Spherical Coordinate Form of Laplace's Equation

Theorem

Laplace's equation can be expressed in spherical coordinates as:

$\dfrac 1 {r^2} \map {\dfrac \partial {\partial r} } {r^2 \dfrac {\partial V} {\partial r} } + \dfrac 1 {r^2 \sin^2 \theta} \dfrac {\partial^2 V} {\partial \phi^2} + \dfrac 1 {r^2 \sin \theta} \map {\dfrac \partial {\partial \theta} } {\sin \theta \dfrac {\partial V} {\partial \theta} } = 0$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Laplace's equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Laplace's equation