Laplace's Equation

$$ \nabla^2 f = \nabla \cdot \nabla f = 0 $$

Where

• \(f\) is a twice-differentiable function,
• \(\nabla\) is the gradient operator,
• \(\nabla^2\) is the Laplace operator, and
• \(\nabla \cdot \) is the divergence operator.

In Different Coordinate Systems #

In rectangular coordinates: #

$$ \scriptsize \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} = 0$$

In cylindrical coordinates: #

$$ \scriptsize \nabla^2 f = \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \phi^2} + \frac{\partial^2 f}{\partial z^2} = 0$$

In spherical coordinates: #

$$ \scriptsize \nabla^2 f = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta}\frac{\partial^2 f}{\partial \varphi^2} = 0$$

Sources #

• Wikipedia
• Griffith’s Introduction to Electrodynamics’ back cover