What is the meaning of the operator $\nabla_y$?

Question

I am trying to understand what this notation means... $\nabla_y$. For example:

Specifically I guess I am confused how about why $\nabla_y v(y)=r^{n-1}\nabla_x u(x+r^{n-1}y)$

Answer 1

The subscript is to stress which variable we are taking the gradient with respect to. So $\nabla_y v(y) = \nabla_y u(x + r^{n - 1}y)$ is the gradient with respect to $y$ of the composition of functions $u$ with $y \mapsto x + r^{n - 1}y$. Thus, the chain rule applies, and we get $r^{n - 1}\nabla u(x + r^{n - 1}y)$.

The $\nabla_x$ appearing in the text is to stress that this last gradient is with respect to $x$, so that in particular, we do not have to do any more chain-ruling with this. So $\nabla_x u(x + r^{n - 1}y)$ is just the gradient of the function $u$, evaluated at the point $x + r^{n - 1}y$ (rather than the gradient of the function of $y$, $u(x + r^{n - 1}y)$, which involves (as is calculated here) the chain rule.