Einstein in 1916 developed the following notation scheme when manipulating expressions involving vectors, matrices, or tensors in general.
The rules of summation convention are:
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Each index can appear at most twice in any term.
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Repeated indices are implicitly summed over.
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Each term must contain identical non-repeated indices.
For example:
![](https://sgp1.digitaloceanspaces.com/awe/awedmin/u/eintein_summation_1.png)
is a valid expression but
![](https://sgp1.digitaloceanspaces.com/awe/awedmin/u/eintein_summation_2.png)
is not a valid expression in summation convention, since the index j appears three times in the first term.
Einstein summation implies the following;
For any vector quantity, V,
![](https://sgp1.digitaloceanspaces.com/awe/awedmin/u/eintein_summation_3.png)
And the dot product of two vector quantity V and F is,
![](https://sgp1.digitaloceanspaces.com/awe/awedmin/u/eintein_summation_4.png)
In a Cartesian basis, the gradient of a scalar (φ) and the divergence of a vector D can be variously written as,
![](https://sgp1.digitaloceanspaces.com/awe/awedmin/u/eintein_summation_5.png)
We can also easily get the matrix multiplication using the Einstein summation convection;
If an N × N matrix C is the product of an N × M matrix A and an M × N matrix B
![](https://sgp1.digitaloceanspaces.com/awe/awedmin/u/eintein_summation_6.png)
The Kronecker delta symbol δij is defined as,
![](https://sgp1.digitaloceanspaces.com/awe/awedmin/u/eintein_summation_7.png)
And Levi-Civita` permutation symbol εijk is,
![](https://sgp1.digitaloceanspaces.com/awe/awedmin/u/eintein_summation_8.png)
Notice that ‘1’ is for cyclic permutation of x, y, and z and ‘-1’ is for anti-cyclic permutation of x, y, and z. Zero is when, either i=j, j=k, or k=i.
That is, if we define x=1, y=2 and z=3 then Levi- Civita symbol holds following,
![](https://sgp1.digitaloceanspaces.com/awe/awedmin/u/eintein_summation_9.png)
All other components are zero.
Using εijk we can write index expressions for the cross product and curl. The ith component of the cross product is given by,
Now, [U✕V]1 = 𝜺1jkUjVk = 𝜺123U2V3 + 𝜺132U3V2 + all other terms zero
This gives [U✕V]1 = U2V3 - U3V2
Similarly,
[U✕V]2 = U3V1 - U1V3
And [U✕V]3 = U1V2 - U2V1
Some other standard relations defined are,
![](https://sgp1.digitaloceanspaces.com/awe/awedmin/u/eintein_summation_11.png)
𝝏krj means 𝝏k/rj
Additionally, 𝜺ijk𝜺ist = 𝜹js𝜹kt - 𝜹jt𝜹ks
𝜺ijk = - 𝜺jik
▽ = 𝝏i and for any vector A, ▽A = 𝝏iAi.