Exam 15: Vector Fields

Find the area of the lateral surface over the curve C in the xy-plane and under the surface $z = f ( x , y )$ where Lateral surface area $= \int _ { c } f ( x , y ) d s$ . $f ( x , y ) = 9$ , C: line from $( 0,0 )$ to $( 12,13 )$

Evaluate $\int_{c} \mathrm{F} \cdot d r$ where $F(x, y)=x^{2} y \mathbf{i}+x y^{\frac{3}{2}} \mathbf{j}$ and $\mathbf { r } ( t ) = ( 1 + 8 \cos t ) \mathbf { i } + \left( 9 \cos ^ { 2 } t \right) \mathbf { j } , 0 \leq t \leq \frac { \pi } { 2 }$

$\text { Use Stokes's Theorem to evaluate } \int _ { C } \overrightarrow { \mathbf { F } } \cdot d \overrightarrow { \mathbf { r } } \text {.Use a computer algebra system to }$ verify your results. Note: C is oriented counterclockwise as viewed from above. (x,y,z)=36xz+y+36xy S:z=121--,z\geq0

Evaluate $\iint f(x, y, z) d S$ , where f(x,y,z)=++ S:z=+=16,0\leqz\leq16.

$\text { Let } \mathbf { E } = x \mathbf { i } + y \mathbf { j } + 2 z \mathbf { k }$ be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere $z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ and its circular base in the $x y$ -plane.

Calculate the line integral along $\int_{C} \mathbf{F} \cdot d \mathbf{r}$ for $\mathbf{F}(x, y)=4 x^{2} \mathbf{i}+4 x y \mathbf{j}$ and C is any path starting at the point $( 0,0 )$ and ending at $( 5,5 )$ .

$\text { Find the mass of the surface lamina } S \text { of density } \rho \text {. }$ $S : \sqrt { 64 - x ^ { 2 } - y ^ { 2 } } , \rho ( x , y , z ) = k z$

$\text { Let } \mathbf { F } ( x , y , z ) = \left( x y ^ { 2 } + z \right) \mathbf { i } + \left( x ^ { 2 } y + z \right) \mathbf { j } + e \mathbf { k } \text { and let } S \text { be the surface bounded by }$ $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and $z = 4$ . Verify the Divergence Theorem by evaluating $\iint_{S} \mathbf{F} \cdot \mathbf{N} d s$ as a surface integral and as a triple integral. Round your answer to two decimal places.

$\text { Evaluate } \int _ { C _ { 1 } } y ^ { 3 } d x + \left( 27 x - x ^ { 3 } \right) d y \text {, where } C _ { 1 } \text { is the unit circle given by }$ $\mathbf { r } ( t ) = \cos t \mathbf { i } + \sin t \mathbf { j } , 0 \leq t \leq 2 \pi$

Use Green's Theorem to calculate the work done by the force $\overrightarrow { \mathbf { F } } \text { on a particle that is }$ moving counterclockwise around the closed path C. (x,y)=5xy+(x+y) C:+=16

Find the work done by a person weighing 180 pounds walking exactly one revolution up a circular helical staircase of radius 4 feet if the person rises 14 feet.

$\text { Find } I _ { z } \text { for the lamina } z = x ^ { 2 } + y ^ { 2 } , 0 \leq z \leq 3$ with uniform density of 1 Use a computer algebra system to verify your result.

Find the divergence at $( 4,0,0 ) \text { for the vector field }$ $\mathbf { F } ( x , y , z ) = e ^ { x } \sin y \mathbf { i } - e ^ { x } \cos y \mathbf { j } + z ^ { 2 } \mathbf { k }$

$\text { Use Stokes's Theorem to evaluate } \int _ { C } \overrightarrow { \mathbf { F } } \cdot d \overrightarrow { \mathbf { r } }$ Use a computer algebra system to verify your results. Note: $C$ is oriented counterclockwise as viewed from above. (x,y,z)=xyz+y+z S:z=,0\leqx\leq3,0\leqy\leq3

Determine the tangent plane for the hyperboloid $x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 121 \text { at } ( 11,0,0 )$

$\text { Use Stokes's Theorem to evaluate } \int _ { C } \mathbf { F } \cdot d r \text { where }$ $\mathbf { F } ( x , y , z ) = - \ln \sqrt { x ^ { 2 } + y ^ { 2 } } \mathbf { i } + \arctan \frac { x } { y } \mathbf { j } + \mathbf { k }$ and $S$ is $z = 25 - 2 x - 3 y$ over $r = 2 \sin 2 \theta$ in the first octant. Use a computer algebra system to verify your result.

Determine whether or not the vector field is conservative. $\vec { F } ( x , y ) = 35 x ^ { 6 } y ^ { 6 } \hat { \mathbf { i } } + 30 x ^ { 7 } y ^ { 5 } \hat { \mathbf { j } }$

Find the work done by the force field $\overrightarrow { \mathbf { F } } \text { on a particle moving along the given path. }$ $\overrightarrow { \mathbf { F } } ( x , y ) = - y \hat { \mathbf { i } } - x \hat { \mathbf { j } } , \quad C : y = \sqrt { 25 - x ^ { 2 } }$ from $( 5,0 )$ to $( - 5,0 )$

Use Green's Theorem to calculate the work done by the force $\overrightarrow { \mathbf { F } } ( x , y ) = \left( 18 x ^ { 2 } + y \right) \mathbf { i } + 19 x y ^ { 2 } \mathbf { j }$ on a particle that is moving counterclockwise around the closed path $C$ where $C$ is the boundary of the region lying between the graphs of $y = \sqrt { x } , y = 0$ , and $x = 9$ . Round your answer to two decimal places.

Evaluate the line integral using the Fundamental Theorem of Line Integrals. Use a computer algebra system to verify your results. $\int _ { C } ( 7 y \hat { \mathbf { i } } + 7 x \hat { \mathbf { j } } ) \cdot d \overrightarrow { \mathbf { r } }$ C: a smooth curve from $( 0,0 )$ to $( 2,4 )$