Del operator in different coordinate

[KFC]

Homework Statement

I would like to transform the Del operator form rectangular coordinate system to spherical coordinate system. The find the Laplace operator in spherical coordinate.


2. The attempt at a solution
1) In rectangular coordinate system, Del operator is given by

[tex]
\nabla = \frac{\partial }{\partial x}\hat{x} + \frac{\partial }{\partial y}\hat{y} + \frac{\partial }{\partial z}\hat{z}
[/tex]

I know if you want to transform a vector in one coordinate system to a corresponding vector in other coordinate system, you got to know the transformation (matrix). For spherical coordinate system, the transformation matrix is given

[tex]
M = \left(
\begin{matrix}
\sin\theta\cos\varphi & \sin\theta\sin\varphi & \cos\theta \\
\cos\theta\cos\varphi & \cos\theta\sin\varphi & -\sin\theta \\
-\sin\varphi & \cos\varphi & 0
\end{matrix}
\right)
[/tex]

So that any vector [tex]\vec{r} = u_x \hat{x} + u_y \hat{y} + u_z \hat{z}[/tex] will be transformed as

[tex]
\left(
\begin{matrix}
u_r \\ u_\theta \\ u_\varphi
\end{matrix}
\right)
=
M
\left(
\begin{matrix}
u_x \\ u_y \\ u_x
\end{matrix}
\right)
[/tex]

Similarly, I apply the same transformation to Del operator

[tex]
\left(
\begin{matrix}
\frac{\partial}{\partial r} \\ \\ \frac{\partial}{\partial \theta} \\ \\ \frac{\partial}{\partial \varphi}
\end{matrix}
\right)
=
M
\left(
\begin{matrix}
\frac{\partial}{\partial x} \\ \\ \frac{\partial}{\partial y} \\ \\ \frac{\partial}{\partial z} \end{matrix}
\right)
[/tex]

But if you multiply all terms out, for example, the first component of the result vector

[tex]
\frac{\partial}{\partial r} =
\sin\theta\cos\varphi \frac{\partial}{\partial x} + \sin\theta\sin\varphi \frac{\partial}{\partial y} + \cos\theta \frac{\partial}{\partial z}
[/tex]

I don't know what to do next. How can I get the following result?

[tex]\nabla = \boldsymbol{\hat r}\frac{\partial}{\partial r} + \boldsymbol{\hat \theta}\frac{1}{r}\frac{\partial}{\partial \theta} + \boldsymbol{\hat \varphi}\frac{1}{r \sin\theta}\frac{\partial}{\partial \varphi}.[/tex]

2) Suppose the Del operator in spherical coordinate is in above form. To find Laplace operator, just dot product the Del operator with itself

[tex]\nabla\cdot\nabla = \nabla^2 = \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial \theta}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right) + \frac{1}{r \sin\theta}\frac{\partial}{\partial \varphi}\left(\frac{1}{r \sin\theta}\frac{\partial}{\partial \varphi}\right)[/tex]

But I remember the Laplace operator in spherical coordinate is

[tex] \nabla^2 = {1 \over r^2} {\partial \over \partial r} \left( r^2 {\partial \over \partial r} \right) + {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 \over \partial \phi^2}.[/tex]

I don't know what's wrong here! :(

 

[NoMoreExams]

Doesn't it basically amount to solving simultaneous equations to get it in that form, for 1)

 

[NoMoreExams]

Here's how I basically started:

[tex] x = r sin(\theta) cos(\phi) [/tex]
[tex] y = r sin(\theta) sin(\phi) [/tex]
[tex] z = r cos(\theta) [/tex]

So let's say we are trying to get the first term, let's differentiate each piece

[tex] \frac{\partial}{\partial x} = sin(\theta) cos(\phi) \frac{\partial}{\partial r} + ...[/tex] some terms with partials w.r.t. theta and phi

Similarly for y and z we get

[tex] \frac{\partial}{\partial y} = sin(\theta) sin(\phi) \frac{\partial}{\partial r} + ... [/tex]

[tex] \frac{\partial}{\partial z} = cos(\theta) \frac{\partial}{\partial r} + ... [/tex]

Now look at your definition for gradient in Cartesian and just multiply and group so for our [tex] \frac{\partial}{\partial r} [/tex] term we get

[tex] sin(\theta) cos(\phi) \frac{\partial}{\partial r} \cdot r sin(\theta) cos(\phi) + sin(\theta) sin(\phi) \frac{\partial}{\partial r} \cdot r sin(\theta) sin(\phi) + cos(\theta) \frac{\partial}{\partial r} \cdot r cos(\theta) [/tex]

Now if you group and use trig identities you are left with:

[tex] \frac{\partial}{\partial r} r [/tex]

 

[NoMoreExams]

For 2), do your parentheses indicate "argument of" or multiplication? I'm fairly certain in theirs they mean argument of