Definition:Dot Product/Einstein Summation Convention
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Definition
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.
Sources
- 1992: Frederick W. Byron, Jr. and Robert W. Fuller: Mathematics of Classical and Quantum Physics ... (previous) ... (next): Volume One: Chapter $1$ Vectors in Classical Physics: $1.3$ The Scalar Product