17Calculus Partial Differential Equations - Laplace Equation
17Calculus
Laplace Equation
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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 |
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Michel vanBiezen - 2-D Laplace Equation
video by Michel vanBiezen |
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Really UNDERSTAND Differential Equations
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