Elementary topology in Euclidean space. Differential calculus of vector valued functions of a vector variable. Implicit and inverse function theorems, regular surfaces. Optimization, Lagrange multipliers, multivariable Taylor polynomials. Integral calculus with the Jordan measure. Fubini’s theorem, change of variables. Line and surface integrals. Vector calculus in two- and three-dimensions. Green’s theorem, Divergence theorem, Stokes’ theorem. Fourier series. This course is recommended for students interested in proof-based multivariable calculus with balanced emphasis between theory and applications.