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.[1]
Introduction
Statement of convention
According to this convention, when an index variable appears twice in a single
The upper indices are not
In general relativity, a common convention is that
- the Greek alphabet is used for space and time components, where indices take on values 0, 1, 2, or 3 (frequently used letters are μ, ν, ...),
- the Latin alphabet is used for spatial components only, where indices take on values 1, 2, or 3 (frequently used letters are i, j, ...),
In general, indices can range over any
An index that is summed over is a summation index, in this case "i ". It is also called a
An index that is not summed over is a
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.[2] 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 § Superscripts and subscripts versus only subscripts below.
Vector representations
Superscripts and subscripts versus only subscripts
In terms of covariance and contravariance of vectors,
- upper indices represent components of contravariant vectors (vectors),
- lower indices represent components of covectors).
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, as in:
where v is the vector and v i are its components (not the ith covector v), w is the covector and wi are its components. The basis vector elements are each column vectors, and the covector basis elements are each row covectors. (See also § Abstract description; duality, below and the examples)
In the presence of a
A basis gives such a form (via the
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
In the above example, vectors are represented as n × 1 matrices (column vectors), while covectors are represented as 1 × n 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)."
- Covectors are row vectors: Hence the lower index indicates which column you are in.
- Contravariant vectors are column vectors: 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
As for covectors, they change by the
The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality. For example, V ⊗ V, the tensor product of V with itself, has a basis consisting of tensors of the form eij = ei ⊗ ej. Any tensor T in V ⊗ V can be written as:
V *, the dual of V, has a basis e1, e2, ..., en which obeys the rule
Common operations in this notation
In Einstein notation, the usual element reference for the -th row and -th column of matrix becomes . We can then write the following operations in Einstein notation as follows.
Inner product
Using an
This can also be calculated by multiplying the covector on the vector.
Vector cross product
Again using an orthogonal basis (in 3 dimensions), the cross product intrinsically involves summations over permutations of components:
εijk is the Levi-Civita symbol, and δil is the generalized Kronecker delta. Based on this definition of ε, there is no difference between εijk and εijk but the position of indices.
Matrix-vector multiplication
The product of a matrix Aij with a column vector vj is:
This is a special case of matrix multiplication.
Matrix multiplication
The matrix product of two matrices Aij and Bjk is:
equivalent to
Trace
For a square matrix Aij, the trace is the sum of the diagonal elements, hence the sum over a common index Aii.
Outer product
The outer product of the column vector ui by the row vector vj yields an m × n matrix A:
Since i and j 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, gμν. For example, taking the tensor Tαβ, one can lower an index:
Or one can raise an index:
See also
- Tensor
- Abstract index notation
- Bra–ket notation
- Penrose graphical notation
- Levi-Civita symbol
- DeWitt notation
Notes
- 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 introductionof this article. Elements of a basis of vectors may carry a lower numerical index and an upper abstract index.
References
- ) on 2006-08-29. Retrieved 2006-09-03.
- ^ "Einstein Summation". Wolfram Mathworld. Retrieved 13 April 2011.
Bibliography
- Kuptsov, L. P. (2001) [1994], "Einstein rule", Encyclopedia of Mathematics, EMS Press.
External links
![](http://upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png)
- Rawlings, Steve (2007-02-01). "Lecture 10 – Einstein Summation Convention and Vector Identities". Oxford University. Archived from the original on 2017-01-06. Retrieved 2008-07-02.
- "Understanding NumPy's einsum". Stack Overflow.