Multivariable Calculus and Matrix Algebra, MATH2010

Second Exam, Thursday, February 28, 2002.

You may use one sheet of handwritten notes, but no other sources. The exam consists of four questions, and lasts fifty minutes. Please work all four problems. Please show all work clearly and in reasonable detail. Answers without appropriate supporting work or requested explanations may not receive full credit.

1.

(20 points) Sketch several representative vectors in the vector field

<I>F= y² <I>i- 3x <I>j.

You should draw at least two vectors in each quadrant, you should draw vectors at different distances from the origin, and you should draw some vectors that do not correspond to points on the axes.

2.

(40 points)

(a)

(20 points)  Let <I>F=(4xy-z²)<I>i+ (2x²+3z)<I>j+ (3y-2xz)<I>k. Show that <I>F is conservative.

(b)

(20 points) Let C be the curve given by (t,t²,t⁴) for $0 \leq t \leq 1$. Evaluate $\oint_C {\mathbf F}d{\mathbf r}$, where F was given in part (2a).

3.

(20 points) Evaluate $\int_C e^{x+y} ds$ along the straight line path between the origin and the point (2,1).

4.

(20 points) Define the line integral

$$I := \oint_C 3\ln(y^2+x^2) dx,$$

where C is a closed curve made up of two parts:

• Represented in polar coordinates by $r=4\sin(\theta)\cos(\theta)$, $\frac{\pi}{6}\leq \theta \leq \frac{\pi}{3}$, traversed in the counterclockwise direction.
• Followed by the boundary of the circle of radius $\sqrt{3}$, from $\frac{\pi}{3}$ back down to $\frac{\pi}{6}$.

(Note that when $\theta=\frac{\pi}{6}$, we get the point $(\frac{3}{2},\frac{\sqrt{3}}{2})$, and when $\theta=\frac{\pi}{3}$, we get the point $(\frac{\sqrt{3}}{2},\frac{3}{2})$. See the picture on the lecture screen.)

Show that

$$I = \int_{\pi/6}^{\pi/3} a \sin^2 \theta \cos \theta + b \sin \theta d\theta$$

for some constants a and b. What are the values of a and b?