Einstein summation convention

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The Einstein summation convention is a tensor notation which is commonly used to implicitly define a sum. The convention states that as soon as one index is repeated in a term that implies a sum over all possible values for that index.

Here are two examples:

$\frac{\partial u_i}{\partial x_i} \equiv \sum_{i=1}^3 \frac{\partial u_i}{\partial x_i} \equiv \frac{\partial u_1}{\partial x_1} + \frac{\partial u_2}{\partial x_2} + \frac{\partial u_3}{\partial x_3}$

$u_j\frac{\partial u_i}{\partial x_j} \equiv \sum_{j=1}^3 u_j\frac{\partial u_i}{\partial x_j} \equiv u_1\frac{\partial u_i}{\partial x_1} + u_2\frac{\partial u_i}{\partial x_2} + u_3\frac{\partial u_i}{\partial x_3}$

Einstein summation convention

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