'A concise and engaging exposition of the mathematics necessary for physics students.' Juan Maldacena, Institute for Advanced Study '... an outstanding addition to the existing stock of books on mathematical methods. It is rigorous, yet readable, and up to date, covering topics like differential forms, which are more and more in use in many areas of physics. An invaluable part of the book that contributes greatly to its pedagogical mission is the vast collection of exercises and their solutions.' R. Shankar, Yale University 'This book takes the physics student along a well-planned trip through mathematics from high school geometry to graduate-level tensor calculus. The key concepts are introduced with a degree of care and precision that is unusual in a book for physicists - but the precision is well motivated, so not at all intimidating. The book is up to date in its contents, especially as it includes the calculus of differential forms, now an essential tool in the professional physicist's toolbox.' Michael Stone, University of Illinois 'It contains all the material required in the undergraduate curriculum and emphasises a pedagogical strategy that presents a unified approach of concepts and methods, whereas most mathematics books for physicists tend to concentrate on methods.' Mike Glazer, Acta Crystallographica