Page 2 - Intro to Tensor Calculus

PREFACE

This is an introductory text which presents fundamental concepts from the subject
areas of tensor calculus, differential geometry and continuum mechanics. The material
presented is suitable for a two semester course in applied mathematics and is flexible
enough to be presented to either upper level undergraduate or beginning graduate students

majoring in applied mathematics, engineering or physics. The presentation assumes the
students have some knowledge from the areas of matrix theory, linear algebra and advanced
calculus. Each section includes many illustrative worked examples. At the end of each
section there is a large collection of exercises which range in difficulty. Many new ideas
are presented in the exercises and so the students should be encouraged to read all the

exercises.
The purpose of preparing these notes is to condense into an introductory text the basic
definitions and techniques arising in tensor calculus, differential geometry and continuum

mechanics. In particular, the material is presented to (i) develop a physical understanding
of the mathematical concepts associated with tensor calculus and (ii) develop the basic
equations of tensor calculus, differential geometry and continuum mechanics which arise
in engineering applications. From these basic equations one can go on to develop more
sophisticated models of applied mathematics. The material is presented in an informal

manner and uses mathematics which minimizes excessive formalism.
The material has been divided into two parts. The first part deals with an introduc-
tion to tensor calculus and differential geometry which covers such things as the indicial
notation, tensor algebra, covariant differentiation, dual tensors, bilinear and multilinear

forms, special tensors, the Riemann Christoffel tensor, space curves, surface curves, cur-
vature and fundamental quadratic forms. The second part emphasizes the application of
tensor algebra and calculus to a wide variety of applied areas from engineering and physics.
The selected applications are from the areas of dynamics, elasticity, fluids and electromag-

netic theory. The continuum mechanics portion focuses on an introduction of the basic
concepts from linear elasticity and fluids. The Appendix A contains units of measurements
from the Syst`me International d’Unit`es along with some selected physical constants. The
e
Appendix B contains a listing of Christoffel symbols of the second kind associated with

various coordinate systems. The Appendix C is a summary of useful vector identities.

J.H. Heinbockel, 1996