A mathematician unacquainted with tensor calculus is at a serious disadvantage in several fields of pure and applied mathematics. He is cut off from the study of Riemannian geometry and the general theory of relativity. Even in Euclidean geometry and Newtonian mechanics (particularly the mechanics of continua) he is compelled to work in notations which lack the compactness of tensor calculus. This book is intended as a general brief introduction to tensor calculus. As treatments of tensor calculus directed towards relativity are comparatively numerous, relativity has been excluded almost completely, and the aplications to classical mathematical physics emphasized. However, by using a metric which may be indefinite, an adequate basis for applications to relativity has been given.