Definition:Dot Product/Einstein Summation Convention

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Let $\mathbf a$ and $\mathbf b$ be vector quantities.

The dot product of $\mathbf a$ and $\mathbf b$ can be expressed using the Einstein summation convention as:

\(\ds \mathbf a \cdot \mathbf b\) \(:=\) \(\ds a_i b_j \delta_{i j}\)
\(\ds \) \(=\) \(\ds a_i b_i\)
\(\ds \) \(=\) \(\ds a_j b_j\)

where $\delta_{i j}$ is the Kronecker delta.