Using summation notation to prove the Leibniz rule for the gradient of product

Question

Use summation notation to prove that $\vec\nabla (fg) = f \vec\nabla g + g\vec\nabla f$ where $f$ and $g$ are scalar functions.

So I'm assuming that I need to start by writing this in the summation notation. However, I am not familiar with how to write the gradient of a scalar in this notation. I know how to write dot product and cross, but how do we write using a scalar?

I think I'm confused on the idea of using a gradient on a scalar function? A scalar function is described as $(x_1,x_2,x_3)$, and the gradient is just the $(d/dx)$ operator on the function, so $(\frac{d}{dx} x_1, \frac{d}{dx} x_2, ..) $ Is this correct..?

Answer 1

Let $[f]_i$ denote the $i$th component of the scalar function $f$. Then we need to prove that \begin{equation*} [\nabla(fg)]_i=[f\nabla g+g\nabla f]_i \end{equation*} But $\nabla f=(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2},..., \frac{\partial f}{\partial x_n})$, so $[\nabla f]_i=\frac{\partial f}{\partial x_i}$. Using the product rule it follows that \begin{equation*} [\nabla(fg)]_i=\frac{\partial}{\partial x_i}(fg)=f\frac{\partial g}{\partial x_i}+g\frac{\partial f}{\partial x_i}=[f\nabla g+g\nabla f]_i \end{equation*} which is the result.

Answer 2

For what it's worth:

Let $f,g: \mathbb{R}^n\rightarrow \mathbb{R}$, thus: \begin{align} \nabla(fg)&= \begin{pmatrix} \partial_{x_1} \\ ..\\ \partial_{x_n}\end{pmatrix}(fg)\\&= \begin{pmatrix} \partial_{x_1} (fg)\\ ..\\ \partial_{x_n}(fg)\end{pmatrix} =\begin{pmatrix} g\partial_{x_1}f + f\partial_{x_1}g \\ ..\\ g\partial_{x_n}f + f\partial_{x_n}g \end{pmatrix} = f \begin{pmatrix} \partial_{x_1} g\\ ..\\ \partial_{x_n}g\end{pmatrix} +g \begin{pmatrix} \partial_{x_1} f\\ ..\\ \partial_{x_n}f\end{pmatrix} \\&= f \nabla g + g \nabla f \end{align}