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  • As an aside, if you want to allow non-unit $\mathbf{v}$, you get a better operator if you do not normalize: for example, you want to have $$\nabla_{2 \mathbf{v}} f = 2 \nabla_{\mathbf{v}} f$$ An example of the benefit is that this version still satisfies the formula $$ \nabla_{\mathbf{v}} f = \mathbf{v} \cdot \nabla f$$
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