MATH529: L12 BVPs in spherical coordinates

Overview

• Spherical coordinate Laplace BVP

Spherical coordinates

• $x=r\sin \theta \cos \varphi$, $y=r\sin \theta \sin \varphi$, $z=r\cos \theta$

• In spherical coordinates traversed distances are $dr,rd\theta ,r\sin \theta d\varphi$

  ${\displaystyle d{s}^2=d{r}^2+r^{2}d{\theta }^2+r^2{\sin }^2\theta d^2\varphi =\left[\begin{array}{lll}dr& d\theta & d\varphi \end{array}\right]\left[\begin{array}{lll}1& 0& 0\\ 0& r^2& 0\\ 0& 0& r^2{\sin }^2\theta \end{array}\right]\left[\begin{array}{l}dr\\ d\theta \\ d\varphi \end{array}\right]}$

• Nabla operator:

  ${\displaystyle \begin{array}{l}\nabla =\frac{\partial }{\partial r}{𝒆}_r+\frac{1}{r}\frac{\partial }{\partial \theta }{𝒆}_{\theta }+\frac{1}{r\sin \theta }\frac{\partial }{\partial \varphi }{𝒆}_{\varphi }\end{array}}$

• Gradient of scalar function $f(r,\theta ,\varphi )$

  ${\displaystyle \nabla f=\frac{\partial f}{\partial r}{𝒆}_r+\frac{\partial f}{r\partial \theta }{𝒆}_{\theta }+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi }{𝒆}_{\varphi }=(\frac{\partial f}{\partial r},\frac{\partial f}{r\partial \theta },\frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi })}$

Differential operators in spherical coordinates

• Divergence of vector function $𝑽(r,\theta ,\varphi )=(u(r,\theta ,\varphi ),v(r,\theta ,\varphi ),w(r,\theta ,\varphi ))$

  ${\displaystyle \nabla \cdot 𝑽(r,\theta ,\varphi )=\frac{1}{r^2\sin \theta }[\frac{\partial }{\partial r}(r^2\sin \theta u)+\frac{\partial }{\partial \theta }(r\sin \theta v)+\frac{\partial }{\partial \varphi }(rw)]}$

• Laplacian of scalar function $f(r,\theta )$, $\Delta f={\nabla }^{2}f=\nabla \cdot \nabla f$

  ${\displaystyle \Delta f=\frac{1}{r^2\sin \theta }[\frac{\partial }{\partial r}(r^2\sin \theta \frac{\partial f}{\partial r})+\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial f}{\partial \theta })+\frac{\partial }{\partial \varphi }(\frac{1}{\sin \theta }\frac{\partial f}{\partial \varphi })]}$ ${\displaystyle \Delta f=\frac{{\partial }^{2}f}{\partial r^2}+\frac{2}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}f}{\partial {\theta }^2}+\frac{\cot \theta }{r^2}\frac{\partial f}{\partial \theta }+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}f}{\partial {\varphi }^2}}$

Laplace equation in spherical coordinates

• Dirichlet problem: find $u(r,\theta )$ in sphere of radius $a$

  ${\displaystyle \frac{{\partial }^{2}u}{\partial r^2}+\frac{2}{r}\frac{\partial u}{\partial r}+\frac{1}{r^2}\frac{{\partial }^{2}u}{\partial {\theta }^2}+\frac{\cot \theta }{r^2}\frac{\partial u}{\partial \theta }=0\mathrm{for}r<a,u(a,\theta )=f\left(\theta \right)}$

• Separation of variables: $u(r,\theta )=R\left(r\right)\Theta \left(\theta \right)\Rightarrow$

  ${\displaystyle \frac{R^{\text{'}\text{'}}}{R}+\frac{2}{r}\frac{R^{\text{'}}}{R}+\frac{1}{r^2}\frac{{\Theta }^{\text{'}\text{'}}}{\Theta }+\frac{\cot \theta }{r^2}\frac{{\Theta }^{\text{'}}}{\Theta }=0\Rightarrow r^2(\frac{R^{\text{'}\text{'}}}{R}+\frac{2}{r}\frac{R^{\text{'}}}{R})=-\frac{{\Theta }^{\text{'}\text{'}}}{\Theta }-\cot \theta \frac{{\Theta }^{\text{'}}}{\Theta }=\lambda }$