# Math 221: Chapter 15 Exam Study Guide

- Evaluate the line integral. Be sure to check to see if the vector field is conservative and independent path. Three parts.
- Find a vector valued function
**r**(u,v) for each of the indicated surfaces. Specify any restrictions on the parameters and indicate the coordinate system used. Three parts.
- Evaluate the surface integral.
- Parametrize a piecewise smooth curve and then evaluate the line integral.
- Use Green's Theorem to evaluate the line integral.
- Find the flux through the surface.
- Use the divergence theorem to find the outward flux through the solid.
- Use Stoke's theorem to evaluate the line integral.
- Given a parametric surface, find
**r**_{u} and **r**_{v}. Then find a vector **N** that is normal to the surface. Find parametric equations of the normal line at the indicated point. Find the equation of the tangent plane at the indicated point, and approximate area of the surface at the indicated point.

## Notes

- You will have Maxima available for the entire exam. Use it to its fullest, but realize that sometimes it is easier to do things by hand.
- You may also use winplot or dpgraph.
- You will be given a Maxima file with some basic definitions to save time and errors recreating them. Remember that Maxima is case-sensitive, so Norm is different than norm. Here are the definitions that will be provided:

load("vect")$

Curl(F):=ev(express(curl(F)),diff)$

Div(F):=ev(express(div(F)),diff)$

Grad(f):=ev(express(grad(f)),diff)$

Cross(u,v):=express(u~v)$

Norm(u):=sqrt(u.u)$

ratprint:false$

fpprintprec:5$

## Points per problem

# |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
Total |

Pts |
18 |
18 |
8 |
8 |
8 |
8 |
8 |
8 |
20 |
104 |