Hey all, I have to take exams in several areas--one in analysis. The exam will also include things from undergraduate analysis. Here are the specific topics that may pop up: For advanced Calculus: 1. Integration of functions of several variables: line and volume integrals in 2D, line, surface, and volume integrals in 3D. 2. Differentiation: gradient, curl, divergence, Jacobian. Connection between rotation-free vector fields and potential fields. 3. Partial integration, Green’s theorems, Stokes’ theorem, Gauss’ theorem. The consequences of these theorems for vector fields that are divergence or rotation free. 4. The concepts max, min, sup, inf, lim sup, lim inf, lim. 5. Convergence criteria for sequences and series. For Higher Analysis: It will cover metric and normed spaces, banach spaces, hilbert spaces(separable spaces only), and Measure theory. So what books would you recommend for the higher analysis section and the advanced calc. section. I'm more curious about the advanced calc topics. Would Spivak's Calculus on Manifolds be a good choice? Any others? Thanks for the help!