This book should serve as a second course in ODEs for a traditional undergraduate path, or as a first course for advanced undergraduates and early graduate students. Most notably, a full sequence of calculus and linear algebra are necessary prerequisites.

Topics covered include:

• Complex numbers and functions

• Linear algebra (including Jordan form, matrix functions, and the spectral mapping theorem)

• Infinite-dimensional vector spaces, function spaces, and inner product spaces

• The Fourier series, Fourier and Laplace transforms

• The matrix exponential solution to systems of linear ODEs

• Duhamel's principle for nonhomogeneous linear ODEs

• Variation of parameters for nonhomogeneous linear ODEs

• Nonlinear maps, and linearization

• Closed-form solutions to nonlinear ODEs (exact equations and integrating factors)

• Euler, Runge-Kutta, and Galerkin methods for approximating solutions

• Chaos theory, and the Hartman-Grobman theorem for stability of equilibrium points

• The Lie series, and an introduction to PDEs

Both proofs-based, and purely computational exercises are included, with the intent that the course could be administed with-or-without a formal proofs course as a prerequisite.

Furthermore, video lectures (each a guided read-through) corresponding to the entire book may be found on my YouTube channel (see below).

This book should serve as a first course in PDEs for a traditional undergraduate path. Most notably, a full sequence of calculus, linear algebra, and ODEs are necessary prerequisites.

Topics covered include:

• Construction of PDEs from physical principles

• Domains and boundaries in arbitrary dimension

• A review of ODEs solution methods

• Outer products and tensor fields

• The Taylor series for scalar fields

• The Fourier series and transform for scalar fields

• Flows, Hamiltonians, and the method of characteristics

• Construction of the Laplace, heat, and wave equations

• Separation of variables

• Fourier methods for the solution to the Laplace, heat, and wave equations

• Classification of elliptic, parabolic, hyperbolic PDEs

• Integral transforms, the Dirac delta function, and Green's functions

• Eigenvalues and eigenfunctions

• The spectral theorem for Hermitian PDE operators

• The Cauchy-Kovalevskaya theorem and power series methods

• Finite difference and Galerkin methods

Video lectures (each a guided read-through) corresponding to the entire book may be found on my YouTube channel (see below).