answer.

Test the vector field F to determine if it is conservative. - $\mathbf { F } = \left( \frac { 8 \mathrm { x } ^ { 7 } \mathrm { y } ^ { 8 } } { \mathrm { z } ^ { 7 } } \right) \mathrm { i } + \left( \frac { 8 \mathrm { x } ^ { 8 } \mathrm { y } ^ { 7 } } { \mathrm { z } ^ { 7 } } \right) \mathrm { j } - \left( \frac { 7 \mathrm { x } ^ { 8 } \mathrm { y } ^ { 8 } } { \mathrm { z } ^ { 8 } } \right) \mathbf { k }$

Find the equation for the plane tangent to the parametrized surface S at the point P. - $S \text { is the parabolic cylinder } \mathbf { r } ( x , z ) = x \mathbf { i } + 10 x ^ { 2 } \mathbf { j } + z \mathbf { k } \text {; } P \text { is the point corresponding to } ( x , z ) = ( - 4 , - 4 ) \text {. }$

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 $x ^ { 2 } + y ^ { 2 } = 4$ . -F = -xi + yj

Using Green's Theorem, calculate the area of the indicated region. -The area bounded above by $y = 8$ and below by $y = \frac { 8 } { 25 } x ^ { 2 }$

Find the flux of the vector field F across the surface S in the indicated direction. - $\mathrm { F } = 3 \mathrm { xi } + 3 \mathrm { yj } + 5 \mathbf { k }$ ; $\mathrm { S }$ is "nose" of the paraboloid $\mathrm { z } = 5 \mathrm { x } ^ { 2 } + 5 \mathrm { y } ^ { 2 }$ cut by the plane $\mathrm { z } = 4$ ; direction is outward