Fall 1997

Final Exam

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1.

The diagram in the figure below shows the contour map for a circular island. Sketch the vertical cross-section of the island through the center. Your sketch should show concavity clearly.

2.

Let $\vec{a}$ and $\vec{b}$ be any two nonzero vectors in 3-space, which are not parallel to one another. Write expressions for vectors representing the following:

(a)

A unit vector parallel to $\vec{a}$.

(b)

A vector perpendicular to $\vec{a}$ and $\vec{b}$.

(c)

What is the value of $\vec{a} \cdot (\vec{a} \times \vec{b})$? Why?

3.

In an electric circuit, two resistances, R₁ and R₂, are hooked up so that their combined resistance, R, is given by R = 1/R₁ + 1/R₂.

(a)

Find $\partial R/\partial R_1$.

(b)

Suppose $\partial R/\partial R_1 = 0.1$ for some values of R₁ and R₂. What does this tell you, in terms of resistances?

4.

Suppose that T(x,y) = x² + 2y² - x is the temperature at the point (x,y). If you are standing at the point (-2,1) and decide to proceed in the direction of the point (1,-3), will the temperature be increasing or decreasing at the moment you begin?

5.

Let $h(x,y) = x^3 + y^3 + 3xy + \frac{1}{8}$.

(a)

Determine all local maxima, minima, and saddle points. Are the local extrema also global extrema?

(b)

Match the function h(x,y) in part (a) to the plot of its level curves. Explain.

6.

We want to compute the volumes of the objects below. Give the integrand and the limits of integration in each case. (No reasons need be given. Note that $\rho, \phi, \theta$ are spherical coordinates and that $r, \theta, z$ are cylindrical coordinates.)

(a)

A wedge of a cantaloupe, cut from a perfect sphere.

$$\mbox{Volume} \thinspace = \int_a^b \int_c^d \int_e^f g(\rho, \phi, \theta) d\rho d\phi d\theta$$

$g(\rho, \phi, \theta) =$

$a = \qquad \qquad \qquad b =$

$c = \qquad \qquad \qquad d =$

$e = \qquad \qquad \qquad f =$

(b)

A spherical bead with a cylindrical hole cut through the center.

$$\mbox{Volume} \thinspace = \int_a^b \int_c^d \int_e^f g(r, \theta, z) dz dr d\theta$$

$g(r, \theta, z) =$

$a = \qquad \qquad \qquad b =$

$c = \qquad \qquad \qquad d =$

$e = \qquad \qquad \qquad f =$

7.

Consider the plane 2x + y - 5z = 7 and the line with parametric equation $\vec{r} = \vec{r}_0 + t\vec{u}$.

(a)

Give a value of $\vec{u}$ which makes the line perpendicular to the plane.

(b)

Give a value of $\vec{u}$ which makes the line parallel to the plane.

(c)

Give values for $\vec{r}_0$ and $\vec{u}$ which make the line lie in the plane.

8.

Let

$$\vec{F} = 3x \hat{i} + 5y \hat{j} \thinspace \mbox{and} \thinspace \vec{G} = 3y \hat{i} + 5x \hat{j}.$$

Let C₁ be the circle with center (2,2) and raius 1, oriented counterclockwise. Let C₂ be the path consisting of the straight line segment from (0,4) to (0,1), followed by the straight line segment from (0,1) to (3,1). Find the following line integrals. Explain your reasoning.

(a)

$\int_{C_1} \vec{F} \cdot d\vec{r}$

(b)

$\int_{C_2} \vec{F} \cdot d\vec{r}$

(c)

$\int_{C_1} \vec{G} \cdot d\vec{r}$

(d)

$\int_{C_2} \vec{G} \cdot d\vec{r}$

9.

Consider the vector field

$$\vec{F}(\vec{r}) = \frac{\vec{r}}{r^3}, \quad \vec{r} \neq \vec{0},$$

where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ and $r = \vert\vert\vec{r}\vert\vert$.

(a)

Show that div $\vec{F} = 0$ for $\vec{r} \neq \vec{0}$.

(b)

Let S be the surface of a ball centereed at the origin. LEt $\vec{F}$ be the same vector field as in part (a). Do the Divergence Theorem and your answer to part (a) imply that $\int_S \vec{F} \cdot d\vec{A} = 0$? Explain why or why not.

10.

Suppose W is the object consisting of two solid cylinders meeting at right angles at the origin. One cylinder is centereed on the y-axis, between y = -5 and y = 5 with radius 2 and the other is centered on the x-axis between x = -5 and x = 5 with radius 2. Let S be the whole surface of W except for the circular end of the cylinder centered at (0,5,0). The boundary of S is a circle, C; the surface S is oriented outward. Let $\vec{F} = (3x^2 + 3z^2)\hat{j}$. You are also told that $\vec{F} = \mbox{curl}(z^3\hat{i} + y^3\hat{j}-x^3\hat{k})$.

(a)

Suppose you want to calculate $\int_S \vec{F} \cdot d\vec{A}$.Write down two other integrals which have the same value as $\int_S \vec{F} \cdot d\vec{A}$. Justify your answer.

(b)

Find $\int_S \vec{F} \cdot d\vec{A}$ by whatever method is easiest.