 This book is designed as a companion to the graduate level physics texts on classical mechanics, electricity, magnetism, and quantum mechanics. It grows out of a course given at Columbia University and taken by virtually all first year graduate students as a fourth basic course, thereby eliminating the need to cover this mathematical material in a piecemeal fashion within the physics courses. The two volumes into which the book is divided correspond roughly to the two semesters of the full..year course. The consolidation of the mathematics needed for graduate physics into a single course permits a unified treatment applicable to many branches of physics. At the same time the fragments of mathematical knowledge possesed by the student can be pulled together and organized in a way that is especially relevant to physics. The central unifying theme about which this book is organized is the concept of a vector space. To demonstrate the role of mathematics in physics, we have included numerous physical applications in the body of the text, as well as many problems of a physical nature.
Although the book is designed as a textbook to complement the basic physics courses, it aims at something lTIOre than just equipping the physicist with the mathematical techniques he needs in courses. The mathematics used in physics has changed greatly in the last forty years. It is certain to change even more rapidly during the working lifetime of physicists being educated today. Thus, the physicist must have an acquaintance with abstract mathematics if he is to keep up with his own field as the mathematical language in which it is expressed changes. It is one of the purposes of this book to introduce the physicist to the language and the style of mathematics as well as the content of those particular subjects which have contemporary relevance in physics.
The book is essentially self..contained, assuming only the standard underel\graduate preparation in physics and mathematics; that is, intermediate mechanics, electricity and magnetism, introductory quantum mechanics, advanced calculus and differential equations. The level of mathematical rigor is generally comparable to that typical of mathematical texts, but not uniformly so. The degree of rigor and abstraction varies with the subject. The topics treated are of varied subtlety and mathematical sophistication, and a logical completeness that is illuminating in one topic would be tedious in another. 


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