Tensor operations and Einstein summation convention

If we want to specify a component of a tensor in a certain coordinate system, we use the index notation. Each tensor order requires its own index. Usually, lowercase letters are used as indices, starting with i (uj, Aij, Algebraic rules are frequently written down in index notation. Although the components themselves depend on the coordinate system, the rules are nevertheless valid in all systems. 

Everywhere in this book, we use a right-handed Cartesian coordinate system with perpendicular coordinate axes and unit vectors of length 1. If this is not done, the notation becomes much more cumbersome. Nevertheless, this step has to be done sometimes e. g., when dealing with large deformations (see section 3.1). 

If several coordinate systems are used, they are distinguished by adding primes to the indices. Thus, a representation in the xx-x -x- coordinate system may be written as a, Aij, and Cijki, changing to a /, Ai j , and Ciij k i in the X11-X21-X31 coordinate system. It is important to add the prime to each index because tensors might be written using indices mixed from different systems e. g., Aij or Ai j. 

If all components of a tensor are to be described, this is done by adding parentheses to the tensor written in index notation, for example, (a ), Aij), Cijki). Implicitly, it is assumed that each index runs from 1 to 3. In second-order tensors, the first index denotes the row and the second index denotes the column of the component in the matrix notation. The components of the tensor 

One of the most important tensor operations is the product. We can write the so-called (single ) contraction of two tensors as 

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A.4 Tensor operations and Einstein summation convention 453 and the value of the order below (4, B, ) Other typographic conventions... 

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