First-order differential equations -- Second-order differential equations -- The Laplace transform -- Series solutions -- Numerical approximation of solutions -- Vectors and vector spaces -- Matrices and systems of linear equations -- Determinants -- Eigenvalues, diagonalization, and special matrices -- Systems of linear differential equations -- Qualitative methods and systems of nonlinear differential equations -- Vector differential calculus -- Vector integral calculus -- Fourier series -- The Fourier integral and Fourier transforms -- Special functions, orthogonal expansions, and wavelets -- The wave equation -- The heat equation -- The potential equation -- Geometry and arithmetic of complex numbers -- Complex functions -- Complex integration -- Series representations of functions -- Singularities and the residue theorem -- Conformal mappings -- Counting and probability -- Statistics.