CFD Online Logo CFD Online URL
www.cfd-online.com
[Sponsors]
Home > Wiki > Einstein summation convention

Einstein summation convention

From CFD-Wiki

(Difference between revisions)
Jump to: navigation, search
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:
:<math>
:<math>
\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}
 +
</math>
 +
 +
:<math>
 +
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}
</math>
</math>

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}
My wiki