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

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:

\nabla \cdot F = \frac{\partial F ^{0} ( x _{0} , x _{1} , x _{2} )}{\partial x _{0}} + \frac{\partial F ^{1} ( x _{0} , x _{1} , x _{2} )}{\partial x _{1}} + \frac{\partial F ^{2} ( x _{0} , x _{1} , x _{2} )}{\partial x _{2}} = \partial_{i} F^{i} (x_{0}, x_{1}, x_{2}) = \frac{\partial F ^{i} ( x _{0} , x _{1} , x _{2} )}{\partial x ^{i}}

It is common to just omit the variables of the function, so we tend to just say:

\nabla \cdot F = \partial_{i} F^{i}

or equivalently when referring just to the operator:

\nabla \cdot = \partial_{i}

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Einstein notation for partial derivatives