Laplacian Definition and 152 Threads

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols








{\displaystyle \nabla \cdot \nabla }
,






2




{\displaystyle \nabla ^{2}}
(where






{\displaystyle \nabla }
is the nabla operator), or



Δ


{\displaystyle \Delta }
. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δf(p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f(p).
The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics, where the operator gives a constant multiple of the mass density when it is applied to the gravitational potential due to the mass distribution with that given density. Solutions of the equation Δf = 0, now called Laplace's equation, are the so-called harmonic functions and represent the possible gravitational fields in regions of vacuum.
The Laplacian occurs in differential equations that describe many physical phenomena, such as electric and gravitational potentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. The Laplacian represents the flux density of the gradient flow of a function. For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplacian of the chemical concentration at that point; expressed symbolically, the resulting equation is the diffusion equation. For these reasons, it is extensively used in the sciences for modelling a variety of physical phenomena. The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology. In image processing and computer vision, the Laplacian operator has been used for various tasks, such as blob and edge detection.

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  1. davidbenari

    Proving some Dirac-Delta identity that uses Laplacian

    Homework Statement Prove that ##(x^2+y^2+z^2)\nabla^2[\delta(x)\delta(y)\delta(z)]=6\delta(x)\delta(y)\delta(z)## Homework Equations ##\delta''(x)/2=\delta(x)/x^2## The Attempt at a Solution I have obtained this: ##6\delta(x)\delta(y)\delta(z) +...
  2. cvex

    How to get the laplacian of a scalar field?

    Hi, I am trying to calculate the laplacian of a scalar field but I might actually need something else. So basically I am applying reaction diffusion on a 2d image. I am reading the neighbours, multiplying them with these weights and then add them. This works great. I don't know if what I am...
  3. M

    Why Is the Index Shifted in Laplacian Eigenvalues?

    hi pf! I am reading a text and am stuck at a part. this is what is being said: If ##g## is a graph we have ##L(g) + L(\bar{g}) = nI - J## where ##J## is the matrix of ones. Let ##f^1,...f^n## be an orthogonal system of eigenvectors to ##L(g) : f^1 = \mathbb{1}## and ##L(g)f^i = \lambda_i...
  4. N

    Why Does the Laplacian of 1/Vector r Equal Zero?

    Homework Statement Show that \nabla^{2}\left(\frac{1}{\overrightarrow{r}}\right)=0Homework Equations The Attempt at a Solution Let \nabla=\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z} and \overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}...
  5. S

    Laplacian of a vector function

    Problem: The vector function A(r) is defined in spherical polar coordinates by A = (1/r) er Evaluate ∇2A in spherical polar coordinates Relevant equation: I'm assuming I have to use the equation 1671 on this website But I haven't got a clue as to how I would apply it since, for example, I...
  6. S

    Does the Generalised Laplacian satisfy a certain relation?

    Hi, I was wondering if the following relation holds: $$ \frac{1}{r^{D-1}} \frac{\partial}{\partial r} \left( r^{D-1} \frac{\partial}{\partial r} \right) \psi = \frac{1}{r^{\frac{D-1}{2}}} \frac{\partial ^2}{\partial r^2} \left( r^{\frac{D-1}{2}} \right) \psi $$ I've seen that the LHS evaluates...
  7. K

    Poisson Summation in Heat Equation (Polar Coordinates)

    Homework Statement I'm currently trying to follow a derivation done by Shankar in his "Basic Training in Mathematics" textbook. The derivation is on pages 343-344 and it is based on the solution to the two dimensional heat equation in polar coordinates, and I'm not sure how he gets from one...
  8. H

    Laplacian term in Navier-Stokes equation

    I am trying to derive part of the navier-stokes equations. Consider the following link: http://www.gps.caltech.edu/~cdp/Desktop/Navier-Stokes%20Eqn.pdf Equation 1, without the lambda term, is given in vector form in Equation 3 as \eta\nabla^2\mathbf{u}. However, when I try to get this from...
  9. C

    How is the Laplacian Applied to Retarded Potential?

    Homework Statement See attachment. Homework Equations The Attempt at a Solution I'm not understanding how the laplacian is creating those 3 terms in 5.4.5. I just understand the basics that laplacian on f(x,y) = d2f/dx2 + d2f/dy2. Can someone elaborate? Thanks in advance. EDIT: Just...
  10. laramman2

    Why does the Laplacian operator still maintain its unit vectors i, j, k?

    When two vectors are dotted, the result is a scalar. But why here http://www.cobalt.chem.ucalgary.ca/ziegler/educmat/chm386/rudiment/mathbas/vectors.htm , the del-squared still maintains its unit vectors i, j, k? Isn't it this way ∇2 = (∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2) and not (i∂2/∂x2 + j∂2/∂y2 +...
  11. I

    Vector Calculus - Laplacian on Scalar Field

    A scalar field \psi is dependent only on the distance r = \sqrt{x^{2} + y^{2} + z^{2}} from the origin. Show: \partial_{x}^{2}\psi = \left(\frac{1}{r} - \frac{x^{2}}{r^{3}}\right)\frac{d\psi}{dr} + \frac{x^{2}}{r^{2}}\frac{d^{2}\psi}{dr^{2}} I've used the chain and product rules so...
  12. C

    Integration and Laplacian in polar coordinates

    Homework Statement I have a function y that is axisymmetric, so that y=y(r). I want to solve for r such that ∇2y(r) = Z. Can anyone tell me if I'm following the right procedure? I'm not sure since there are two "∂/∂r"s present... Homework Equations ∇2 = (1/r)(∂/∂r)(r*(∂/∂r)) +...
  13. T

    MHB Eigenvalues of Laplacian are non-negative

    Hi, I need to learn the following proof and I'm having trouble getting my head round it. Any help would be appreciated. Show that if vector x in R^n with components x=(x1,x2,...,xn), then x.Lx=0.5 sum(Aij(xi-xj)^2) where A is the graphs adjacency matrix, L is laplacian. Then use this result to...
  14. J

    Understanding Matrix Calculus: Laplacian, Hessian, and Jacobian Explained

    Hellow! I was studying matrix calculus and learned new things as: \frac{d\vec{y}}{d\vec{x}}=\begin{bmatrix} \frac{dy_1}{dx_1} & \frac{dy_1}{dx_2} \\ \frac{dy_2}{dx_1} & \frac{dy_2}{dx_2} \\ \end{bmatrix} \frac{d}{d\vec{r}}\frac{d}{d\vec{r}} = \frac{d^2}{d\vec{r}^2} = \begin{bmatrix}...
  15. A

    Potential, field, Laplacian and Spherical Coordinates

    Homework Statement Say I am given a spherically symmetric potential function V(r), written in terms of r and a bunch of other constants, and say it is just a polynomial of some type with r as the variable, \frac{q}{4\pi\varepsilon_o}P(r), and we are inside the sphere of radius R, so r<R…...
  16. S

    Integrals featuring the laplacian and a tensor

    Ok, so I'd like some advice on doing integrals that involve a laplacian and a tensor for example =\int\frac{\delta}{\delta A_{\mu}}\frac{1}{4M^{2}}(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho})\frac{\partial^{2}}{\partial x^{2}}(\partial^{\rho}A^{\sigma}-\partial^{\sigma}A^{\rho}) where...
  17. Einj

    Laplacian in toroidal coordinates

    Hi everyone, I would like to write the Laplacian operator in toroidal coordinate given by: $$ \begin{cases} x=(R+r\cos\phi)\cos\theta \\ y=(R+r\cos\phi)\sin\theta \\ z=r\sin\phi \end{cases} $$ where r and R are fixed. How do I do? More generally how do I find the Laplacian under a...
  18. A

    Laplacian of 1/r Explodes at Origin

    Ok, there are a couple of other threads about this, but they don't seem to answer my question. If I take the double derivative of 1/r, I'll get 2/r^3, but if I take the laplacian, I get something different. Why? Namely: \frac{d}{dr}\frac{d}{dr}(\frac{1}{r}) = \frac{d}{dr}...
  19. M

    LaPlacian joint probability density function.

    A joint pdf is given as pxy(x,y)=(1/4)^2 exp[-1/2 (|x| + |y|)] for x and y between minus and plus infinity. Find the joint pdf W=XY and Z=Y/X. f(w,z)=∫∫f(x,y)=∫∫(1/4)^2*e^(-(|x|+|y|)/2)dxdy -∞<x,y<∞ Someone told me I can not use Jacobian because of the absolute value. Is that true? So far this...
  20. M

    What is the joint pdf for W=XY and Z=Y/X in the LaPlacian distribution?

    A joint pdf is given as pxy(x,y)=(1/4)^2 exp[-1/2 (|x| + |y|)] for x and y between minus and plus infinity. Find the joint pdf W=XY and Z=Y/X. f(w,z)=∫∫f(x,y)=∫∫(1/4)^2*e^(-(|x|+|y|)/2)dxdy -∞<x,y<∞ Someone told me I can not use Jacobian because of the absolute value. Is that true...
  21. M

    Learn About the Laplacian Property

    Hello all, I came across this rather mysterious, for me of course, statement. It is stated that, starting from the solution of the Laplacian under definite boundary conditions and for a given geometry, if a portion of the domain is altered in such a way that the maximum distance between...
  22. B

    Expansion in terns of Laplacian eigenfunctions

    Hi ! I am currently studying String theory in Polchinski's book. In section 6.2, eq. 6.2.2, he takes an arbitrary function X(\sigma) defined on a Riemann Surface M. Then he expands it on a complete set of eigenfunctions of the laplacian, X(\sigma)=\sum_I x_I X_I(\sigma) with \Delta X_I =...
  23. M

    Laplacian in Slanted Coordinates

    Whoops, I figured it out!
  24. C

    Spectra of the Laplacian (Matrix)

    Hi all I was reading this paper about spectral clustering: Ng et all. (Nips 2001) http://ai.stanford.edu/~ang/papers/nips01-spectral.pdf Short question: What is the connection between the eigenvalues of a Laplacian? Long question: I got stuck on why the process makes sense...
  25. M

    How Can the Laplacian Relate to Einstein Manifolds?

    I am researching a hypothesis and looking for anyone who is familiar with differential topology (specifically Einstein manifolds). I have access to the Besse book Einstein Manifolds but am also looking for any open questions in differential topology that I am not aware of. I am attempting to...
  26. S

    Gradient and Laplacian of three functions

    Hello, I've been reading up on Smoothed Particle Hydrodynamics. While reading some papers I found some math that I do not know how to do because I never took multi variable calculus. I need the gradient and laplacian of all three of the following functions ( h is a constant )...
  27. S

    Bochner-Weitzenbock formula (-> Laplacian)

    Hi! I'm trying to understand a proof for the Bochner-Weitzenbock formula. I'm sorry I have to bother you with such a basic question but I've worked at this for more than an hour now, but I just don't get the very first step, i.e.: Where we are in a complete Riemannian manifold, f \in...
  28. P

    MHB Dirichlet problem for the laplacian in the strip

    I'm looking for all functions $u$ harmonic in $S$ and continuous in $\overline S$ such that $$u(a,y)=u(b,y)=0,\forall y$$ and $$\lim_{|y|\rightarrow +\infty} u(x,y)=0$$ where $S$ is the strip $\{a<\operatorname{Re}(z)<b\}$ My strategy is the following. I know that if $g$ is continuous on...
  29. C

    Laplacian and harmonic functions

    Homework Statement The hyperbolic coordinate sysem onthe first quadrant in R^2 is defined by the change of variables K(u,v)=(x(u,v),y(u,v))=(ve^u,ve^(-u)) u is in R,and v>0, find all harmonic functions on the first quadrant in R^2 which are constant on all rectangular hyperbolas xy=c , c is a...
  30. P

    Laplacian in Spherical Coordinates

    Homework Statement Homework Equations All above. The Attempt at a Solution Tried the first few, couldn't get them to work. Any ideas, hopefully for each step?
  31. E

    Del vs. Laplacian Operator : Quick Question

    Just to clarify: The del operator's a vector and the laplacian operator is just a scalar?
  32. A

    The correct domain of self-adjointness for the Laplacian

    The "correct" domain of self-adjointness for the Laplacian Consider the Hilbert space L^2(\mathbb R^d), and consider the Laplacian operator \Delta on this space. We want to find a domain, D(\Delta) \subset L^2(\mathbb R^d), such that this guy is a self-adjoint operator. We have been talking...
  33. iVenky

    Is the Laplacian a Vector or Scalar?

    Here's the link that I read for Laplacian- http://hyperphysics.phy-astr.gsu.edu/hbase/lapl.html It looks as if the laplacian is scalar but the point is we know that ∇x∇xA= ∇(∇.A) - ∇2A This means that laplacian should be vector in nature which contradicts what was given in the link...
  34. M

    Singularity in Laplacian operator

    I am trying to understand the following basic problem, \partial_{xx} f^\alpha (x) = \alpha (\alpha-1) \frac{1}{f^{2-\alpha}} \partial_x f + \alpha \frac{1}{f^{1-\alpha}} \partial_{xx} f So it is not hard to see that if f tends to zero the laplacian becomes undefined (im not sure if i...
  35. M

    Laplacian over a radial function for charge density

    As you probably can see from the above shot, I'm determining charge density via the Laplacian over the potential (phi). I understand the mathematical steps, just confused on the factor of 4pi that pops up in the denominator. I think I understand why you would do that and here's my reasoning...
  36. C

    From London Equations to Penetration Depth(Integrate Laplacian)

    (In SI units) Start with London's 2nd equation in Superconductivity, curl J = 1/(μ*λ²), and Ampere's curl B = μ*j. Then we curl both side curl curl B = μ* curl J and we do the substitution. So curl curl B = 0 - del²B which is the laplacian operator. My question is...how to integrate...
  37. M

    Can complex analysis be used to solve PDEs other than the Laplacian?

    Hey all, I was reading up on Harmonic functions and how every solution to the laplace equation can be represented in the complex plane, so a mapping in the complex domain is actually a way to solve the equation for a desired boundary. This got me wondering: is this possible for other PDEs...
  38. A

    Laplace Operator: Vector Dot Product & 2nd Derivative

    Hi guys The Laplace Operator The Laplace operator is defined as the dot product (inner product) of two gradient vector operators: When applied to f(x,y), this operator produces a scalar function: My question is how a vector dot product ( del operator vector dot product...
  39. N

    Chain-rule issue on Laplacian equation

    Homework Statement "The flow of a fluid past a wedge is described by the potential ψ(r,θ) = -crαsin(αθ), where c and α are constants, and (r,θ) are the cylindrical coordinates of a point in the fluid (the potential is independent of z). Verify that this function satisfies Laplace's...
  40. K

    Eigenvalues for a 400x400 normalized laplacian of a graph

    This is related to spectral graph theory. I am getting the following eigenvalues for a 400x400 matrix which is a normalized laplacian matrix of a graph. The graph is not connected. So why am i getting a> a negative eigenvalue. b> why is not second eigenvalue 0? ... I used colt(java) and octave...
  41. M

    Electric potential inside and outside spherical capacitator using laplacian

    Homework Statement Find the electric potential inside and outside a spherical capacitor, consisting of two hemispheres of radius 1 m. joined along the equator by a thin insulating strip, if the upper hemisphere is kept at 220 V and the lower hemisphere is grounded Homework Equations...
  42. R

    PDE Help - Eigenfunction of a LaPlacian

    [b]1. A function "v" where v(x,y,z)≠0 is called an eigenfunction of the Laplacian Δ (on some region Ω - with specified homogenous BC) if v satisifies the BC ad also Δv=-λv on Ω for some number λ. Part A: Give an example of an eigenfunction of Δ when Ω is the cube [0,∏]3 with Dirichlet BCs...
  43. C

    About Dirichlet problem of Laplacian equation in 3D sperical coordinates

    i wanted to solve an electrostatic problem that includes an infinite conical conductor whose vertex is the origin of the spherical coordinates, and i knew that the general solution of laplacian equation in spherical coordinates is: u(r,θ)=Ʃ(An*rn+Bn*r-n-1), n>=0 however, the boundary conditions...
  44. U

    Laplacian of the metric

    When reading about the ideas behind ricci flow, I've often read that the ricci tensor is proportional to the laplacian of the metric, but only in harmonic coordinates. Can someone explain this to me? What laplacian operator would one use to show this as there are many different laplacians in...
  45. A

    What is the Laplacian of the scalar potential with an extra term?

    On page 35 of Jackson's Classical Electrodynamics, he calculates the Laplacian of a scalar potential due to a continuous charge distribution. In the expression for the potential, the operand of the Laplacian is \frac{1}{|r-r'|}, where r is the the point where the potential is to be...
  46. X

    Converting the Laplacian into polar coordinates

    I need to convert the Laplacian in two dimensions to polar coordinates. \nabla^2 u=\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} I am having problems with computing the second derivatives using the chain rule. For example, the first derivative with respect to x...
  47. D

    Vector laplacian and del squared confusion.

    Hello all, I am reading a research paper and have found the equation below: http://latex.codecogs.com/gif.latex?\mathbf{z}%20=%20\mathbf{a}%20-%20%28\nabla^2E%28\mathbf{t}%29%29^{-1}\Delta%20E%28\mathbf{t}%29%29 in which E is some function with the variable t being the vector input, and a...
  48. M

    Laplacian on Riemannian manifolds

    hi friends :) is there someone who has studied the spectrum of a Riemannian Laplacian? I have a question on this subject. Thank you very much for answering me.
  49. M

    How to Derive the Finite Difference Laplacian in Various Coordinates?

    Updated: Finite difference of Laplacian in spherical Homework Statement I understand the problem a little better now and am revising my original plea for help. I don't actually need to integrate the expression. Integration was just one technique of arriving at a finite difference...
  50. Q

    Laplacian VS gradient of divergence

    i don't really understand the difference :( ∇2V versus ∇ (∇ . V) ? can anyone give me a simple example to showcase the application difference? thanks!
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