# Math 221: Chapter 16 Study Guide

1. Evaluate the line integral along the parametrically defined curve.
2. Evaluate the line integral wrt to the arclength parameter s along the parametically defined curve.
3. Evaluate the line integral F dot dr along the curve.
4. Evalute the line integral on a line segment between two points.
5. Find div F and curl F.
6. Find the potential function for the conservative vector field F.
7. Find the potential function for the conservative vector field F.
8. Show that the integral is independent of path and find its value.
9. Use Green's theorem to evaluate the line integral.
10. Evaluate the surface integral over the surface represented by a vector valued function r.
11. Find the flux of the vector field F across the surface.
12. Use the Divergence Theorem to find the flux of F across the surface σ with outward orientation.
13. Use Stoke’s Theorem to evaluate the integral.
14. Locate all relative maxima, relative minima, and/or saddle points.
15. Given a parametrically defined surface r(u,v), find the principal unit normal vector, the equation of the tangent plane, and the parametric equations of the normal line at the indicated point.
16. Identify the quadric surface. The equations may be given in rectangular coordinates, cylindrical coordinates, or spherical coordinates. They may also be a parametrically defined surface. Eleven parts.

## Notes

• Questions 1-7 are to be done without a computer and turned in before getting the second part of the exam.
• For questions 8-16, you are welcome to use Derive, Winplot, or DPGraph.
• For the problems where you use the computer, make any sketchs of any regions needed.
• There is a maximum of 207 possible points, but the test is only worth 200 points, so you may miss some problems without adversely affecting your grade. If you do exceptionally well, you may score above 100%.

## Points per problem

 # Pts 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Total 10 10 10 10 10 10 10 15 15 15 15 15 15 10 15 22 207