Laplace's Equation

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The Laplace's Equation is a second-order partial differential equation.

[math]\displaystyle{ \nabla ^2f(x,y,z)=0\quad\iff\quad\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right)f(x,y,z)=0 }[/math]
[math]\displaystyle{ \nabla ^2f(x,y,z)=0 \quad\iff\quad \left({1 \over r^2} {\partial \over \partial r} (r^2 {\partial \over \partial r} + {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left(\sin \theta {\partial \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 \over \partial \varphi^2}\right) f(r,\theta,\phi)=0 }[/math]


References

2015

1999