Laplace operator

From Wikipedia, the free encyclopedia
  (Redirected from Laplacian)
Jump to: navigation, search






\Delta f = \nabla^2 f = \nabla \cdot \nabla f






where the latter notations derive from formally writing \nabla = \left ( \frac{\partial}{\partial x_1} , \dots , \frac{\partial}{\partial x_n} \right ). Equivalently, the Laplacian of ƒ is the sum of all the unmixed second partial derivatives in the Cartesian coordinates x_i :

\Delta f = \sum_{i=1}^n \frac {\partial^2 f}{\partial x^2_i}










\int_{\partial V} \nabla u \cdot \mathbf{n}\, dS = 0,


\int_V \operatorname{div} \nabla u\, dV = \int_{\partial V} \nabla u \cdot \mathbf{n}\, dS = 0.


\operatorname{div} \nabla u = \Delta u = 0.


Density associated to a potential[edit]


q = -\epsilon_0\Delta\varphi\,

where \epsilon_0 is the electric constant.


\int_{\partial V} \mathbf{E}\cdot \mathbf{n}\, dS = \int_V \operatorname{div}\mathbf{E}\,dV=\frac1{\epsilon_0}\int_V q\,dV.


-\int_V \operatorname{div}(\operatorname{grad}\phi)\,dV =  \frac1{\epsilon_0}\int_V q\,dV.


\operatorname{div}(\operatorname{grad}\phi) = -\frac{1}{\epsilon_0}q


Energy minimization[edit]

Another motivation for the Laplacian appearing in physics is that solutions to  \Delta f = 0 in a region U are functions that make the Dirichlet energy functional stationary:

 E(f) = \frac{1}{2} \int_U \Vert \nabla f \Vert^2 \,dx.

To see this, suppose f\colon U\to \mathbb{R} is a function, and u\colon U\to \mathbb{R} is a function that vanishes on the boundary of U. Then

 \frac{d}{d\varepsilon}\Big|_{\varepsilon = 0} E(f+\varepsilon u) 
= \int_U \nabla f \cdot \nabla u \, dx
= -\int_U u \Delta f\, dx

where the last equality follows using Green's first identity. This calculation shows that if  \Delta f = 0, then E is stationary around f. Conversely, if E is stationary around f, then \Delta f=0 by the fundamental lemma of calculus of variations.

Coordinate expressions[edit]

Two dimensions[edit]

The Laplace operator in two dimensions is given by

\Delta f = \frac{\partial^2f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}


In polar coordinates,

 \Delta f 
&= {1 \over r} {\partial \over \partial r}
  \left( r {\partial f \over \partial r} \right) 
+ {1 \over r^2} {\partial^2 f \over \partial \theta^2}\\
&= {\partial^2 f \over \partial r^2}
+{1 \over r} {\partial f \over \partial r}

Three dimensions[edit]


In Cartesian coordinates,

\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.

In cylindrical coordinates,

 \Delta f 

In spherical coordinates:

 \Delta f 
= {1 \over r^2} {\partial \over \partial r}
  \left(r^2 {\partial f \over \partial r} \right) 


In general curvilinear coordinates ( \xi^1, \xi^2, \xi^3 ):

\nabla^2 = \nabla \xi^m \cdot \nabla \xi^n {\partial^2 \over \partial \xi^m \partial \xi^n} + \nabla^2 \xi^m {\partial \over \partial \xi^m },

where summation over the repeated indices is implied.

N dimensions[edit]


 \Delta f
+ \frac{N-1}{r} \frac{\partial f}{\partial r}
+ \frac{1}{r^2} \Delta_{S^{N-1}} f

where \Delta_{S^{N-1}} is the Laplace–Beltrami operator on the (N−1)-sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as

\frac{1}{r^{N-1}} \frac{\partial}{\partial r} \Bigl(r^{N-1} \frac{\partial f}{\partial r} \Bigr).


Spectral theory[edit]


-\Delta f = \lambda f.



Laplace–Beltrami operator[edit]


\Delta f = \mathrm{tr}(H(f))\,\!



 \Delta f = d^* d f\,


\Delta \alpha = d^* d\alpha + dd^*\alpha.\,





\frac {1}{c^2}{\partial^2 \over \partial t^2 }
{\partial^2 \over \partial z^2 }.


See also[edit]


  1. ^ Evans 1998, §2.2
  2. ^ Gilbarg & Trudinger 2001, Theorem 8.6
  3. ^ Gilbarg & Trudinger 2001, Corollary 8.11


External links[edit]