# Einstein summation convention

(Difference between revisions)
 Revision as of 09:50, 28 November 2005 (view source)Jola (Talk | contribs)← Older edit Revision as of 10:04, 28 November 2005 (view source)Jola (Talk | contribs) Newer edit → Line 1: Line 1: 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. 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 is an example: + Here are two examples: :$:[itex] \frac{\partial u_i}{\partial x_i} \equiv \sum_{i=1}^3 \frac{\partial u_i}{\partial x_i} \equiv \frac{\partial u_1}{\partial x_1} + \frac{\partial u_2}{\partial x_2} + \frac{\partial u_3}{\partial x_3} \frac{\partial u_i}{\partial x_i} \equiv \sum_{i=1}^3 \frac{\partial u_i}{\partial x_i} \equiv \frac{\partial u_1}{\partial x_1} + \frac{\partial u_2}{\partial x_2} + \frac{\partial u_3}{\partial x_3} +$ + + :$+ u_j\frac{\partial u_i}{\partial x_j} \equiv \sum_{j=1}^3 u_j\frac{\partial u_i}{\partial x_j} \equiv u_1\frac{\partial u_i}{\partial x_1} + u_2\frac{\partial u_i}{\partial x_2} + u_3\frac{\partial u_i}{\partial x_3}$ [/itex]

## Revision as of 10:04, 28 November 2005

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:

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