TABLE OF CONTENTS In index notation, a vector u is represented as u i, a two-dimensional matrix A is represented as A ij, and an N-dimensional matrix B is represented as B ij m, where the number of subscripts i through m is equal to N. The same index may not appear more than twice in an index array or across index arrays in a product.
Einstein's summation convention reckons that, if a subscript appears twice in a variable or across a product of variables, then summation is implied over that subscript in its range. For example, under this convention
The vector or matrix nature of an expression is determined by the number of its free indices, that is, the indices that appear only once. For example, A iju j is a vector, whereas u iu j and u iA ijl are two-dimensional matrices.
Kronecker's delta ? ij represents the N N identity or unit matrix; ? ij = 0 when i ? j and ? ij = 1 when i = j. On the basis of this definition we obtain
If x i is a set of N independent variables, then
The Einstein summation convention is implied in the last case.
The alternating matrix ? ijk, where the indices take the values 1, 2,...
