Exam 16: Vector Calculus

Assuming that $S$ satisfies the conditions of the Divergence Theorem and the scalar functions and components of the vector fields have continuous second order partial derivatives, find $\iint _ { s} 4 \mathbf { a } \cdot \mathbf { n } d S$ , where $\mathbf { a }$ is the constant vector.

The temperature at the point $( x , y , z )$ in a substance with conductivity $K = 4$ is $u ( x , y , z ) = 2 x ^ { 2 } + 2 y ^ { 2 }$ . Find the rate of heat flow inward across the cylindrical $y ^ { 2 } + z ^ { 2 } = 2,0 \leq x \leq 7$ . Select the correct answer.

Evaluate $\int _ { C } y z d y + x y d z$ , where $C$ is given by $x = 10 \sqrt { t } , y = 3 t , z = 10 t ^ { 2 } , 0 \leq t \leq 1$ . Round your answer to two decimal place.

Find the work done by the force field $\mathbf { F }$ on a particle that moves along the curve $C$ . $\mathbf { F } ( x , y ) = ( 5 x + 5 y ) \mathbf { i } + 4 x y \mathbf { j } ; \quad C : \mathbf { r } ( t ) = 3 t ^ { 2 } \mathbf { i } + t ^ { 2 } \mathbf { j } , 0 \leq t \leq 1$

Use Stoke's theorem to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r } , \mathbf { F } ( x , y , z ) = 6 y x ^ { 2 } \mathbf { i } + 2 x ^ { 3 } \mathbf { j } + 6 x y \mathbf { k }$ , $C$ is the curve of intersection of the hyperbolic paraboloid $z = y ^ { 2 } - x ^ { 2 }$ and the cylinder $x ^ { 2 } + y ^ { 2 } = 25$ oriented counterclockwise as viewed from above.

Evaluate the line integral. 6yzxds C:x=t,y=t,z=t0\leqt\leq\pi

Use Stokes' Theorem to evaluate $\iint _ { S } \operatorname { cur } \mathbf { F } \cdot d \mathbf { S }$ . $\mathbf { F } ( x , y , z ) = 9 x y \mathbf { i } + 6 y z \mathbf { j } + 9 z ^ { 2 } \mathbf { k }$ $S$ is the part of the ellipsoid $4 x ^ { 2 } + 4 y ^ { 2 } + 49 z ^ { 2 } = 196$ lying above the $x y$ -plane and oriented with normal pointing upward.

A plane lamina with constant density $\rho ( x , y ) = 12$ occupies a region in the $x y$ -plane bounded by a simple closed path $C$ . Its moments of inertia about the axes are $l _ { x } = - \frac { \rho } { 3 } \int _ { C } y ^ { 3 } d x \text { and } l _ { y } = \frac { \rho } { 3 } \int _ { C } x ^ { 3 } d y$ Find the moments of inertia about the axes, if $C$ is a rectangle with vertices $( 0,0 ) , ( 4,0 )$ , $( 4,5 )$ and $( 0,5 ) .$

Find the work done by the force field $\mathbf { F }$ on a particle that moves along the curve $C$ . Select the correct answer. $\mathbf { F } ( x , y ) = ( 5 x + 5 y ) \mathbf { i } + 4 x y \mathbf { j } ; \quad C : \mathbf { r } ( t ) = 3 t ^ { 2 } \mathbf { i } + t ^ { 2 } \mathbf { j } , 0 \leq t \leq 1$

Let $S$ be the cube with vertices $( \pm 1 , \pm 1 , \pm 1 )$ . Approximate $\iint _ { S } \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 7 z ^ { 2 } }$ by using a Riemann sum as in Definition 1, taking the patches $S _ { y j }$ to be the squares that are the faces of the cube and the points $P _ { i j }$ to be the centers of the squares.

Determine whether or not vector field is conservative. If it is conservative, find a function $f$ such that $\mathbf { F } = \nabla f$ . $\mathbf { F } ( x , y , z ) = 25 y z e ^ { 5 x x _ { i } } \mathbf { i } + 5 e ^ { 5 x x _ { j } } \mathbf { j } + 25 x y e ^ { 5 x z } \mathbf { k }$

Let $\mathbf { r } = x \mathbf { i } + y \mathbf { j } + z \mathbf { k }$ and $r = | \mathbf { r } |$ . Find $\nabla \cdot ( \mathbf { r } )$

Evaluate $\iint _ { S } f ( x , y , z ) d S$ $f ( x , y , z ) = z ; S$ is the part of the torus with vector representation $\mathbf { r } ( u , v ) = ( 5 + 3 \cos v ) \cos u \mathbf { i } + ( 5 + 3 \cos v ) \sin \imath \mathbf { j } + 3 \sin v \mathbf { k } , 0 \leq u \leq 2 \pi , 0 \leq v \leq \frac { \pi } { 2 } .$ Select the correct answer.

Show that $\mathbf { F }$ is conservative and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use this result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any path from $A \left( x _ { 0 } , y _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } \right)$ . $\mathbf { F } ( x , y ) = \left( 16 x y ^ { 2 } - 16 x y ^ { 3 } \right) \mathbf { i } + \left( 16 x ^ { 2 } y - 24 x ^ { 2 } y ^ { 2 } \right) \mathbf { j } ; A ( 1,0 )$ and $B ( 1,0 )$

Let $S$ be the cube with vertices $( \pm 1 , \pm 1 , \pm 1 )$ . Approximate $\iint _ { S } \sqrt { x ^ { 2 } + 2 y ^ { 2 } + 7 z ^ { 2 } }$ by using a Riemann sum as in Definition 1, taking the patches $S _ { i j }$ to be the squares that are the faces of the cube and the points $P _ { i j }$ to be the centers of the squares.

Find the curl of the vector field. $\mathbf { F } ( x , y , z ) = ( x - 9 z ) \mathbf { i } + ( 7 x + y + 2 z ) \mathbf { j } + ( x - 16 y ) \mathbf { k }$

Show that $\mathrm { F }$ is conservative, and find a function $f$ such that $\mathbf { F } = \nabla f$ , and use the result to evaluate $\int _ { C } \mathbf { F } \cdot d \mathbf { r }$ , where $C$ is any curve from $A \left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ to $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 8 x y \mathbf { i } + \left( 4 x ^ { 2 } + 16 y ^ { 3 } z ^ { 2 } \right) \mathbf { j } + 8 y ^ { 4 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( 2 , - 2,2 )$ Select the correct answer.

Find the area of the part of the surface $y = 8 x + z ^ { 2 }$ that lies between the planes $x = 0 , x = 4 , z = 0$ , and $z = 1$ . Select the correct answer.

Use Green's Theorem and/or a computer algebra system to evaluate $\int _ { C } x ^ { 2 } y d x - x y ^ { 2 } d y$ , where $C$ is the circle $x ^ { 2 } + y ^ { 2 } = 64$ with counterclockwise orientation.

Below is given the plot of a vector field $\mathbf { F }$ in the $x y$ -plane. (The $z$ -component of $\mathbf { F }$ is 0 .) By studying the plot, determine whether $\operatorname { div } \mathbf { F }$ is positive, negative, or zero.