Page 18 - Introduction to Continuum Mechanics

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Tensors

As mentioned in the introduction, all laws of continuum mechanics must be formulated in
terms of quantities that are independent of coordinates. It is the purpose of this chapter to
introduce such mathematical entities. We shall begin by introducing a short-hand notation
- the indicial notation - in Part A of this chapter, which will be followed by the concept of
tensors introduced as a linear transformation in Part B. The basic field operations needed for
continuum formulations are presented in Part C and their representations in curvilinear
coordinates in Part D.
Part A The Indicial Notation

2A1 Summation Convention, Dummy Indices

Consider the sum

We can write the above equation in a compact form by using the summation sign:

It is obvious that the following equations have exactly the same meaning as Eq. (2A1.2)

etc.

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