Spring1998

Final Exam

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The following problems are from spring 1998's final exam. Working them should give you a good review of the course as well as give you a feel for the final.

Part I: Short Questions

Answer each of the following short questions. Give a brief justification for your answer.

1.

Here is a table of values of a differentiable function f(x,y):

-1 0 1

1.0 0.7 0.1 -0.5

y 1.2 4.8 4.2 3.6

1.4 8.9 8.3 7.7

Does the gradient vector of f at (0,1.2) point into a) the first quadrant, b) the second quadrant, c) the third quadrant, or d) the fourth quadrant. Justify your answer.

2.

If $\vec{u}$ is a non-zero vector, write an expression for a vector of length 1 which points in the opposite direction to $\vec{u}$. Justify your answer.

3.

Find f_H if $f(H,T) = \frac{2H + T}{(5-H)^3}$. Justify your answer.

4.

Which of the following diagrams represents the curve $x = \cos t, y = \sin t, 0 \le t \le \pi$? Justify your answer.

5.

Which of the vector fields below is the gradient field of f(x,y) = x² + y²? Justify your answer.

6.

True or False? If C is a circle in the plane, and if f(x,y) is not constant when constrained to C, then there must be at least one point on C where grad f is perpendicular to C. Justify your answer.

7.

True or False? If $\vec{F}$ is a gradient field, $\vert\vert\vec{F}\vert\vert = 1$everywhere, and if C is a circle of radius 1 centered at the origin, oriented clockwise, then $\int_C \vec{F} \cdot d\vec{r} = 2\pi$. Justify your answer.

8.

True or False? The angle between $-\hat{i} + 2\hat{k}$ and $\hat{i} + \hat{j} + \hat{k}$ is greaterthan $\pi/2$. Justify your answer.

Part II: Longer Questions

9.

The ideal gas law says that

PV = RT

for a certain fixed amount of a gas (called a mole of gas), where P is the pressure (in atmospheres), V is the volume (in cubic meters), T is the temperature (in degrees Kelvin) and R is a positive constant.

(a)

Find $\frac{\partial P}{\partial T}$ and $\frac{\partial P}{\partial V}$.

(b)

A mole of a certain gas is at a temperature of 298 degrees Kelvin, a pressure of 1 atmosphere and a volume of 0.0245 m³. Calculate $\frac{\partial P}{\partial V}$ for this gas. Give the units of your answer and explain what it means in practical terms.

10.

Consider the volume between a cone centered along the psitive z-axis, with vertex at the origin and containing the point (0,1,1) and a sphere of radius 3 centered at the origin.

(a)

Using spherical coordinates, write a triple integral which represents this volume.

(b)

Evaluate the integral.

11.

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.

12.

Consider the vector field

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

(a)

Find the flux of $\vec{F}$ out of the sphere of radius 1 centered at the origin.

(b)

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

(c)

Use parts (a) and (b), and the Divergence Theorem, to calculate the flux of $\vec{F}$ out of a cube of side 3 centered at the origin. [Note: it is possible to answer this question without a long involved calculation.]

13.

Let $\vec{G} = z^3 \hat{i} + y^3 \hat{j} - x^3 \hat{k}$.

(a)

Calculate curl $\vec{G}$.

(b)

Let C be a simple closed curve lying in the xy-plane. Use your answer to part (a) and Stokes' theorem to show that $\int_C \vec{G} \cdot d\vec{r} = 0$. To receive full credit you must state Stokes' Theorem accurately and explain carefully how it applies.