No Title

Math 223 Name
Fall 1997
Test 4

Directions: Please read each question carefully and show all your work in doing each question. No partial credit will be given if you do not show your work. If you have any questions about the test, please ask me. Your work should of course be your own. Good Luck!

1.

(10 points) On the gradient field below sketch and label 4 possible contours of an associated potential function.

2.

(15 points) Suppose the wind exerts a force on a sailboat given by the vector field
$\vec{F} = y \vec{i} + 4x \vec{j}$ .Find the amount of work done by the wind to move the sailboat along the path y = x² from (0,0) to (1,1).

3.

(15 points) Consider the function f(x,y,z) = x³ - 4x² y + y⁴. Find $\displaystyle \int_{C} \nabla f \cdot d \vec{r}$ where C is part of a helix given by $x = 5 \cos{t}$ , $y = 5 \sin{t}$ ,and $z = \frac{t}{2 \pi}$ for $0 \leq t \leq 5 \pi$ .

4.

(20 points) Consider the vector field $\vec{F} = 2xy \vec{i} + (x^{2} + \frac{1}{y}) \vec{j}$ .

(a): Compute $curl \vec{F}$ . Can you determine from your computation whether or not $\vec{F}$ has a potential function? Explain your conclusion.
(b): Try to construct a potential function for $\vec{F}$ . Is the result of your attempt consistent with your answer above? Why or why not?

5.

(15 points) Given $\vec{F} = -x^{2} y \vec{i} + y^{2}x \vec{j}$ ,find $\displaystyle \int_{C} \vec{F} \cdot d \vec{r}$ where C is a circle of radius 2 centered at the origin with counterclockwise orientation.

6.

(10 points) Without actually computing the flux, determine whether the flux of $\vec{F} = \vec{j} + \vec{k}$ through the surmface z = 4 for $0 \leq x \leq 2$ and $0 \leq y \leq 2$ with upward orientation is positive, negative, or zero. Explain your reasoning.

7.

(15 points) Evaluate $\displaystyle \int_{S} \vec{F} \cdot d \vec{A}$ where $\vec{F} = -x \vec{i} - y \vec{j}$ and S is the surface given by
f(x,y) = x² + y² for $1 \leq x \leq 2$ and $0 \leq y \leq 1$ with upward orientation.

Vector Calculus
8/21/1998