Exam 17: Integrals and Vector Fields

Find the potential function f for the field F. - $\mathbf { F } = \left( \mathrm { e } ^ { 2 } - \sin \mathrm { x } \right) \mathbf { i } + 2 x y \mathrm { e } ^ { 2 } \mathbf { j } + \mathbf { k }$

Find the limit. - $r ( t ) = - 3 t i + 2 t j + 2 t k ; 0 \leq t \leq 1$

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. -F = zi + xyj + zyk; D: the solid cube cut by the coordinate planes and the planes x = 2, y = 2, and z = 2

Find the divergence of the field F. - $\mathbf { F } = x z ^ { 6 } \mathbf { i } + 6 y \mathbf { j } - 4 z ^ { 7 } \mathbf { k }$

Apply Green's Theorem to evaluate the integral. - $\oint _ { C } \left( y ^ { 2 } + 4 \right) d x + \left( x ^ { 2 } + 6 \right) d y$ $C$ : The triangle bounded by $x = 0 , x + y = 1 , y = 0$

$\text { Find the required quantity given the wire that lies along the curve } r \text { and has density } \delta .$ -Moment of inertia $\mathrm { I } _ { \mathrm { Z } }$ about the $\mathrm { z }$ -axis, where $\mathbf { r } ( \mathrm { t } ) = ( 5 \cos \mathrm { t } ) \mathbf { i } + ( 5 \sin \mathrm { t } ) \mathbf { j } , 0 \leq \mathrm { t } \leq \frac { \pi } { 2 } ; \delta = 2 ( 1 + \sin 2 \mathrm { t } )$

Calculate the flow in the field F along the path C. - $\mathbf { F } = 8 \mathbf { i } + 3 \mathbf { j } + 8 \mathbf { k } ; \mathrm { C }$ is the curve $\mathbf { r } ( \mathrm { t } ) = 6 \cos 7 \mathrm { ti } + 6 \sin 7 \mathrm { t } \mathbf { j } + 3 \mathrm { tk } , 0 \leq t \leq \frac { 6 } { 7 } \pi$

Find the surface area of the surface S. - $S$ is the portion of the surface $3 x + 4 z = 4$ that lies above the rectangle $5 \leq x \leq 7$ and $4 \leq y \leq 5$ in the $x y$ -plane.

Use Stokes' Theorem to calculate the circulation of the field F around the curve C in the indicated direction. - $\mathbf { F } = 2 \mathrm { y } \mathbf { i } + 8 x \mathbf { j } + \mathrm { z } ^ { 3 } \mathbf { k } ; \mathrm { C } :$ the counterclockwise path around the perimeter of the triangle in the $x - y$ plane formed from the $x$ -axis, $y$ -axis, and the line $y = 5 - 4 x$

Calculate the flux of the field F across the closed plane curve C. - $\mathbf { F } = \mathbf { x } \mathbf { i } + \mathbf { y } \mathbf { j }$ ; the curve $\mathrm { C }$ is the circle $( \mathrm { x } + 5 ) ^ { 2 } + ( \mathrm { y } - 9 ) ^ { 2 } = 81$

Using Green's Theorem, find the outward flux of F across the closed curve C. - $\mathbf { F } = \left( - \mathrm { y } - \mathrm { e } ^ { \mathrm { y } } \cos \mathrm { x } \right) \mathbf { i } + \left( \mathrm { y } - \mathrm { e } ^ { \mathrm { y } } \sin \mathrm { x } \right) \mathbf { j } ; \mathrm { C }$ is the right lobe of the lemniscate $\mathrm { r } ^ { 2 } = \cos 2 \theta$ that lies in the first quadrant.

Find the flux of the curl of field F through the shell S. - $\mathbf { F } = ( 4 \mathrm { y } + 2 ) \mathbf { i } - 5 \mathrm { xj } + \left( \mathrm { e } ^ { \mathrm { z } } - 1 \right) \mathbf { k } ; \mathrm { S } : \mathbf { r } ( \mathrm { r } , \theta ) = 2 \sin \varphi \cos \theta \mathbf { i } + 2 \sin \varphi \sin \theta \mathbf { j } + 2 \cos \varphi \mathbf { k } , 0 \leq \theta \leq 2 \pi \text { and } 0 \leq \varphi \leq \frac { \pi } { 2 }$

Find the divergence of the field F. - $F = - 3 x ^ { 4 } \mathbf { i } - 8 y ^ { 4 } \mathbf { j } + 6 z ^ { 4 } \mathbf { k }$

Calculate the flow in the field F along the path C. - $\mathrm { F } = \nabla \left( x \mathrm { y } ^ { 2 } \mathrm { z } ^ { 3 } \right) ; \mathrm { C } \text { is the line segment from } ( 6,1,1 ) \text { to } ( 7,1 , - 1 )$

Evaluate the line integral along the curve C. - $\int _ { C } ( y + z ) d s , C$ is the straight-line segment $x = 0 , y = 5 - t , z = t$ from $( 0,5,0 )$ to $( 0,0,5 )$

Find the potential function f for the field F. - $\mathbf { F } = - \left( \frac { \mathrm { xyz } } { \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { 3 / 2 } } \right) \mathbf { i } + \left( \frac { \mathrm { z } } { \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { 1 / 2 } } \right) \mathbf { j } + \left( \frac { \mathrm { y } } { \left( 1 + \mathrm { x } ^ { 2 } \right) ^ { 1 / 2 } } \right) \mathbf { k }$

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. - $\mathbf { F } = \ln \left( \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } \right) \mathbf { i } + \tan ^ { - 1 } \left( \frac { \mathrm { x } } { \mathrm { y } } \right) \mathrm { j } ; \mathrm { C } \text { is the region defined by the polar coordinate inequalities } 3 \leq \mathrm { r } \leq 8 \text { and } 0 \leq \theta \leq \pi$

Solve the problem. - $\text { What can be said about the flux of } \mathbf { F } = \frac { x \mathbf { i } + \mathrm { y } \mathbf { j } + \mathrm { zk } } { \left( \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } } \right) ^ { 3 } } \text { across a sphere centered at the origin? Will differing }$ radii change the flux?

Find the equation for the plane tangent to the parametrized surface S at the point P. - $\mathrm { S } \text { is the sphere } \mathbf { r } ( \theta , \varphi ) = 7 \cos \theta \sin \varphi \mathbf { i } + 7 \sin \theta \sin \varphi \mathbf { j } + 7 \cos \varphi \mathbf { k } ; \mathrm { P } \text { is the point corresponding to } ( \theta , \varphi ) = \left( \frac { \pi } { 3 } , \frac { \pi } { 4 } \right) \text {. }$

Solve the problem. -Assuming $\mathrm { C }$ is a closed path, what is special about the integral $\int _ { C } 9 x ^ { 8 } y ^ { 7 } d x + 7 x ^ { 9 } y ^ { 6 } d y$ ? Give reasons for your answer.