3.3-3.4 (HO L8) Tensors, Kronecker delta PDF

Preview

Summary

Lecture notes...

Description

3.3 Tensors In index notation, a scalar quantity has no free indices, while a vector quantity has one free index. The concept of a coordinate-invariant quantity can be generalized to objects with an arbitrary number of indices, known as tensors. A tensor with two indices (called a second-rank tensor) can be represented by a 3 x 3 matrix,

with Aij denoting the element in the ith row and jth column. Matrix multiplication can then be represented, in index notation, as

Any product of two vectors or tensors that share a dummy index, e.g. aij uj, is called an inner product (the generalization of a scalar product).

3.4. The Kronecker delta The Kronecker delta is a special tensor, defined as -(3.5)

Represented as a matrix, it corresponds to the identity matrix

Properties of the Kronecker delta (1) It is symmetric: (2) If i or j is a free index, dij acts as a substitution operator. e.g.



dij aj = ai

-(3.6)

Hence the action of dij on aj is to replace the index j with the index i. (3) dij can be used to represent the scalar product thus:...