Proving Laplacian in Cylindrical Coordinates

[quasar987]

I'm supposed to prove the laplacian in cylindrical coord. is what it is. I tried tackling the problem in two ways and none work! I have no idea what's the matter. The first way is to calculate d²f/dr² , d²f/dO² and d²f/dz² and isolate d²f/dx² , d²f/dy² and d²f/dz². In cylindrical coord.,

x=rcosO
y=rsinO
z=z

(I'm not writting partial derivatives because in latex it's a pain in the @)

$$df/dr = df/dx dx/dr + df/dy dy/dr$$
$$\Rightarrow d^2f/dr^2 = d^2f/dx^2 dx/dr + df/dx d^2x/dr^2 + d^2f/dy^2 dy/dr + df/dy d^2y/dr^2$$
$$d^2f/dr^2 = d^2f/dx^2 cos^2\theta +d^2f/dy^2 sin^2\theta$$

In the same way,

$$d^2f/d\theta^2 = r^2 (d^2f/dx^2 d^2f/dx^2)$$

Hence, it would seem that the laplacian in cylindrical is just (1/r²)d²f/dO² + d^2f/dz^2.

 

[Tide]

Have you tried

$$\frac {\partial}{\partial x} = \frac {\partial r}{\partial x} \frac {\partial}{\partial r} + \frac {\partial \theta}{\partial x} \frac {\partial}{\partial \theta}$$

etc.?

 

[HallsofIvy]

You are doing it backwards!
You know that the Laplacian is
$$\frac{\partial^2U}{/partial x^2}+ \frac{\partial^2U}{\partial y^2}+ \frac{\partial^2U}{\partial z^2}$$
in Cartesian coordinates so you should be writing
$$\frac{\partialU}{\partial z}=\frac{\partial U}{\partial r}\frac{\partial r}{\partial x}+ \frac{\partial U}{\partial \theta}\frac{\partial \theta}{\partial x}+ \frac{\partial U}{\partial z}\frac{\partial z}{\partial x}$$
etc. so that you can substitute into that.

 

[quasar987]

Yeah that's the "second way" I was referring to! Why do you call "backward" the first way? It's just as good imo. It's not guarenteed that you'll be able to isolate the cartesian laplancian, but if you can, shouldn't it give the right answer too ?!

 

[quasar987]

Ok, after working out the algebra completely and correcting a few mistake I had made in the second way (the one advetised by HallsofIvy), it works. But I'm still very interested to know why the "backwards" way doesn't work.