Explaining the Einstein Summation Convention in Simple Terms

The Einstein summation convention, a shorthand notation used in mathematics and physics, significantly simplifies expressions involving sums over indices. This convention allows mathematicians and physicists to write complex equations more concisely, without the need to explicitly write summation symbols.

Key Points

Implicit Summation

When an index appears twice in a single term with one as a subscript and one as a superscript, it indicates a summation over all possible values of that index. For example, if i ranges from 1 to n, then:

AIBI sum;_{i1}^{n} ABI

Here, AI and BI are components of two different tensors or vectors.

Indices

The indices can represent dimensions or components in a vector space. They can also indicate different objects such as vectors or matrices. This convention helps in dealing with these objects without writing out the summation explicitly.

No Summation for Free Indices

If an index appears only once, it is not summed over. For example, the expression AIBJ indicates a component-wise multiplication with no summation over J.

Examples

Matrix-Vector Multiplication

Instead of writing:

aIcI sum;_{I} aIcI

you can simply write:

wI aIcI

This indicates that wI is the result of multiplying a vector mathbf{a} with a vector mathbf{c} without the need for an explicit summation.

Three-Dimensional Tensor

A three-dimensional tensor text{T} can be written as the sum of the product of its components and the unit tensor in a certain basis:

T TIJK hat{e}IJK

Vector Products

The cross product can also be written in this shorthand notation:

mathbf{u} times mathbf{v} u_i v_j hat{e}_{ij}

Summary

The Einstein summation convention streamlines notation in tensor calculus and physics, allowing for cleaner and more efficient equations. It is especially useful in tensor calculus and vector products where dealing with multiple indices is common.

In simple terms, the convention is just avoiding the sum symbol. By following this rule, we can avoid writing out explicit summations, making equations more readable and concise.

Really easy--if you have repeated indices on two tensors or matrices, it is assumed that you sum over them:

AIJ XJ sum;_J AIJ XJ