Why chain rule isn't working here?

Question

Let $(x,y,z)$ and $(x',y',z')$ be two points in space. Let the distance between them be $r$. Also let:

$x-x'=\xi$

$y-y'=\eta$

$z-z'=\zeta$

Since $(x,y,z)$ and $(x',y',z')$ are two separate points, changing only the point $(x,y,z)$ will not change the point $(x',y',z')$. Therefore:

$\dfrac{\partial x'}{\partial x}=\dfrac{\partial x}{\partial x'}=0$

Then:

$\dfrac{\partial r}{\partial x}=\dfrac{dr}{dr^2}\dfrac{\partial r^2}{\partial x}=\dfrac{1}{2r}\dfrac{\partial(\xi^2 +\eta^2 +\zeta^2)}{\partial \xi}\dfrac{\partial \xi}{\partial x}=\dfrac{1}{2r}.2 \xi.\dfrac{\partial(x-x')}{\partial x}=\dfrac{\xi}{r}$

and

$\dfrac{\partial r}{\partial x'}=\dfrac{dr}{dr^2}\dfrac{\partial r^2}{\partial x'}=\dfrac{1}{2r}\dfrac{\partial(\xi^2 +\eta^2 +\zeta^2)}{\partial \xi}\dfrac{\partial \xi}{\partial x'}=\dfrac{1}{2r}.2 \xi.\dfrac{\partial(x-x')}{\partial x'}=-\dfrac{\xi}{r}$

Now let's find $\dfrac{\partial r}{\partial x'}$ by another way using chain rule:

$\dfrac{\partial r}{\partial x'}=\dfrac{\partial r}{\partial x}\dfrac{\partial x}{\partial x'}=\dfrac{\xi}{r}.0=0$

I guess I am getting different answers because there is something wrong in the application of chain rule here. Can someone explain why am I getting different answers?

Answer 1

The thing you have to bear in mind about partial derivatives is they assume something is held constant, but $\partial_u v\partial_v w = \partial_u w$ is only guaranteed when the partial derivatives on the left-hand side hold the same things constant. Define $\alpha:=\left(\xi,\,\eta,\,\zeta\right)$. In your first calculation, you show $(\partial_{x'}x)_{\alpha}=0$, where the subscript indicates what's held constant. But in your final calculation, you try $(\partial_{x'}r)_{?}=(\partial_x r)_{x'}(\partial_{x'}x)_\alpha$, about which nothing follows from the chain rule.

Answer 2

What you've written in the yellow box is nonsense.

An expression like $\frac{\partial x}{\partial x'}$ only makes sense when $x'$ is thought of as one of several independent variables, including presumably $y'$ and $z'$, when $x$ is as a function of all three of them (in that setting, $\frac{\partial x}{\partial x'}$ is defined by holding $y'$ and $z'$ constant while letting $x'$ vary, and taking the limit of a very carefully formulated difference quotient).

Similarly, and expression like $\frac{\partial x'}{\partial x}$ only makes sense when $x$ is thought of as one of several independent variables, including presumably $y$ and $x$, and when $x'$ is as a function of all three of them.

Obviously these two situations are incompatible, so any kind of relation or equation between $\frac{\partial x}{\partial x'}$ and $\frac{\partial x'}{\partial x}$ makes no sense, and any assumption of an equation like $\frac{\partial x}{\partial x'} = 0$ makes no sense.