Spring1998

Final Exam


Main menu / Final Exams / Spring 1998 Final Solutions

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):

      x  
    -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 fH 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

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

(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.


Vector Calculus
8/20/1998