Einstein summation convention

The Einstein summation convention implies that when an index occursmore than once in the same expression, the expression is implicitlysummed over all possible values for that index. Therefore, in order touse the summation convention, it must be clear from the contextover what range indices should be summed.

The Einstein summation convention is illustrated in the below examples.

1.
Let $\\{e_{i}\\}_{i=1}^{n}$ be a orthogonal basis in $\\mathbb{R}^{n}$ .Then the inner product of the vectors $u=u^{i}e_{i}=(\\sum_{i=1}^{n})u^{i}e_{i}$ and $v=v^{i}e_{i}=(\\sum_{i=1}^{n})v^{i}e_{i}$ , is
$\\displaystyle u\\cdot v$ $\\displaystyle=$ $\\displaystyle u^{i}v^{j}e_{i}\\cdot e_{j}$
$\\displaystyle=$ $\\displaystyle\\delta_{ij}u^{i}v^{j}.$
2.
Let $V$ be a vector space with basis $\\{e_{i}\\}_{i=1}^{n}$ anda dual basis $\\{e^{i}\\}_{i=1}^{n}$ . Then, for a vector $v=v^{i}e_{i}$ and dual vectors $\\alpha=\\alpha_{i}e^{i}$ and $\\beta=\\beta_{i}e^{i}$ , we have
$\\displaystyle(\\alpha+\\beta)(v)$ $\\displaystyle=$ $\\displaystyle\\alpha_{i}v^{i}+\\beta_{j}v^{j}$
$\\displaystyle=$ $\\displaystyle(\\alpha_{i}+\\beta_{i})v^{i}.$
This example shows that the summation convention is “distributive” ina natural way.
3.
Chain rule. Let $F:\\mathbb{R}^{m}\\to\\mathbb{R}^{n}$ , $x\\mapsto(F^{1},\\cdots,F^{n})$ , and $G:\\mathbb{R}^{n}\\to\\mathbb{R}^{p}$ , $y\\mapsto(G^{1}(y),\\cdots,G^{p}(y))$ be smooth functions. Then
$\\frac{\\partial(G\\circ F)^{i}}{\\partial x^{j}}(x)=\\frac{\\partial G^{i}}{%\\partial y^{k}}\\big{(}F(x)\\big{)}\\frac{\\partial F^{k}}{\\partial x^{j}}(x),$
where the right hand side is summed over $k=1,\\ldots,n$ .

An index which is summed is called a dummy index or dummy variable.For instance, $i$ is a dummy index in $v^{i}e_{i}$ . An expression does not depend ona dummy index, i.e., $v^{i}e_{i}=v^{j}e_{j}$ .It is common that one must change the name of dummy indices. Forinstance, above, in Example 2 when we calculated $u\\cdot v$ , it was necessary to change the index $i$ in $v=v^{i}e_{i}$ to $j$ so that it would not clash with $u=u^{i}e_{i}$ .

When using the Einstein summation convention, objects are usually indexedso that when summing, one index will always be an “upper index” andthe other a “lower index”.Then summing should only take place over upper and lower indices.In the above examples, we have followedthis rule. Therefore we did not write $\\delta_{ij}u^{i}v^{j}=u^{i}v^{i}$ in the first example since $u^{i}v^{i}$ has two upper indices. This is consistent; it is not possibleto take the inner product of two vectors without a metric, which is here $\\delta_{ij}$ .The last example illustrates that when we consider $k$ as a “lowerindex” in $\\frac{\\partial G^{i}}{\\partial y^{k}}$ , then the chain ruleobeys this upper-lower rule for the indices.