Part I: Real numbers and limits: Numbers and logic Infinity Sequences Subsequences Functions and limits Composition of functions Part II: Topology: Open and closed sets Compactness Existence of maximum Uniform continuity Connected sets and the intermediate value theorem The Cantor set and fractals Part III: Calculus: The derivative and the mean value theorem The Riemann integral The fundamental theorem of calculus Sequences of functions The Lebesgue theory Infinite series $\sum_{n=1}^\infty a_n$ Absolute convergence Power series The exponential function Volumes of $n$-balls and the gamma function Part IV: Fourier series: Fourier series Strings and springs Convergence of Fourier series Part V: The calculus of variations: Euler's equation First integrals and the Brachistochrone problem Geodesics and great circles Variational notation, higher order equations Harmonic functions Minimal surfaces Hamilton's action and Lagrange's equations Optimal economic strategies Utility of consumption Riemannian geometry Noneuclidean geometry General relativity Partial solutions to exercises Greek letters Index.