Convergence of infinite series / Taylors theorem and its applications / Series expansions / Proof of d'Alembert's ratio test / Alternating & other series / Complex numbers / Complex plane: Argand diagrams / Complex variables & series / Eulers formula / Partial derivatives / Differentials / Change of variables / Taylor series / Stationary points / Lagrange multipliers / Differentiation of integrals / Scalars & vectors / Products of vectors / Applications to geometry / Differentiation & integration / Determinants / Vectors in n Dimensions / Matrices & linear transformations / Square Matrices / The eigenvalue equation / Diagonalisation of matrices / Line integrals / Double integrals / Curvilinear co-ordinates in three dimensions / Triple or volume integrals / Scalar & vector fields / Line, surface, & volume integrals / The divergence theorem / Stokes theorem / Fourier series / Complex Fourier series / Fourier transforms / First-order equations / Linear ODEs with constant coefficients / Eulers equation / Series solutions / Eigenvalue equations / Legendres equation / Bessels equation / Some important PDEs in physics / Separation of variables: Cartesian co-ordinates / Separation of variables: polar co-ordinates / The wave equation: d'Alembert's solution / Euler Equations / Boundary conditions and uniqueness.