Laplacian operator in different coordinates

[captain]

how do you write the laplacian operator in spherical coordinates and cylindrical coordinates from a cartesian basis?

 

[Kummer]

how do you write the laplacian operator in spherical coordinates and cylindrical coordinates from a cartesian basis?

Cylindrical: Use the substitution $$r=\sqrt{x^2+y^2}$$ and $$\theta = \tan^{-1} \frac{y}{x}$$ assuming this is valid on this region.

This leads to,
$$\nabla^2 u = \frac{\partial ^2 u}{\partial r^2} +\frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2 }\frac{\partial ^2 u}{\partial \theta ^2} + \frac{\partial ^2 u}{\partial z^2}=0$$
For the most part $$z$$ coordinate is not taken and that term vanished.

Spherical: Using Spherical Coordinate substitutions:
$$\nabla^2 u = \frac{1}{r^2} \left\{ \frac{\partial (r^2u_r)}{\partial r}+\csc^2 \theta \frac{\partial ^2 y}{\partial \theta^2}+\csc \theta \frac{\partial (\sin \phi u_{\phi})}{\partial \phi} \right\} = 0$$

 

[MathematicalPhysicist]

in general (which is something you learn in vector analysis for physicists):
$$\nabla^2=(\frac{h_3h_2}{h_1}\frac{\partial}{\partial u_1}\frac{\partial h_1}{\partial u_1},\frac{h_3h_1}{h_2}\frac{\partial}{\partial u_2}\frac{\partial h_2}{\partial u_2},\frac{h_1h_2}{h_3}\frac{\partial}{\partial u_3}\frac{\partial h_3}{\partial u_3})$$ or something like this.
where:
r=xi+yj+zk
and h_i=|dr/du_i|
i.e you take the norm of the vector.

 

[CompuChip]

Cheat sheet (well not really cheating, unless you like deriving these things)

 

[dextercioby]

Simply use the rules of change of variables in partial differentials. For example

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

and then sub everyting in terms of the spherical coordinates. Then compute the 2-nd partial wrt to x and the same for y and z.

 

[wasia]

in general (which is something you learn in vector analysis for physicists):
$$\nabla^2=(\frac{h_3h_2}{h_1}\frac{\partial}{\partial u_1}\frac{\partial h_1}{\partial u_1},\frac{h_3h_1}{h_2}\frac{\partial}{\partial u_2}\frac{\partial h_2}{\partial u_2},\frac{h_1h_2}{h_3}\frac{\partial}{\partial u_3}\frac{\partial h_3}{\partial u_3})$$ or something like this.
where:
r=xi+yj+zk
and h_i=|dr/du_i|
i.e you take the norm of the vector.

Either I am confused at the moment, or it is not right. In general case it should be

$$\nabla^2 \Phi= \frac{1}{h_1h_2h_3} \left[ \frac{\partial}{\partial u_1} \left( \frac{h_2h_3}{h_1} \frac{\partial \Phi}{\partial u_1} \right) + \frac{\partial}{\partial u_2} \left( \frac{h_3h_1}{h_2}\frac{\partial \Phi}{\partial u_2} \right) +\frac{\partial}{\partial u_3} \left(\frac{h_1h_2}{h_3}\frac{\partial \Phi}{\partial u_3}\right) \right]$$

Source: Hobson, Mathematical methods for Physics and Engineering, pg. 374.