Final Exam, Fall 1998, Math 223

Main menu / Final Exams / Fall 1998 Final Solutions Answer the questions in the space provided. You must show your work or explain your solution; otherwise points may be deducted. If you make and unnecessary approximation in your solution to a problem, your answer will be judged on its accuracy. Points may be deducted for poor or inappropriate approximation. The points for each problem are as follows: 1 and 2 (12 pts), 3 (6 pts), 4-7 (10 pts each), 8 and 9 (15 pts each).

1.

Label the following statements T (True), F (False), or C (Can't tell without further information.) No reasons needed. Assume that $\vec{F}$ is an arbitrary vector field and f(x,y,z) is an arbitrary function.

(a)

Curl $\vec{F}$ is a vector field.

(b)

Flux of a vector field is a vector.

(c)

grad f is parallel to the level surface f(x,y,z) = c.

(d)

$\int_C \vec{F} \cdot d\vec{r} = 0$ whenever C is a closed curve.

(e)

$(\vec{v} \times \vec{w}) \times \vec{v}$ is the zero vector for any vectors $\vec{v}$ and $\vec{w}$.

(f)

div (grad f) is a vector.

(g)

$\int_C \mbox{grad} \, f \cdot d\vec{r} = 0$ if C is a closed curve.

(h)

The gradient of a scalar function is a scalar.

(i)

A line integral is a scalar.

(j)

A directional derivative is a vector.

(k)

$- \frac{\vec{u}}{\vert\vert \vec{u} \vert\vert}$ is a unit vector if $\vec{u} \neq \vec{0}$.

(l)

$\frac{\partial f}{\partial x}$ is a vector.

2.

No partial credit, but you must show your work. Put your answer in the space provided.

(a)

If $\vec{u} = \vec{i} + 2\vec{j} - \vec{k}$ and $\vec{w} = -\vec{i} + \vec{j}$ find $\vec{u} \cdot \vec{w}$.

(b)

If f(x,y) = e^xy find grad f.

(c)

If $\vec{F} = 3\vec{r}$ (in three dimensions) find div $\vec{F}$.

(d)

If $\vec{F} = 3\vec{r}$ (in three dimensions) find curl $\vec{F}$.

3.

According to the Theory of Relativity, the energy, E, of a body of mass m moving with speed v is given by the formula

$$E = mc^2 \left( \frac{1}{\sqrt{1 - v^2/c^2}} - 1 \right).$$

The speed, v, is non-negative and less than the speed of light, c, which is a constant.

(a)

Find $\frac{\partial E}{\partial v}$.

(b)

Explain what you would expect the sign of $\frac{\partial E}{\partial v}$ to be and why.

4.

Write a triple integral representing the volume above the cone $z = \sqrt{x^2 + y^2}$ and below the sphere of radius 2 centered at the origin. Include limits of integration, but do not evaluate:

(a)

Use cylindrical coordinates.

(b)

Use spherical coordinates.

5.

The sketch shows level curves of the function f(x,y).

(a)

On the diagram above, sketch a vector at P in the direction of grad f.

(b)

The length of the vector grad f at P is

longer than

shorter than

the same length as

the length of grad f at Q? (Circle one) Give a reason.

(c)

If C is a curve going from P to Q, evaluate

$$\int_C \, \mbox{grad} \, f \cdot d\vec{r}.$$

Show your reasoning.

6.

The vector field, $\vec{F}$, sketched above, has no z-component and is independent of z.

(a)

Do you think that $\vec{F}$ is a conservative (that is, path independent) vector field? Justify your answer using the sketch.

(b)

Do you think that curl $\vec{F} \cdot \vec{k}$ is positive, negativeor zero? Explain.

(c)

Give a possible formula for $\vec{F}$.

7.

Curves C₁ and C₂ are parameterized as follows C₁ is (x(t), y(t)) = (0,t) for $-1 \le t \le 1$ C₂ is $(x(t), y(t)) = (\cos t, \sin t)$ for $\frac{\pi}{2} \le t \le \frac{3\pi}{2}$

(a)

Sketch C₁ and C₂ on the axes below, with arrows showing their orientation. Label your sketch clearly.

(b)

Suppose $\vec{F} = (x + 3y) \vec{i} + y\vec{j}$. Calculate $\int_C \vec{F} \cdot d\vec{r}$, where C is the curve given by C = C₁ + C₂. Show your work.

8.

Consider the vector fields $\vec{F} = \vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$ and $\vec{G} = 2\vec{j} + 3\vec{k}$. Evaluate the following. No partial credit, but you must show your reasoning.

(a)

$\int_{C_1} \vec{F} \cdot d\vec{r}$ where C₁ is the unit circle in the xy-plane, orineted counterclockwise.

(b)

$\int_{C_2} \vec{G} \cdot d\vec{r}$ where C₂ is the y-axis from the origin to the point (0,10,0).

(c)

$\int_{S_1} \vec{F} \cdot d\vec{A}$ where S₁ is the cylinder x² + y² = 1 with $0 \le z \le 1$, oriented outward.

(d)

$\int_{S_2} \vec{G} \cdot d\vec{A}$ where S₂ is the unit sphere oriented outward.

9.

Let $\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$ and let $\vec{F}$ be the vector field given by

$$\vec{F} = \frac{\vec{r}}{\vert\vert \vec{r} \vert\vert^3}.$$

(a)

Calculate the flux of $\vec{F}$ out of the unit sphere x² + y² + z² = 1 oriented outward. Show your work.

(b)

Calculate div $\vec{F}$. Show your work and simplify your answer completely.

(c)

Use your answers to part (a) and (b) to calculate the flux of $\vec{F}$ out of a box of side 10 centered at the origin with sides parallel to the coordinate planes. (The box is also oriented outward.) Give reasons for your answer.