Math Help - cylindrical coordinates

1. cylindrical coordinates

The relation between three-dimentional cartesian coordinates $(X,Y,Z)$ and cylindrical coordinates $(\rho,\phi,z)$ is given by
$X=\rho cos\phi$
$Y=\rho sin\phi$
$Z=z$

Calculate the partial derivatives $\frac{\partial \rho}{\partial X}$, $\frac{\partial \phi}{\partial Y}$ and $\frac{\partial z}{\partial Z}$ in terms of $\rho$, $\phi$, $z$.

My current problem here is like a puzzle. I certainly can solve the question and find the answer. For example, I did find $\frac{\partial \rho}{\partial X}=cos\phi$.Same as solution
However, my problem occured when I want to calculate $\frac{\partial X}{\partial \rho}$,if I just use $X=\rho cos\phi$, I would get $cos\phi$ as well!
I think it is weird, because I thought $\frac{\partial X}{\partial \rho}$ supposed to equal to the reciprocal of $\frac{\partial \rho}{\partial X}$, which means it supposed to be $\frac{1}{cos \phi}$.
So pretty sure $\frac{\partial \rho}{\partial X}=cos\phi$ is the right answer(same as solution),but why the recipocal rule does not apply here? I'm so confused about this puzzle. Same as two other questions.

2. What was your process by which you obtained

$\displaystyle\frac{\partial\rho}{\partial x}?$

3. Oddly enough, both $\frac{\partial \rho}{\partial X}= cos(\theta)$ and $\frac{\partial X}{\partial \rho}= cos(\theta)$ are correct.
Your "mistake" is in thinking that $\frac{\partial \rho}{\partial X}$ and $\frac{\partial X}{\partial \rho}$ must be reciprocals. For partial derivatives, that simply is not true.

4. Originally Posted by HallsofIvy
Oddly enough, both $\frac{\partial \rho}{\partial X}= cos(\theta)$ and $\frac{\partial X}{\partial \rho}= cos(\theta)$ are correct.
Your "mistake" is in thinking that $\frac{\partial \rho}{\partial X}$ and $\frac{\partial X}{\partial \rho}$ must be reciprocals. For partial derivatives, that simply is not true.
Thanks a lot.
I felt the reason must be because their are partial derivatives, therefore they are not reciprocals.
But I wonder the reason, is there any mathematical way can explain that? Thanks a lot.