Final Exam, Fall 1998, Math 223

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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) = exy 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

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

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 thanshorter 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

\begin{displaymath}\int_C \, \mbox{grad} \, f \cdot d\vec{r}.
\end{displaymath}

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

\begin{displaymath}\vec{F} = \frac{\vec{r}}{\vert\vert \vec{r} \vert\vert^3}.
\end{displaymath}

(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.


Vector Calculus
1999-04-26