$c\frac{\text{d}\text{e}\text{l}\text{t}}{\text{d}\text{e}\text{l}}=\frac{1}{r^2}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r^2\frac{\text{d}\text{e}\text{l}\text{t}}{\text{d}\text{e}\text{l}\text{r}})+\frac{1}{r^{2}sin}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}}(sin\frac{\text{d}\text{e}\text{l}\text{t}}{\text{d}\text{e}\text{l}})+\frac{1}{r^{2}si{n}^2}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}}(\frac{\text{d}\text{e}\text{l}\text{t}}{\text{d}\text{e}\text{l}})+q_v$

$$ is constant:

$\frac{\text{d}\text{e}\text{l}\text{t}}{\text{d}\text{e}\text{l}}=a[\frac{1}{r}\frac{de{l}^2(rt)}{del{r}^2}+\frac{1}{r^{2}sin}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}}(sin\frac{\text{d}\text{e}\text{l}\text{t}}{\text{d}\text{e}\text{l}})+\frac{1}{r^{2}si{n}^2}\frac{de{l}^{2}t}{\text{d}\text{e}\text{l}{}^2}]+\frac{q_v}{c}$

a$)$ Please derive the heat conduction equation under the spherical coordinate using the

control volume analysis on the volume unit shown above.

b$)$ Derive the equation again through the Lame coefficient.