Mat 241 Chapter 16 Exam B

Mat 241 Chapter 16 exam BKEY

Fall, 2013 Name ___________________________

Directions: Show all work for each question and make sure your answers are clearly identified. You may use the back side of pages if needed.

#1. (5 pts) Use the Divergence Theorem to calculate the surface integral with the given vector field

on the extremely uncool surface of the bounded solid created by the functions:

Assume outward pointing orientation.

#2. Suppose we wish to compute the work done on a particle as it traverses the helical path shown in the figure defined by:

This requires a line integral.

If we are lucky, the field is a gradient field and is therefore independent of path (conservative). Use the vector field, , for questions A, B, and C.

A. (2pts) Compute to show that is a conservative vector field.

B. (2pts) Find a potential function,, such that .

C.(2pts) Compute the work using your result from part B and the fundamental theorem of line integrals:

#3. (6pts) I went over to the tutor center with the following very nasty vector field

I wanted to compute the work along the very simple curve, C, parameterized by:

where C is oriented counterclockwise as viewed from the positive z – axis. A tutor named Peter said, “Hey, you can use Stokes’ Theorem” with the following surface.

Looking at the surface a light-bulb went off in my head! I can use any surface which the curve bounds with Stokes’ Theorem.

Part1: (3 points) Instead of Peter’s ridiculous surface, choose a VERY simple surface and compute the work around the curve.

Since we really just have a circle in the xy-plane, let’s choose the surface to be z = 0. Our space curve certainly contains it.

Then, the normal to the surface z = 0 pointing upward is given by:

So by Stokes’ Theorem;

Part2: (3 points) Compute the line integral directly using.

#4. (5pts)

I have decided to divulge the top-secret equation that draws the Hershey Kiss candy. Here it is:

Use Green’s Theorem in reverse to determine the cross-sectional area of the Kiss!

Sorry, guys….wrong Kiss

We need to select a vector field, , such that . There are three choices which our book presents:

We’ll use the first.

Then the line integral value will be equivalent to the area of the Kiss.