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.