Introduction to Tensor Analysis and the Calculus of Moving Surfaces

Tensor calculus is a must for researchers dealing with natural phenomena as well as for highly qualified engineers working with man-made technological equipment. This textbook was conceived as a self-contained introduction to tensor calculus and grown out of the lecture notes for a course continually taught by the author at Drexel University. The text contains over 150 exercises and is divided into four parts. The first part focuses on tensors in Euclidean spaces, and includes chapters on subjects such as covariant differentiation. The second part focuses on tensors in embedded surfaces, and includes chapters on the curvature tensor and Gauss's theorem. The third part covers applications of tensor calculus and contains chapters on equations of classical mechanics, equations of continuum mechanics, and Einstein's Theory of Relativity. The fourth and final chapter explains the rules and applications of the calculus of moving surfaces. Though the book's approach is informal and avoids a formalization of the subject, it maintains a respectable level of rigor and reflects the author's deep passion for the subject. Moreover, it focuses on concrete objects and appeals to the reader's intuition with regard to such fundamental concepts as the Euclidean space, surface, and length. It is intended for advanced undergraduate students (and first-year graduate students) in technical fields, and assumes a solid understanding of linear algebra and multivariable calculus.

Pavel Grinfeld is currently a professor of mathematics at Drexel University, teaching courses in linear algebra, tensor analysis, numerical computation, and financial mathematics. Drexel is interested in recording Grinfeld's lectures on tensor calculus and his course is becoming increasingly popular. The author has published in a number of journals including 'Journal of Geometry and Symmetry in Physics' and 'Numerical Functional Analysis and Optimization'. Grinfeld received his PhD from MIT under Gilbert Strang.