Chapter 13.5, Problem 9P

Find the steady-state distribution of temperature in a cube of side 10 if the temperature is ${100}^{°}$ on the face $z=0$ and $0^{°}$ on the other five faces. Hint: Separate Laplace’s equation in three dimensions in rectangular coordinates, and follow the methods of Section 2. You will want to expand 100 in the double Fourier series ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{\displaystyle \underset{m=1}{\overset{\infty }{\sum }}{a}_{nm}}}\sin \frac{n\pi x}{l}\sin \frac{m\pi y}{l}$

The coefficients $a_{nm}$ are determined by using the orthogonality of the functions $\sin (n\pi x/l)\sin (m\pi y/l)$ over the square, that is, ${\displaystyle {\int }_0^l{\displaystyle {\int }_0^l\sin }}\frac{n\pi x}{l}\sin \frac{m\pi y}{l}\sin \frac{p\pi x}{l}\sin \frac{q\pi y}{l}dxdy=0$ unless