1. ## Laplacian Operator

Here is a exercise that our teacher gave us. He said to try it out. I don't understand it. Help would be greatly appreciated.

Thanks

2. Suppose you have a bijective, analytic mapping $\displaystyle f:\mathbb{R}^3 \rightarrow \mathbb{R}^3$, given by

$\displaystyle (x,y,z) \mapsto (r(x,y,z), s(x,y,z), t(x,y,z))$.

Then

$\displaystyle \frac{\partial}{\partial x} = \frac{\partial r}{\partial x}\frac{\partial }{\partial r}+\frac{\partial s}{\partial x}\frac{\partial }{\partial s}+\frac{\partial t}{\partial x}\frac{\partial }{\partial t}$

$\displaystyle \frac{\partial}{\partial y} = \frac{\partial r}{\partial y}\frac{\partial }{\partial r}+\frac{\partial s}{\partial y}\frac{\partial }{\partial s}+\frac{\partial t}{\partial y}\frac{\partial }{\partial t}$

$\displaystyle \frac{\partial}{\partial z} = \frac{\partial r}{\partial z}\frac{\partial }{\partial r}+\frac{\partial s}{\partial z}\frac{\partial }{\partial s}+\frac{\partial t}{\partial z}\frac{\partial }{\partial t}$

Apply this theorem (which is the chain rule in 3 dimensions) to the cylindrical coordinate transformation.