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..?
Best Answer
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.