# Final exam review page

**Review session. **Josh will hold a review session next week 6-7 in SC 221 (during the usual section time). Please email Josh with questions/topics that you'd especially like to have discussed.

**Review material. **The exam will cover material from the entire semester.

**Continuity.**- limits, algebra of limits, epsilon-delta proofs of limits, continuity
- Topology of R^n, least upper bound property, supremum, infimum; open, closed, interior, exterior, boundary; open covers, compactness, Heine-Borel
- continuity theorems: intermediate value theorem, boundedness theorem, maximum value theorem

**Differentiability and differentiation.**- Derivative definition (in 1-d and higher dimensions), algebra of derivatives, chain rule; mean value theorem; directional derivatives, partial derivatives, continuous partials theorem, C^r function; maximum values and the derivative
- Inverse function and implicit function theorems
- Manifolds, tangent spaces, manifold recognition, Lagrange multipliers.

**The integral and integration.**- Defining the integral, partitions, upper/lower sums, upper/lower integral, measure 0 and content 0, integrability criterion theorem.
- Tools for integration: fundamental theorem of calculus, Fubini's theorem, change of variables.

**Stokes theorem.**- Forms on R^n and differential forms on open subsets of R^n. elementary forms, wedge product, pullbacks, exterior derivatives, closed and exact forms; k-cubes and k-chains and the boundary map; integration of k-forms over k-chains.
- Stokes' theorem and applications. Winding numbers, fundamental theorem of algebra; Green's theorem.

List of terminology you should know. List of theorems you should know.

**Practice problems. **You might like to try the first practice problems as a warm up before doing practice exams 1 and 2. practice problems, practice exam 1, practice exam 2

solutions 1, solutions 2 (there is a discrepancy between practice exam 1 problem 3(c) and the solutions problem. sorry about that!)

solutions to assorted practice problems