Einstein notation

The Einstein summation convention is a convention for the notation of mathematical expressions within the Ricci calculus and provides an index notation dar. This calculus is used in tensor analysis, differential geometry, and in particular in theoretical physics. The summation convention was introduced in 1916 by Albert Einstein. With it, the sum of characters to improve the overview are simply omitted and instead summing over repeated indices.

Motivation

In the matrix and tensor sums are often formed using indices. For example, is the matrix product of two matrices and components:

Here is summed over the index of from 1 to. If more than one matrix multiplication, scalar products, or other sums on an invoice, this can quickly become confusing. With the Einstein summation convention the bill from above is then:

Formal Description

In the simplest case, the summation convention applies: About repeated indices in a product is summed. In the theory of relativity is considered as an additional rule: Summing only if the index occurs both as an upper ( contravariant ) and lower than ( covariant ) index.

The summation convention is reduced especially the writing effort. In some cases it helps to highlight existing relationships and symmetries that are not easily discernible in the conventional summation notation.

Examples

Without observing the index position

In the following examples are for matrices with entries and for matching vectors.

• Standard scalar product.
• Application of a matrix to a vector:
• Product of several ( here 4) matrices.
• Trace of a matrix A:

Taking into account the index position

• The product of two tensors with tensor and is
• Application of a tensor with components to the sum of the vectors to get the vector.
• A tensor t in an environment has the representation