Einstein summation convention

The Einstein summation convention implies that when an index occurs more than once in the same expression, the expression is implicitly summed over all possible values for that index. Therefore, in order to use the summation convention, it must be clear from the context over what range indices should be summed.

The Einstein summation convention is illustrated in the below examples.

  1. 1.

    Let {ei}i=1n be a orthogonal basis in n. Then the inner product of the vectors u=uiei=(i=1n)uiei and v=viei=(i=1n)viei, is

    uv = uivjeiej
    = δijuivj.
  2. 2.

    Let V be a vector spaceMathworldPlanetmath with basis {ei}i=1n and a dual basisMathworldPlanetmath {ei}i=1n. Then, for a vector v=viei and dual vectors α=αiei and β=βiei, we have

    (α+β)(v) = αivi+βjvj
    = (αi+βi)vi.

    This example shows that the summation convention is “distributive” in a natural way.

  3. 3.

    Chain ruleMathworldPlanetmath. Let F:mn, x(F1,,Fn), and G:np, y(G1(y),,Gp(y)) be smooth functionsMathworldPlanetmath. Then


    where the right hand side is summed over k=1,,n.

An index which is summed is called a dummy index or dummy variable. For instance, i is a dummy index in viei. An expression does not depend on a dummy index, i.e., viei=vjej. It is common that one must change the name of dummy indices. For instance, above, in Example 2 when we calculated uv, it was necessary to change the index i in v=viei to j so that it would not clash with u=uiei.

When using the Einstein summation convention, objects are usually indexed so that when summing, one index will always be an “upper index” and the other a “lower index”. Then summing should only take place over upper and lower indices. In the above examples, we have followed this rule. Therefore we did not write δijuivj=uivi in the first example since uivi has two upper indices. This is consistent; it is not possible to take the inner product of two vectors without a metric, which is here δij. The last example illustrates that when we consider k as a “lower index” in Giyk, then the chain rule obeys this upper-lower rule for the indices.

Title Einstein summation convention
Canonical name EinsteinSummationConvention
Date of creation 2013-03-22 13:32:03
Last modified on 2013-03-22 13:32:03
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 15A69
Related topic Summation
Defines dummy index