Index notation to prove vector identity

Question

How can I use index notation to prove this identity? I have not been able to find any good resources on using index notation. \begin{equation} \nabla (fg)= f\nabla g + g \nabla f \end{equation}

Answer 1

You can write nabla simbol like \begin{equation*} \nabla = \partial_{i}\mathbf{e}^{i}, \end{equation*} where $\partial_{i}$ is the partial derivative with respect the coordinate $i$ and $\mathbf{e}^{i}$ is the $i$-th unit vector. So, is $f$ and $g$ are scalar funtions, then \begin{align*} \nabla(fg) &= \partial_{i}\mathbf{e}^{i}(fg) \\ &= \left\{ f\partial_{i}(g)+g\partial_{i}(f) \right\}\mathbf{e}^{i} \\ &= f\partial_{i}(g)\mathbf{e}^{i}+g\partial_{i}(f)\mathbf{e}^{i} \\ &= f\nabla g+g\nabla f. \end{align*}

Answer 2

Since $\nabla f = \left( \frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n} \right)$, you can just verify the identity component-wise.

To evaluate the $k^\mathrm{th}$ component of $\nabla (fg)$, take the partial derivative with respect to $x_k$. But then this is just the usual product rule:

$$ \frac{\partial}{\partial x_k} (fg) = \frac{\partial f}{\partial x_k} g + f \frac{\partial g}{\partial x_k} $$

I'm not sure if this is what you mean by "index notation".