User:Egm6321.f11.team2.Xia/RP7.3

Given
Sphericalcoordinate using Math/Physics convention:

$$\displaystyle \begin{cases} x_{1}=r\cos\overline{\theta}\cos\varphi=\xi_{1}\cos\xi_{2}\cos\xi_{3}\\ x_{2}=r\cos\overline{\theta}\sin\varphi=\xi_{1}\cos\xi_{2}\sin\xi_{3}\\ x_{3}=r\sin\overline{\theta}=\xi_{1}\sin\xi_{2}\end{cases}$$

(7.3.1) Where $$\displaystyle \overline{\theta}=\frac{\pi}{2}-\theta $$

Find
Find $$\displaystyle \triangle u$$ in this coordinate.

Solution
From 7.3.1, we can obtain

$$\displaystyle \begin{cases} x_{1}=r\sin\theta\cos\varphi=\xi_{1}\cos\xi_{2}\cos\xi_{3}\\ x_{2}=r\sin\theta\sin\varphi=\xi_{1}\cos\xi_{2}\sin\xi_{3}\\ x_{3}=r\cos\theta=\xi_{1}\sin\xi_{2}\end{cases} $$ (7.3.2) Thus $$\displaystyle \begin{cases} dx_{1}=\mathrm{cos}\phi\, r\,\mathrm{cos}\theta\, d\theta+\mathrm{cos}\phi\,\mathrm{sin}\theta\, dr-\mathrm{sin}\phi\, r\,\mathrm{sin}\theta\, d\phi\\ dx_{2}=\mathrm{sin}\phi\, r\,\mathrm{cos}\theta\, d\theta+\mathrm{sin}\phi\,\mathrm{sin}\theta\, dr+\mathrm{cos}\phi\, r\,\mathrm{sin}\theta\, d\phi\\ dx_{3}=\mathrm{cos}\theta\, dr-r\,\mathrm{sin}\theta\, d\theta\end{cases}$$ (7.3.3) Since $$ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}$$, then $$\displaystyle ds^{2}={dr}^{2}+{r}^{2}\,{d\theta}^{2}+{r}^{2}\,{\mathrm{sin}^2\theta}\,{d\phi}^{2}$$ (7.3.4) So $$\displaystyle h_{1}=1,\; h_{2}=r,\; h_{3}=r\sin\theta$$ (7.3.5) The Laplacian operator is as follows: $$\displaystyle \Delta u = \frac{1}{h_1h_2h_3}\sum_{i=1}^{3} \frac{\partial}{\partial \xi_i}\left [ \frac{h_1h_2h_3}{h_i^2}\frac{\partial u }{\partial \xi_i} \right ] $$ (7.3.6) hence $$\displaystyle \Delta u=\frac{1}{r^{2}\sin\theta}\left(\frac{\partial}{\partial r}\left(\frac{r^{2}\sin\theta}{1}\frac{\partial u}{\partial r}\right)+\frac{\partial}{\partial\theta}\left(\frac{r^{2}\sin\theta}{r^{2}}\frac{\partial u}{\partial\theta}\right)+\frac{\partial}{\partial\phi}\left(\frac{r^{2}\sin\theta}{r^{2}\sin^{2}\theta}\frac{\partial u}{\partial\phi}\right)\right)$$ (7.3.7)

$$\displaystyle \Delta u=\frac{1}{r^{2}\sin\theta}\left(\frac{\partial}{\partial r}\left(r^{2}\sin\theta\frac{\partial u}{\partial r}\right)+\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial u}{\partial\theta}\right)+\frac{\partial}{\partial\phi}\left(\frac{1}{\sin\theta}\frac{\partial u}{\partial\phi}\right)\right)$$ (7.3.8)