When looking through this document, I was confused by Equation $(4)$. It is stated there that a function $f(x,y)$, with $x,y\in\mathbb{R}^n$ and $x=x(r),y=y(r)$ for some $r\in\mathbb{R}$, has the derivative
$$ \frac{\partial f}{\partial r} = \frac{\partial f}{\partial x_i}\frac{\partial x_i}{\partial r} + \frac{\partial f}{\partial y_j}\frac{\partial y_j}{\partial r}$$ using Einstein's notation for summation. But to me, this makes no sense - wouldn't this reduce to $$ \frac{\partial f}{\partial r} = n\cdot\frac{\partial f}{\partial r} + n\cdot\frac{\partial f}{\partial r}$$ via the chain rule? Assuming that the original author made a mistake, would the total derivate $$ \frac{df}{dr} = \frac{\partial f}{\partial x_i}\frac{dx_i}{dr} + \frac{\partial f}{\partial y_j}\frac{dy_j}{dr}$$
be a reasonable fix for the notation? Or am I misunderstanding something?