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.
- 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.
solutions to assorted practice problems