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