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Einstein notation

Einstein notation for partial derivatives

{title2=$\partial_i$}

First we write a <vector field> as:
$$
F(x_0, x_1, x_2) = (F^0(x_0, x_1, x_2), F^1(x_0, x_1, x_2), F^2(x_0, x_1, x_2)) : \R^3 \to \R^3
$$
Note how we are denoting each component of $F$ as $F^i$ with a <raised index>.

Then, the <divergence> can be written in <Einstein notation> as:
$$
\div{F} = \pdv{F^0(x_0, x_1, x_2)}{x_0} + \pdv{F^1(x_0, x_1, x_2)}{x_1} + \pdv{F^2(x_0, x_1, x_2)}{x_2} = \partial_i F^i(x_0, x_1, x_2) = \pdv{F^i(x_0, x_1, x_2)}{x^i}
$$

It is common to just omit the variables of the function, so we tend to just say:
$$
\div{F} = \partial_i F^i
$$
or equivalently when referring just to the <operator>:
$$
\div{} = \partial_i
$$


Divergence in Einstein notation ( $\partial_{i}$ )

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Divergence in Einstein notation (∂i​)

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Divergence in Einstein notation ( $\partial_{i}$ )

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