Exam 17: Integrals and Vector Fields

Solve the problem. -Consider a fluid with a flow field $\mathbf { F } = \mathrm { x } ^ { 5 } \mathrm { y } ^ { 4 } \mathbf { i } + 9 \mathrm { z } ^ { 2 } \mathbf { j } + \mathrm { z } \mathbf { k }$ . A miniature paddlewheel (idealized) is to be inserted into the flow at the point $( 1,1,1 )$ . Find a vector describing the orientation of the paddlewheel axis which produces the maximum rotational speed.

Calculate the circulation of the field F around the closed curve C. - $\mathbf { F } = x y y ^ { 2 } \mathbf { i } + x ^ { 2 } y \mathbf { j } ;$ curve $C$ is the counterclockwise path around $C _ { 1 } \cup C _ { 2 } : C _ { 1 } : r ( t ) 6 \cos t i + 6 \sin t j , 0 \leq t \leq \pi$ $C _ { 2 } : r ( t ) = t i , - 6 \leq t \leq 6$

Evaluate the line integral along the curve C. - $\int _ { C } ( y + z ) d s , C$ is the path from $( 0,0,0 )$ to $( 6 , - 6,1 )$ given by: $C _ { 1 } : r ( t ) = 6 t ^ { 2 } i - 6 t j , 0 \leq t \leq 1$ $C _ { 2 } : \mathbf { r } ( \mathrm { t } ) = 6 \mathbf { i } - 6 \mathbf { j } + ( t - 1 ) \mathbf { k } , 1 \leq \mathrm { t } \leq 2$

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

Find the flux of the curl of field F through the shell S. - $\mathrm { F } = - 5 \mathrm { x } ^ { 2 } \mathrm { yi } + 5 \mathrm { xy } ^2 \mathbf { j } + \mathrm { z } ^ { 5 } \mathbf { k }$ ; $\mathrm { S }$ is the portion of the paraboloid $2 - \mathrm { x } ^ { 2 } - \mathrm { y } ^ { 2 } = \mathrm { z }$ that lies above the $x y -$ plane

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

Find the gradient field of the function. - $\mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { x } ^ { 5 } \mathrm { e } ^ { 7 \mathrm { x } } + \mathrm { y } ^ { 3 } \mathrm { z } ^ { 7 }$

Find the potential function f for the field F. - $\mathrm { F } = 8 \mathrm { x } ^ { 7 } \mathrm { y } ^ { 9 } \mathrm { z } ^ { 3 } \mathbf { i } + 9 \mathrm { x } ^ { 8 } \mathrm { y } ^ { 8 } \mathrm { z } ^ { 3 } \mathbf { j } + 3 \mathrm { x } ^ { 8 } \mathrm { y } ^ { 9 } \mathrm { z } ^ { 2 } \mathbf { k }$

Using Green's Theorem, find the outward flux of F across the closed curve C. - $\mathbf { F } = \left( \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } \right) \mathbf { i } + ( \mathrm { x } - \mathrm { y } ) \mathbf { j } ; C \text { is the rectangle with vertices at } ( 0,0 ) , ( 8,0 ) , ( 8,10 ) , \text { and } ( 0,8 )$

Evaluate the work done between point 1 and point 2 for the conservative field F. - $\mathbf { F } = ( \mathrm { y } + \mathrm { z } ) \mathbf { i } + \mathrm { xj } + \mathrm { xk } ; \mathrm { P } _ { 1 } ( 0,0,0 ) , \mathrm { P } _ { 2 } ( 1,4,7 )$

Solve the problem. -Consider the counterclockwise integral $\int _ { C } f ( x , y ) d x + g ( x , y ) d y$ where $C$ is a closed path in a region where Green's Theorem applies. To evaluate the integral, should one use the flux-divergence form or the circulation-flow form of Green's theorem? Explain.

Calculate the flow in the field F along the path C. - $\mathbf { F } = - \frac { z y } { 6 \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { i } + \frac { z x } { 6 \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } j + \frac { z ^ { 3 } } { 3 \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { k } , C$ is the curve $r ( t ) = 4 \cos 6 t i + 4 \sin 6 t j + 3 t k , 0 \leq t$ $\leq 1$

Using the Divergence Theorem, find the outward flux of F across the boundary of the region D. - $\mathbf { F } = 3 x y ^ { 2 } i + 3 x ^ { 2 } y j + 6 x y \mathbf { k } ;$ D: the region cut from the solid cylinder $x ^ { 2 } + y ^ { 2 } \leq 9$ by the planes $z = 0$ and $z = 6$

Calculate the flux of the field F across the closed plane curve C. - $\mathbf { F } = \mathrm { y } ^ { 2 } \mathbf { i } + \mathrm { x } ^ { 3 } \mathbf { j }$ ; the curve $\mathrm { C }$ is the closed counterclockwise path formed from the semicircle $\mathbf { r } ( \mathrm { t } ) = 3 \cos \mathrm { ti } + 3$ sin $\mathrm { j }$ , $0 \leq t \leq \pi$ , and the straight line segment from $( - 3,0 )$ to $( 3,0 )$

Parametrize the surface S. - $S \text { is the portion of the plane } 3 x - 6 y - 5 z = - 4 \text { that lies within the cylinder } x ^ { 2 } + y ^ { 2 } = 4 \text {. }$

Test the vector field F to determine if it is conservative. - $\mathbf { F } = \left( \frac { e ^ { x } + e ^ { - x } } { y z } \right) i + \left( \frac { e ^ { - x } + e ^ { x } } { y ^ { 2 } z } \right) j + \left( \frac { e ^ { x } + e ^ { - x } } { y z ^ { 2 } } \right) k$

Find the work done by F over the curve in the direction of increasing t. - $\mathbf { F } = - 5 \mathrm { yi } + 5 \mathrm { x } \mathbf { j } + 6 \mathrm { z } ^ { 7 } \mathbf { k } ; \mathrm { C } : \mathbf { r } ( \mathrm { t } ) = \cos \mathrm { ti } + \sin \mathrm { tj } , 0 \leq \mathrm { t } \leq 3$

Find the flux of the curl of field F through the shell S. - $\mathbf { F } = ( x - y ) \mathbf { i } + ( x - z ) \mathbf { j } + ( y - z ) \mathbf { k } ; S$ is the portion of the cone $z = 3 \sqrt { x ^ { 2 } + y ^ { 2 } }$ below the plane $z = 2$

Calculate the flux of the field F across the closed plane curve C. - $\mathbf { F } = \mathrm { e } ^ { 5 \mathrm { x } _ { \mathrm { i } } + \mathrm { e } ^ { 2 } \mathrm { y } } \mathrm { j }$ ; the curve $\mathrm { C }$ is the closed counterclockwise path around the triangle with vertices at $( 0,0 ) , ( 5,0 )$ , and $( 0,1 )$

Solve the problem. -Let $\mathrm { M } = \frac { \mathrm { y } } { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } }$ and $\mathrm { N } = \frac { - \mathrm { x } } { \mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } }$ . Show that $\int _ { \mathrm { C } } \mathrm { Mdx } + \mathrm { Ndy } z \int _ { \mathrm { R } } \left( \frac { \partial \mathrm { N } } { \partial \mathrm { x } } - \frac { \partial \mathrm { M } } { \partial \mathrm { y } } \right) \mathrm { dx } \mathrm { dy }$ , where $\mathrm { R }$ is the region bounded by the unit circle $C$ centered at the origin. Why is Green's Theorem failing in this case?