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Which way is appropriate for defining del in index notation: ##\nabla \equiv \partial_i()\vec{e_i}## or ##\nabla \equiv \vec{e_i}\partial_i()##. The two cannot be generally equivalent. Quick example.

Let ##\vec{v}## and ##\vec{w}## be vectors. Then $$\nabla \vec{v} \cdot \vec{w} = \partial_i(v_j \vec{e_j})\vec{e_i} \cdot u_k \vec{e_k}\\ = \partial_i(v_j \vec{e_j}) u_i$$ yet using the other definition for del implies $$\nabla \vec{v} \cdot \vec{w} = \vec{e_i} \partial_i(v_j \vec{e_j}) \cdot u_k \vec{e_k}\\=\vec{e_i} v_ju_k (\partial_i(\vec{e_j}) \cdot \vec{e_k}) + \vec{e_i} u_j \partial_i(v_j)$$

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# I Del in index notation

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