- #1
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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! :(
Last edited: