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The Kronecker Delta Function and the Einstein Summation Convention

The Kronecker delta function is defined by the rules:

    $\displaystyle 1 \quad\textrm{if } i = j$  
$\displaystyle \delta_{ij}$ $\displaystyle =$    
    $\displaystyle 0 \quad\textrm{if } i \ne j$  

Using this we can reduce the dot product to the following tensor contraction, using the Einstein summation convention:

$\displaystyle \vA \cdot \vB = \sum_{i = 1}^3 A_i B_i = A_i \delta_{ij} B_j = A_i
B_i$

where we sum repeated indices over all of the orthogonal cartesian coordinate indices without having to write an explicit $ \sum_{i = 1}^3$ . We will henceforth use this convention almost all the time to streamline the notation of certain kinds of (vector and tensor) algebra.

The Kronecker delta function is obviously useful for representing the dot product in a compact way. We can similarly invent a symbol that incorporates all of the details of the ways the unit vectors multiply in the cross product, next.


next up previous contents
Next: The Levi-Civita Tensor Up: &delta#delta;_ij and &epsi#epsilon;_ijk Previous: &delta#delta;_ij and &epsi#epsilon;_ijk   Contents
Robert G. Brown 2017-07-11