4 - Tensor Calculus on Manifolds

Abstract

Tensor calculus was the culmination of pioneering work by B. Christoffel and G. Ricci, in 1869 and 1887–1896, respectively. It reached maturity in a joint publication by Ricci-Curbastro and Levi-Cività in 1900. It is difficult to overstate its importance as a field – general relativity could not exist without it. Tensor calculus plays an essential role in every area of physics; it is also crucial for continuum mechanics and many other engineering sciences, and it, of course, lies at the heart of differential geometry. Section 4.2 of this chapter is purely algebraic (with some algebraic topology in section 4.2.6). The mathematical objects that physicists call tensors are more precisely tensor fields; they are only introduced in section 4.3. Historically, “tensors” were presented to students as exotic mathematical millipedes of the form tj1, …, jpi1, …, ip (or more briefly) that behave in a certain way under change of coordinates (see [4.3, 4.4]). This stemmed from the decision to only define the components of the tensor (see [4.2]). As a result, students could complete an entire course of tensor calculus without finding a satisfactory answer to the question: “But what actually is a tensor?” We will, of course, choose a different approach by defining tensor fields before attempting to study them.