Exam 17: Integrals and Vector Fields

Find the work done by F over the curve in the direction of increasing t. - $\mathbf { F } = x \mathrm { e } ^ { 3 \mathrm { x } ^ { 2 } } \mathbf { i } + \mathrm { e } ^ { 7 \mathrm { y } _ { j } } + \mathrm { e } ^ { 3 \mathrm { z } } \mathbf { k }$ ; the path is $\mathrm { C } _ { 1 } \cup \mathrm { C } _ { 2 }$ where $\mathrm { C } _ { 1 }$ is the straight line from $( 0,0,0 )$ to $( 1,1,0 )$ and $\mathrm { C } _ { 2 }$ is the straight line from $( 1,1,0 )$ to $( 1,1,1 )$

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 $x ^ { 2 } + y ^ { 2 } = 4$ . - $\mathbf { F } = - \frac { \mathrm { y } } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } } \mathbf { i } - \frac { \mathrm { x } } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } } \mathrm { j }$

Calculate the flow in the field F along the path C. - $\mathbf { F } = ( \mathrm { x } - \mathrm { y } ) \mathbf { i } - \left( \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } \right) \mathbf { j } ; \mathrm { C }$ is curve from $( 4,0 )$ to $( - 4,0 )$ on the upper half of the circle $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 16$

Solve the problem. -The velocity field $\mathbf { F }$ of a fluid has a constant magnitude $\mathrm { k }$ and always points towards the origin. Following the smooth curve $y = f ( x )$ from $( a , f ( a ) )$ to $( b , f ( b ) )$ , show that the flow along the curve is $\int _ { C } \mathbf { F } \cdot \mathbf { T } d s = k \left[ \left( a ^ { 2 } + ( f ( a ) ) ^ { 2 } \right) ^ { 1 / 2 } - \left( b ^ { 2 } + ( f ( b ) ) ^ { 2 } \right) ^ { 1 / 2 } \right]$

Solve the problem. - $\text { The flow } \mathrm { F } \text { of a fluid in a plane is illustrated below. At which point or points would } \nabla \times \mathrm { F } \text { point out from the }$

Find the flux of the vector field F across the surface S in the indicated direction. - $F ( x , y , z ) = 7 i + y ^ { 10 } z ^ { 7 } j - y ^ { 7 } z ^{10} k , S \text { is the rectangular surface } x = 0 , - 5 \leq y \leq 5 \text {, and } - 2 \leq z \leq 2 \text {, direction } \mathbf { i }$

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. - $\mathbf { F } = \sin 5 y \mathbf { i } + \cos 7 x \mathbf { j } ; \mathrm { C }$ is the rectangle with vertices at $( 0,0 ) , \left( \frac { \pi } { 7 } , 0 \right) , \left( \frac { \pi } { 7 } , \frac { \pi } { 5 } \right)$ , and $\left( 0 , \frac { \pi } { 5 } \right)$

Calculate the circulation of the field F around the closed curve C. -F = (-x - y)i + (x + y)j , curve C is the counterclockwise path around the circle with radius 3 centered at (3, 6)

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. - $\mathbf { F } = \left( x ^ { 2 } + y ^ { 2 } \right) \mathbf { i } + ( x - y ) \mathbf { j } ; C \text { is the rectangle with vertices at } ( 0,0 ) , ( 8,0 ) , ( 8,5 ) , \text { and } ( 0,5 )$

Find the work done by F over the curve in the direction of increasing t. - $\mathbf { F } = \mathrm { ti } + \frac { 1 } { 2 } \mathrm { j } + \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \mathrm { e } ^ { 2 } \mathrm { t } _ { \mathbf { i } } + \mathrm { e } ^ { 2 } \mathrm { t } \mathbf { j } + \left( - 7 \mathrm { t } ^ { 2 } + \mathrm { t } \right) \mathbf { k } , - 1 \leq \mathrm { t } \leq 1$

Find the flux of the curl of field F through the shell S. - $\mathbf { F } = ( 4 - \mathrm { y } ) \mathbf { i } + ( 3 + \mathrm { x } ) \mathbf { j } + \mathrm { z } ^ { 5 } \mathbf { k } ; \mathrm { S }$ is the upper hemisphere of $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + \mathrm { z } ^ { 2 } = 16$

$\text { Find the mass of the wire that lies along the curve } r \text { and has density } \delta \text {. }$ - $C _ { 1 } : \mathbf { r } ( t ) = ( 5 \cos t ) \mathbf { i } + ( 5 \sin t ) \mathbf { j } , 0 \leq t \leq \frac { \pi } { 2 } \text {; }$ $C _ { 2 } : \mathbf { r } ( \mathrm { t } ) = 5 \mathrm { j } + \mathrm { tk } , 0 \leq \mathrm { t } \leq 1 ; \delta = 8 \mathrm { t } ^ { 3 }$

Find the gradient field of the function. - $f ( x , y , z ) = x ^ { 6 } y ^ { 9 } + \frac { x ^ { 6 } } { z ^ { 8 } }$

Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. - $\mathrm { F } = - \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } \mathrm { i } + \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } \mathrm { j } ; \mathrm { C } \text { is the region defined by the polar coordinate inequalities } 9 \leq \mathrm { r } \leq 10 \text { and } 0 \leq \theta \leq \pi$

Find the work done by F over the curve in the direction of increasing t. - $\mathbf { F } = - \frac { 1 } { 16 \mathrm { x } ^ { 2 } } \mathbf { i } + \frac { 3 \mathrm { z } } { 4 \mathrm { x } } \mathbf { j } + \frac { 1 } { 48 \mathrm { x } ^ { 2 } } \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \frac { \cos 4 \mathrm { t } } { 4 } \mathbf { i } + \frac { \sin 4 \mathrm { t } } { 4 } \mathbf { j } + 3 \mathrm { t } \mathbf { k } , 0 \leq \mathrm { t } \leq \frac { \pi } { 16 }$

Find the flux of the vector field F across the surface S in the indicated direction. - $\mathbf { F } = 5 x \mathbf { i } + 5 y \mathbf { j } + \mathrm { z } ^ { 2 } \mathbf { k } ; \mathrm { S }$ is portion of the cylinder $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } = 25$ between $\mathrm { z } = 0$ and $\mathrm { z } = 5$ ; direction is outward

Using Green's Theorem, find the outward flux of F across the closed curve C. - $\mathbf { F } = - \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } \mathrm { i } + \sqrt { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } } \mathrm { j } ; \mathrm { C } \text { is the region defined by the polar coordinate inequalities } 1 \leq \mathrm { r } \leq 4 \text { and } 0 \leq \theta \leq \pi$

Evaluate the line integral along the curve C. - $\int _ { C } \left( \frac { x ^ { 2 } + y ^ { 2 } } { z ^ { 2 } } \right) d s , C$ is the curve $r ( t ) = ( 2 \sin 6 t ) i + ( 2 \cos 6 t ) j + 5 t k , 2 \leq t \leq 4$

Evaluate the surface integral of G over the surface S. - $S$ is the parabolic cylinder $y = 3 x ^ { 2 } , 0 \leq x \leq 2$ and $0 \leq z \leq 4 ; G ( x , y , z ) = 3 x$

Calculate the area of the surface S. - $S$ is the portion of the plane $7 x + 8 y + 7 z = 4$ that lies within the cylinder $x ^ { 2 } + y ^ { 2 } = 1$ .