Subsection 7.3.3 Laplace’s Equation
The equation of heat flow in steady state is known as Laplace’s equation. i.e., \(u\) does not change with time.
\begin{equation*}
\therefore \frac{\partial u}{\partial t} =0
\end{equation*}
and
\begin{equation*}
\nabla^{2}u = \frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}} =0.
\end{equation*}
which is a cartesian form of Laplace’s equation in three dimensions.
In cylindrical coordinates,
\begin{equation*}
\nabla^{2}u = \frac{\partial^{2}u}{\partial r^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial \theta^{2}}+\frac{\partial^{2}u}{\partial z^{2}} =0.
\end{equation*}
In spherical polar coordinates,
\begin{equation*}
\nabla^{2}u = \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{1}{r^{2}\sin\theta}\frac{\partial^{2}u}{\partial \phi^{2}} =0.
\end{equation*}
