Questions tagged [laplacian]

The properties of the Laplace differential operator, denoted $\Delta$ or $\nabla^2$, and defined as the divergence of the gradient. For Laplace equation, see (harmonic-functions)

Laplacian Operator is a derivative operator which is used to find edges in an image.

Mathematically, the Laplacian operator is defined as: $$\nabla^2 \equiv \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$

  • The Laplacian is a scalar operator. If it is applied to a scalar field, it generates a scalar field.

  • If the scalar Laplacian operator is applied to a vector field, it acts on each component in turn and generates a vector field.

Note that the operator $~\nabla ^2~$ is commonly written as $~\Delta~$ by mathematicians.

Differences with other operators:

The major difference between Laplacian and other operators like Prewitt, Sobel, Robinson and Kirsch is that these all are first order derivative masks but Laplacian is a second order derivative mask. Another difference between Laplacian and other operators is that unlike other operators Laplacian didn’t take out edges in any particular direction but it take out edges in following classification.

$1.\quad$ Inward Edges $\qquad$ $2.\quad$ Outward Edges

Applications:

The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics, and appears in

$1.\quad$ Laplace's equation $$\nabla^2 \phi=0$$

$2.\quad$ The Helmholtz differential equation$$\nabla^2 \phi+k^2 \phi=0$$

$3.\quad $ The wave equation$$\nabla^2 \phi=\frac{1}{c^2}\frac{\partial^2 \phi}{\partial t^2}$$

$4.\quad$ The Schrödinger equation$$i \hbar \frac{\partial }{\partial t}\psi(x,y,z,t)=\left[-\frac{\hbar^2}{2m}\nabla^2+V(x)\right]\psi(x,y,z,t)$$

Laplacian, Various Coordinates :

$1.\quad$ In rectangular coordinates $~(x,y,z)~$: $$\nabla^2\equiv \nabla \cdot \nabla \equiv \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$

$2.\quad$ In cylindrical polar coordinates $~(r,\theta,z)~$:$$\nabla^2\equiv \frac{\partial^2}{\partial r^2}+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}+\frac{\partial^2}{\partial z^2}$$

$3.\quad$ In spherical polar coordinates $~(r,\theta,\phi)~$:$$\nabla^2\equiv \frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}+\frac{\cos \theta}{r^2\sin \theta}\frac{\partial}{\partial \theta}+\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial \phi^2}$$

Note: This tag is also for questions concerning the properties, such as self-adjointness, compactness of inverse, and spectral structure of the Laplace differential operator. The operator, denoted $\Delta$ or $\nabla^2$, is defined as the divergence of the gradient, and generalized to the Laplace-Beltrami and Laplace-deRham operators.

References:

https://en.wikipedia.org/wiki/Laplace_operator

http://mathworld.wolfram.com/Laplacian.html

https://nptel.ac.in/courses/115101005/downloads/lectures-doc/Lecture-5.pdf

1066 questions
141
votes
7 answers

Intuitive interpretation of the Laplacian Operator

Just as the gradient is "the direction of steepest ascent", and the divergence is "amount of stuff created at a point", is there a nice interpretation of the Laplacian Operator (a.k.a. divergence of gradient)?
koletenbert
  • 3,590
  • 4
  • 23
  • 35
65
votes
5 answers

Understanding the Laplace operator conceptually

The Laplace operator: those of you who now understand it, how would you explain what it "does" conceptually? How do you wish you had been taught it? Any good essays (combining both history and conceptual understanding) on the Laplace operator, and…
26
votes
1 answer

Symmetries and eigenvalues of the Laplacian.

Lets consider a domain $\Omega \subseteq \mathbb R^2$ smooth enough, and the eigenvalue for the laplacian \begin{align} -\Delta u &= \lambda u &x\in\Omega\\ u &= 0 &x\in \partial \Omega \end{align} I am interested in an explicit relation between…
16
votes
2 answers

Are spherical harmonics harmonic?

According to Wikipedia, a harmonic function is one which satisfies: $$ \nabla^2 f = 0 $$ The spherical harmonics (also according to Wikipedia) satisfy the relation $$ \nabla^2 Y_l^m(\theta,\phi) = -\frac{l(l+1)}{r^2} Y_l^m(\theta,\phi) $$ which is 0…
vibe
  • 926
  • 4
  • 11
16
votes
0 answers

Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: $$K(t,x,x)\sim (4\pi…
12
votes
2 answers

Proof that $a\nabla^2 u = bu$ is the only homogenous second order 2D PDE unchanged/invariant by rotation

Looking for feedback and maybe simpler intuition for my proof of the theorem, shown below The statement of the theorem: Theorem Among all second-order homogeneous PDEs in two dimensions with constant coefficients, show that the only ones that do…
12
votes
1 answer

Two fluids flowing perpendicular in thermal contact with a Wall [Help to mathematically model]

I will try to describe briefly how I am modelling the problem. (Please bear with the length). The governing equation describing temperature for a block at steady state is $$\nabla^2 T = 0$$ where $\nabla^2 T = \frac{\partial^2}{\partial x^2} +…
12
votes
2 answers

Does the symbol $\nabla^2$ has the same meaning in Laplace Equation and Hessian Matrix?

we know that the Laplace Equation can be written in the form: $$\nabla^2 \Phi=0$$ while in this equation,the symbol $\nabla^2 \Phi$ stand for $\sum_{i=1}^n\frac{\partial^2 \Phi}{\partial x_i^2}$. At the same time, the Hessian Matrix is also denoted…
maple
  • 2,453
  • 2
  • 23
  • 33
12
votes
2 answers

Laplacian of a radial function

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a radial function, i.e. $f(x)=f(r)$ with $r:=\left\|x\right\|_2$. As stated at Wikipedia $$\Delta f=\frac{1}{r^{n-1}}\frac{d}{dr}(r^{n-1}f')$$ What's the most elegant way to prove this fact?
0xbadf00d
  • 12,822
  • 9
  • 29
  • 68
11
votes
4 answers

Show Laplace operator is rotationally invariant

I'm trying to show the Laplace operator is rotationally invariant. Essentially this boils down to showing $$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 f}{\partial u^2} + \frac{\partial^2 f}{\partial…
11
votes
1 answer

Why is Laplacian ubiquitous?

What I am asking here is a moral question. Mathematically moral, don't bother physics. I mean, Euler's number is ubiquitous because, among all the exponentials, it alone is its own derivative with all the consequences we know. I know that the…
Zappa
  • 183
  • 6
11
votes
1 answer

Why is a diffeomorphism an isometry if and only if it commutes with the Laplacian?

I came across the following statement in a book on automorphic forms: In general, on a Riemannian manifold, the Laplace-Beltrami operator $\Delta$ is characterized by the property that a diffeomorphism is an isometry if and only if it leaves…
4461013
  • 145
  • 9
10
votes
2 answers

Fundamental solution to the Poisson equation by Fourier transform

The fundamental solution (or Green function) for the Laplace operator in $d$ space dimensions $$\Delta u(x)=\delta(x),$$ where $\Delta \equiv \sum_{i=1}^d \partial^2_i$, is given by $$ u(x)=\begin{cases} \dfrac{1}{(2-d)\Omega_d}|x|^{2-d}\text{ for }…
Brightsun
  • 6,193
  • 1
  • 21
  • 48
10
votes
3 answers

Neumann problem for Laplace equation on Balls by using Green function

It is well known that for Dirichlet problem for Laplace equation on balls or half-space, we could use the green function to construct a solution based on the boundary data. For instance, one could find a nice proof in Evans PDE book, chapter 2.2, it…
spatially
  • 5,018
  • 3
  • 18
  • 32
9
votes
1 answer

Reference for the relation of the Casimir element to the Laplace Beltrami operator

Wikipedia says, "If $G$ is a Lie group with Lie algebra $\mathfrak {g}$, the choice of an invariant bilinear form on $\mathfrak {g}$ corresponds to a choice of bi-invariant Riemannian metric on $G$. Then under the identification of the universal…
1
2 3
71 72