Does $\frac{\partial^2}{\partial^2 (\vec{r}_2 + \vec{r}_1 )} ?= \frac{\partial^2}{\partial^2(\vec{r}_2)} + \frac{\partial^2}{\partial^2(\vec{r}_1)}$?

Question

Is the following equation true? $$ \frac{\partial^2}{\partial^2 (\vec{r}_2 + \vec{r}_1 )} ?= \frac{\partial^2}{\partial^2(\vec{r}_2)} + \frac{\partial^2}{\partial^2(\vec{r}_1)} ?$$ If not, what is the left side equal to? I think this question falls into the category of matrix calculus.

Answer 1

If you set the bijective change $\begin{cases} u=x+y\\v=x-y\end{cases}$

Then you have to apply the chain rule to $f(x,y)=f(\frac{u+v}2,\frac{u-v}2)$

So you get

$\dfrac{\partial f}{\partial u}=\dfrac{\partial f}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial f}{\partial y}\dfrac{\partial y}{\partial u}=\dfrac 12(f'_x+f'_y)$

$\dfrac{\partial^2 f}{\partial u^2}=\cdots=\dfrac 14(f''_{xx}+2f''_{xy}+f''_{yy})$