Exam 16: Vector Calculus

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 $B \left( x _ { 1 } , y _ { 1 } \right)$ . $\mathbf { F } ( x , y ) = \left( 15 x ^ { 2 } y ^ { 2 } - 14 x y ^ { 4 } \right) \mathbf { i } + \left( 10 x ^ { 3 } y - 28 x ^ { 2 } y ^ { 3 } \right) \mathbf { j } ; A ( 1 , - 2 ) \text { and } B ( 1 , - 1 )$

Find the exact mass of a thin wire in the shape of the helix $x = 2 \sin ( t ) , y = 2 \cos ( t ) , z = 7 t , 0 \leq t \leq 2 \pi$ if the density is 4 .

Use Green's Theorem to find the work done by the force $\mathbf { F } ( x , y ) = x ( x + 3 y ) \mathbf { i } + 3 x y ^ { 2 } \mathbf { j }$ in moving a particle from the origin along the $x$ -axis to $( 1,0 )$ then along the line segment to $( 0,1 )$ and then back to the origin along the $y$ -axis.

A thin wire is bent into the shape of a semicircle $x ^ { 2 } + y ^ { 2 } = 4 , x > 0$ . If the linear density is 3 , find the exact mass of the wire. Select the correct answer.

Use a computer algebra system to compute the flux of $\mathbf { F }$ across $S$ . $S$ is the surface of the cube cut from the first octant by the planes $x = \frac { \pi } { 2 } , y = \frac { \pi } { 2 } , z = \frac { \pi } { 2 }$ $\mathbf { F } ( x , y , z ) = 5 \sin x \cos ^ { 2 } y \mathbf { i } + 5 \sin ^ { 3 } y \cos ^ { 4 } z \mathbf { j } + 5 \sin ^ { 5 } z \cos ^ { 6 } x \mathbf { k }$

Use a computer algebra system to compute the flux of $\mathbf { F }$ across $S . S$ is the surface of the cube cut from the first octant by the planes x=,y=,z= (x,y,z)=3xy+3yz+3zx Select the correct answer.

Find the gradient vector field of the scalar function $f$ . (That is, find the conservative vector field $\mathbf { F }$ for the potential function $f$ of $\mathbf { F }$ .) $f ( x , y , z ) = 8 x y ^ { 2 } + 2 y z ^ { 3 }$

Find the curl of the vector field. $\mathbf { F } ( x , y , z ) = 8 x y \mathbf { i } + 10 y z \mathbf { j } + 5 x z \mathbf { k }$ Select the correct answer.

Evaluate the line integral over the given curve $C$ . $\int _ { C } 3 x y d s$ , where $C$ is the line segment joining $( - 5 , - 7 )$ to $( 3,1 )$

Evaluate the line integral over the given curve $C$ . $\int _ { C } 4 x y d s$ , where $C$ is the line segment joining $( - 2 , - 1 )$ to $( 4,5 )$

Use Green's Theorem to evaluate the line integral along the positively oriented closed curve $C$ . $\oint _ { C } 3 x y d x + 4 x ^ { 2 } d y$ , where $\mathrm { C }$ is the triangle with vertices $( 0,0 ) , ( 3,4 )$ , and $( 0,4 )$ .

A thin wire in the shape of a quarter-circle $\mathbf { r } ( t ) = 6 \cos t \mathbf { i } + 6 \sin t \mathbf { j } , 0 \leq t \leq \frac { \pi } { 2 }$ , has a linear mass density $\pi ( x , y ) = 2 x + 4 y$ . Find the mass and the location of the center of mass of the wire.

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 $B \left( x _ { 1 } , y _ { 1 } , z _ { 1 } \right)$ . $\mathbf { F } ( x , y , z ) = 24 x ^ { 3 } y \mathbf { i } + \left( 6 x ^ { 4 } + 4 y z ^ { 2 } \right) \mathbf { j } + 4 y ^ { 2 } z \mathbf { k } ; A ( 0,0,0 )$ and $B ( 0,0 , - 2 )$ Select the correct answer.

Which plot illustrates the vector field $\mathbf { F } ( x , y , z ) = \mathbf { k }$ ?

Evaluate the surface integral. Round your answer to four decimal places. $\iint _ { S } 8 z d S$ $S$ is surface $x = y ^ { 2 } + 2 z ^ { 2 } , 0 \leq y \leq 1,0 \leq z \leq 1$ .

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

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , that is, find the flux of $\mathbf { F }$ across $S$ . $\mathbf { F } ( x , y , z ) = - 4 z \mathbf { i } + y \mathbf { j } - 3 x \mathbf { k } ; \mathrm { S }$ is the hemisphere $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } ; \mathbf { n }$ points upward

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 $\iint _ { s } 3 \mathbf { a } \cdot \mathbf { n } d S \text {, }$ where a is the constant vector.

Find the mass of the surface $S$ having the given mass density. $S$ is the hemisphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 64 , z \geq 0$ ; the density at a point $P$ on $S$ is equal to the distance between $P$ and the $x y$ -plane. Select the correct answer.