Final Study Guide

The Final Exam will contain questions from Chapter 13 plus questions taken from each of the three examinations.

Formulas: the following formulas should be learned for the exam. Any others will be given on the exam or you can ask me for them during the exam.

1. Green, Stokes and Divergence Theorems. (Formulas derived from these, such as Green's area formulas, need not be memorized).
2. Fundamental Theorem of Line Integrals.
3. Gradient, divergence and curl.
4. The area element dS for surface area and integrals.
5. The arclength element ds for lengths of curves and line integrals with respect to arc length.
6. The Jacobian for the main changes of variable: polar coordinates, cylindrical coordinates and spherical coordinates.

The following summary outlines topics you can expect to appear on the exam. The exam itself will probably contain only a proper subset of these topics.

1. Determine when a vector field is conservative. Find its line integral.
2. Use Green's Theorem to evaluate a line integral. Correct orientations.
3. Use Stokes's Theorem to evaluate a line integral.
4. Evaluate line and surface integrals: find parametrizations of curves and surfaces. This includes flux integrals of vector fields as well as finding the area of surfaces.
5. Double and triple integrals and their relationship to interated integrals: Fubini's Theorem. Find the limits of integration.
6. Find an equation of the tangent plane to F(x,y,z)=constant.
7. Find an equation of the tangent line to f(x,y)=constant.
8. Area of regions enclosed by polar equations: r = f(theta).
9. Max-min problems and the method of Lagrange multipliers.
10. Lines in space: symmetric equations, vector representations.
11. Planes in space: equation or vector representation.
12. Dot and cross products of vectors. Scalar triple product.
13. Parametrize a line from point P to point Q.
14. Parametrize a circle in the plane with center P and radius a>0.
15. Work