Laplasyen

Laplasyen $(\nabla^2 \phi)$, skaler bir $\phi\,$ alanının gradyanı alınarak elde edilen vektörün diverjansıdır. Fizikteki birçok diferansiyel denklem laplasyen içerir.

Kartezyen koordinat sisteminde

 $\nabla^2 \phi = \operatorname{div}\, (\operatorname{grad}\, \phi) = \vec \nabla \cdot (\vec \nabla \phi)$ $= \frac{\partial}{\partial x} \left(\frac{\partial \phi}{\partial x}\right) + \frac{\partial}{\partial y} \left(\frac{\partial \phi}{\partial y}\right) + \frac{\partial}{\partial z} \left(\frac{\partial \phi}{\partial z}\right)$ $= \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}$

Küresel koordinat sisteminde

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

Silindirik koordinat sisteminde

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

Tansör gösterimi

 $\nabla^2 T=T_{,kk}$