Vectors - The Kronecker Delta and Einstein Summation Convention

Sabtu, 08 Februari 2020

The Kronecker Delta tensor is defined

This notation will be useful in our work with vectors.

Consider writing a vector in terms of its rectangular components. Instead of using ellipses: a = a₁e₁ +···+a_ne_n, we could write the expression as a sum:. We can shorten this notation by leaving out the sum: a = aiei, where it is understood that whenever an index is repeated in a term we sum over that index from 1 to n. This is the Einstein summation convention. A repeated index is called a summation index or a dummy index. Other indices can take any value from 1 to n and are called free indices.

Example. Consider the matrix equation: A · x = b. We can write out the matrix and vectors  explicitly.

This takes much less space when we use the summation convention.

a_ijx_j = b_i

Here j is a summation index and i is a free index.



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