diff --git a/src/2_coordinates.md b/src/2_coordinates.md index e53b801be28b60e851323ea8d8eb00101eb1faac..a9d5e8bbe3c030e33f00a8b3c2d761576ed64da2 100644 --- a/src/2_coordinates.md +++ b/src/2_coordinates.md @@ -426,9 +426,9 @@ We have discussed four different coordinate systems: In a similar fashion it can be shown that for spherical coordinates, the Laplace operator acting on a function $\psi({\bf r})$ becomes: - $$\nabla^2 \psi (r,\vartheta,\varphi) = - \frac{1}{r^2} \frac{\partial}{\partial r^2} \left( r^2 \frac{\partial \psi(r,\vartheta,\varphi)}{\partial r} \right) + \frac{1}{r^2\sin^2\vartheta} \frac{\partial^2 \psi(r,\vartheta, \varphi)}{\partial \varphi^2} + \frac{1}{r^2\sin\vartheta} - \frac{\partial}{\partial \vartheta}\left( \sin\vartheta \frac{\partial\psi(r,\vartheta, \varphi)}{\partial \vartheta}\right).$$ + \begin{align} \nabla^2 \psi (r,\vartheta,\varphi) &= + \frac{1}{r^2} \frac{\partial}{\partial r^2} \left( r^2 \frac{\partial \psi(r,\vartheta,\varphi)}{\partial r} \right) \\ &+ \frac{1}{r^2\sin^2\vartheta} \frac{\partial^2 \psi(r,\vartheta, \varphi)}{\partial \varphi^2} \\ &+ \frac{1}{r^2\sin\vartheta} + \frac{\partial}{\partial \vartheta}\left( \sin\vartheta \frac{\partial\psi(r,\vartheta, \varphi)}{\partial \vartheta}\right).\end{align} This is however even more tedious (you do not have to show this). 6. [:grinning:] *Integration and coordinates I*