Laplacian

From Mathematics wiki

1 Cartesian coordinates
2 Curvilinear coordinates
3 Vector Laplacian
4 See also

Cartesian coordinates

$\nabla^2 f(x,y,z)= \frac{\partial^2f} {\partial x^2} + \frac{\partial^2f} {\partial y^2} + \frac{\partial^2f} {\partial z^2}\,$ .

Curvilinear coordinates

Cylindrical coordinates

$\nabla^2 f(\rho, \theta, z) = {1 \over \rho} \frac{\partial }{ \partial \rho} \left( \rho \frac{\partial f }{ \partial \rho} \right) + {1 \over \rho^2} \frac{\partial^2 f }{ \partial \theta^2} + \frac{\partial^2 f }{ \partial z^2 }\,$ .

Spherical coordinates

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

Note the following useful identity:

$\frac{1}{r^2} \frac{\partial }{\partial r}r^2\frac{\partial f}{\partial r} = \frac{1}{r}\frac{\partial^2}{\partial r^2} (r f)$ ,

and therefore the alternate form

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

Laplacian

From Mathematics wiki

Contents

Cartesian coordinates

Curvilinear coordinates

Cylindrical coordinates

Spherical coordinates

Parabolic coordinates

Vector Laplacian

See also

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