Studies in Theoretical Physics, Volume 1: Fundamental mathematical methods is the first of the six-volume series in theoretical physics. It provides the mathematical methods that any physical sciences and engineering undergraduate might need in upper-division courses in classical mechanics, quantum mechanics, and electricity and magnetism. It is a book designed for a two-semester course in Mathematical Methods. The first eight chapters introduce series and sequences, complex numbers, functions and series, vectors, matrices and determinants, and differential and integral calculus of scalar and vector functions, suitable for the first semester of a two-semester sequence. The last eight chapters introduce the calculus of variations, linear algebra, power series substitution method for solving differential equations, partial differential equations, contour integrals, Laplace transforms, Fourier series, and Fourier transforms, suitable for the second semester. Unlike other books, this book has a section at the end of each chapter that focuses on solving problems using Mathematica.