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Wikipedia / Einstein notation

In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916.

Introduction

Statement of convention

According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set

The upper indices are not exponents but are indices of coordinates, coefficients or basis vectors is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see .

In general relativity, a common convention is that

In general, indices can range over any indexing set, including an infinite set. This should not be confused with a typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation.

An index that is summed over is a summation index". It is also called a dummy index" without changing the meaning of the expression (provided that it does not collide with other index symbols in the same term).

An index that is not summed over is a free index.

Application

Einstein notation can be applied in slightly different ways. Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term. When dealing with covariant and contravariant vectors, where the position of an index indicates the type of vector, the first case usually applies; a covariant vector can only be contracted with a contravariant vector, corresponding to summation of the products of coefficients. On the other hand, when there is a fixed coordinate basis (or when not considering coordinate vectors), one may choose to use only subscripts; see below.

Vector representations

Superscripts and subscripts versus only subscripts

In terms of covariance and contravariance of vectors,

They transform contravariantly or covariantly, respectively, with respect to change of basis.

In recognition of this fact, the following notation uses the same symbol both for a vector or covector and its components

are each row covectors. (See also ; duality, below and the examples)

In the presence of a non-degenerate form (an isomorphism, for instance a Riemannian metric or Minkowski metric), one can raise and lower indices.

A basis gives such a form (via the dual basis with a Euclidean metric and a fixed orthonormal basis, one has the option to work with only subscripts.

However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see Covariance and contravariance of vectors.

Mnemonics

matrices matrices (row covectors).

When using the column vector convention:

  • "Upper indices go up to down; lower indices go left to right."

  • "Covariant tensors are row vectors that have indices that are below (co-row-below)."

  • Hence the lower index indicates which column you are in.

  • Hence the upper index indicates which row you are in.

Abstract description

The virtue of Einstein notation is that it represents the invariant quantities with a simple notation.

In physics, a scalar is invariant under transformations of basis. In particular, a Lorentz scalar is invariant under a Lorentz transformation. The individual terms in the sum are not. When the basis is changed, the components of a vector change by a linear transformation described by a matrix. This led Einstein to propose the convention that repeated indices imply the summation is to be done.

As for covectors, they change by the inverse matrix. This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.

The value of the Einstein convention is that it applies to other vector spaces using the tensor product and duality

is the Kronecker delta the row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.

Common operations in this notation

. We can then write the following operations in Einstein notation as follows.

Inner product

The inner product of two vectors is the sum of the products of their corresponding components, with the indices of one vector lowered (see #Raising and lowering indices In the case of an orthonormal basis

Vector cross product

In three dimensions, the cross product of two vectors with respect to a positively oriented

is the Levi-Civita symbol, when treated as a tensor.

Matrix-vector multiplication

This is a special case of matrix multiplication.

Matrix multiplication

The matrix product

Trace

For a square matrix, the trace.

Outer product

The outer product

represent two different indices, there is no summation and the indices are not eliminated by the multiplication.

Raising and lowering indices

Given a tensor, one can raise an index or lower an index by contracting the tensor with the metric tensor

See also

Notes

  1. This applies only for numerical indices. The situation is the opposite for abstract indices. Then, vectors themselves carry upper abstract indices and covectors carry lower abstract indices, as per the example in the introduction of this article. Elements of a basis of vectors may carry a lower numerical index and an upper abstract index.

References

Bibliography

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