This book is a written version of the lecture course I have given over a number of years to first-year graduate students at the University of Illinois on the subject of mathematical methods for physics. The course (and the book) are intended to provide the students with the basic mathematical background which they will need to perform typical calculations in classical and quantum physics. The level is intermediate; the usual undergraduate course in advanced calculus should be an adequate prerequisite and would even provide some overlap (e.g. Fourier series) with the subjects covered in the present work. The treatment is limited to certain standard topics in classical analysis; no attempt is made to cover the method of characteristics, Hilbert space, or group theoretical methods. What I have tried to do is provide a short readable textbook from which the average physics or engineering student can learn the most important mathematical tools he will need in his professional career. The physics which lies behind the mathematical problems is all explained in some detail, so that the treatment should be intelligible also to pure mathematicians and might even provide an introduction to some of the advanced texts by mathematicians on the subject.