DisclaimerPermalink

This page is directly inspired (many parts copied) from my Geometry Course Notes, written by Dr Johans Martens.

IntroductionPermalink

Einstein Summation has been very important in many part of physics, especially where we need to sum up a lot of stuff, like in electromagnetism.

Mathematicians define it more rigorously than physicists.

DefinitionPermalink

Suppose that we have a vector space $V$, and dual vector space $V^*$. A choice of a basis $\mathbf{e}_{\mathbf{1}}, \ldots, \mathbf{e}_{\mathbf{n}}$ for $V$ induces a dual basis $\mathbf{f}^{\mathbf{1}}, \ldots, \mathbf{f}^{\mathbf{n}}$ for $V^*$, determined by the rule $\mathbf{f}^i\left(\mathbf{e}_j\right)=\delta_j^i$, where $\delta_j^i$ is the Kronecker delta.

Any element $\mathbf{a}$ of $V$ or $\mathbf{b}$ of $V^*$ can be written as a linear combination of these basis vectors:

$\mathbf{a}=\sum_i a^i \mathbf{e}_i \text { and } \mathbf{b}=\sum_j b_j \mathbf{f}^j,$ and we have $\mathbf{b}(\mathbf{a})=\sum_i a^i b_i$.

The coefficients of $V*$ and basis vectors of $V$ transform the same way (co-variant) (labelled as Sub-script). Coefficients of V and the basis vectors of V* transform the same way (contra-variant) (labelled as Super-Script).

We can deduce how each the contravariant and covariant vectors transform. First,

This is the result for Contra-variant (Super_Script).

Using this, we have:

We can finally observe that:

This is the result for Co-variant (Sub_Script).

Einstein SummationPermalink

Einstein Summation states that:

Any time an index in a formula occurs twice, once as sub-script and once as super-script, it is implicitly assumed that we sum over all instances of that index.