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.‎