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17Calculus Partial Differential Equations - Laplace Equation

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Laplace Equation

Laplace Equation

Laplace equation can be written in compact form as \(\nabla^2 f = 0\).
In a more clear and expanded form, we can write \[ \nabla^2 f = \dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} + \dfrac{\partial^2 f}{\partial z^2} = 0 \]

This equation can be found in several places in engineering and science including electrostatics, fluid dynamics and is the steady-state heat equation.

We can write this equation in cylindrical and spherical coordinates as follows.

cylindrical coordinates [source Wikipedia]

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

spherical coordinates [source Wikipedia]

\[ \nabla^2 f = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2\sin\theta} \frac{\partial}{\partial\theta} \left( \sin\theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2\sin^2\theta} \frac{\partial^2 f}{\partial \phi^2} = 0 \]

MIT OCW - Laplace Equation

video by MIT OCW

Michel vanBiezen - 2-D Laplace Equation

video by Michel vanBiezen

Really UNDERSTAND Differential Equations

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