91Ó°ÊÓ

The curved surface is$u=\cos \mathrm{θ}$.

The total of the second-order partial derivatives of R, the unknown function, in Cartesian coordinates equals 0, according to Laplace's equation.

Write the Laplace equation in the spherical coordinates.

$$\begin{gathered} {∇}^2=\frac{1}{r^2}\frac{∂}{∂r}\left(r^2\frac{∂u}{∂r}\right)+\frac{1}{r^2\sin \mathrm{θ}}\frac{∂}{∂\mathrm{θ}}\left(\sin \mathrm{θ}\frac{∂u}{∂\mathrm{θ}}\right)+\frac{1}{r^2{\sin }^2\mathrm{θ}}\frac{{∂}^{2}u}{∂{\mathrm{Φ}}^2} \\ =0 \end{gathered}$$

Use separation of variables to separate the equation.

$u=R\left(r\right)\mathrm{Θ}\left(\mathrm{θ}\right)\mathrm{Ψ}\left(\mathrm{Φ}\right)$

Write the general solution as the series of basic functions.

$u=\underset{l=0}{\overset{∞}{∑}}{c}_l{r}^{l}Pl\left(\cos \mathrm{θ}\right)$

First check the odd-even nature of the function.

$$\begin{gathered} f\left(u\right)=\cos \mathrm{θ} \\ f\left(-u\right)=\cos \left(-\mathrm{θ}\right) \\ f\left(-u\right)=\cos \mathrm{θ} \\ f\left(-u\right)=f\left(u\right) \end{gathered}$$

Hence, the function is even.

The curved surface is at $u=\cos \mathrm{θ}$and the equatorial plane is:

$$\begin{gathered} c_l=\underset{0}{\overset{1}{∫}}\cos \mathrm{θ}{P}_l\left(\cos \mathrm{θ}\right)d\mathrm{θ} \\ =\frac{l+1}{2l+1}\underset{0}{\overset{1}{∫}}{P}_{l+1}\left(\cos \mathrm{θ}\right)d\mathrm{θ}+\frac{l}{2l+1}\underset{0}{\overset{1}{∫}}{P}_{l-1}\left(\cos \mathrm{θ}\right)d\mathrm{θ} \end{gathered}$$

Here,

$$\begin{gathered} \underset{0}{\overset{1}{∫}}{P}_{l+1}d\mathrm{θ}=\left\{\begin{array}{l}0,l\text{odd}\\ {\left(-1\right)}^{(l+4)/2}\frac{\left(l+2\right)!}{2^{l+2}{\left[\frac{l+2}{2}!\right]}^2(l+1)},                      \text{if}l\text{is even}\end{array}\right. \\ \underset{0}{\overset{1}{∫}}{P}_{l-1}d\mathrm{θ}=\left\{\begin{array}{l}{\mathrm{δ}}_{1,l},          l\text{odd}\\ {\left(-1\right)}^{(l+2)/2}\frac{\left(l\right)!}{2^l{\left[\frac{l}{2}!\right]}^2\left(l-1\right)},                      \text{if}l\text{is even}\end{array}\right. \end{gathered}$$

Hence,the interior temperature in a hemisphere is given by,

.$u=1-\frac{1}{2}r{P}_1\left(\cos \mathrm{θ}\right)+\frac{7}{8}{r}^3{P}_3\left(\cos \mathrm{θ}\right)-\frac{11}{16}{r}^5{P}_5\left(\cos \mathrm{θ}\right)…$