What is Laplacian: Definition and 152 Discussions

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. I The Vector Laplacian: Understanding the Third Term

Suppose ##A = A_i\mathbf{\hat{e}}_i## and ##B = B_i\mathbf{\hat{e}}_i## are vectors in ##\mathbb{R}^3##. Then \begin{align} \Delta\left(A\times B\right) &= \epsilon_{ijk}\Delta\left(A_jB_k\right)\mathbf{\hat{e}}_i \\ &= \epsilon_{ijk}\left[A_j\Delta B_k + 2\partial_mA_j\partial_mB_k + B_k\Delta...
2. B Solving for the Nth divergence in any coordinate system

Preface We know that, in Cartesian Coordinates, $$\nabla f= \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}$$ and $$\nabla^2 f= \frac{\partial^2 f}{\partial^2 x} + \frac{\partial^2 f}{\partial^2 y} + \frac{\partial^2 f}{\partial^2 z}$$ Generalizing...
3. A Vector analysis question. Laplacian of scalar and vector field

If we define Laplacian of scalar field in some curvilinear coordinates ## \Delta U## could we then just say what ##\Delta## is in that orthogonal coordinates and then act with the same operator on the vector field ## \Delta \vec{A}##?
4. A Prove a formula with Dirac Delta

Why is the Laplacian of ##1/r## in spherical coordinates proportional to Dirac's Delta, namely: ##\left(\frac{\partial^2 }{\partial r^2}+\frac{2}{r}\frac{\partial }{\partial r}\right)\left(\frac{1}{r}\right)=-\frac{\delta(r)}{r^2}## I get that the result is zero.
5. Deriving the Laplacian in spherical coordinates

As a part of my self study, I am trying to derive the Laplacian in spherical coordinates to gain a deeper understanding of the mathematics of quantum mechanics. For reference, this the sphere I am using, where ##r## is constant and ##\theta = \theta (x,y, z), \phi = \phi(x,y)##. Given the...
6. I Graph Representation Learning: Question about eigenvector of Laplacian

Hi, I was reading the following book about applying deep learning to graph networks: link. In chapter 2 (page 22), they introduce the graph Laplacian matrix ##L##: L = D - A where ##D## is the degree matrix (it is diagonal) and ##A## is the adjacency matrix. Question: What does an...
7. Engineering Image Processing: what is the correct form of the Laplacian Filter

Hi, I just have a quick question regarding image processing. What is the correct form of the Laplacian for image processing? I have seen different versions online and don't understand which one is meant to be the conventional one. I know that: \nabla^2 f(x, y) = \frac{\partial^2 f}{\partial...
8. I 9-point Laplacian stencil derivation

Hey there I'm currently taking a course on numerical methods for solving differential equations, and atm we are working with the discrete laplacian operator. In particular the 9-point stencil: However unlike the 5-point stencil, this one is getting to me. I have tried several things, in...
9. A How do I express an equation in Polar coordinates as a Cartesian one.

I got a polar function. $$\psi = P(\theta )R(r)$$ When I calculate the Laplacian: $$\ \vec \nabla^2 \psi = P(\theta)R^{\prime\prime}(r) + \frac{P(\theta)R^{\prime}(r)}{r} + \frac{R(r)P^{\prime\prime}(\theta)}{r^{2}}$$ Now I need to convert this one into cartesian coordinates and then...
10. A Ground state energy of a particle-in-a-box in coordinate scaling

The energy spectrum of a particle in 1D box is known to be ##E_n = \frac{h^2 n^2}{8mL^2}##, with ##L## the width of the potential well. In 3D, the ground state energy of both cubic and spherical boxes is also proportional to the reciprocal square of the side length or diameter. Does this...
11. A Partial differential equation containing the Inverse Laplacian Operator

I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...
12. Poisson's equation: Calculating the Laplacian of an electric potential

First I calculated the electric fields outside of the sphere in terms of the total charge Q. total charge Q: Q = aπR^4 electric field outside: (r>R) E(r) = (1/4πε) Q/r^2 (ε is the vacuum permittivity) electric potential...
13. I What is a good formula for the Laplace operator?

I have found various formulations for the Laplacian and I want to check that they are all really the same. Two are from Wikipedia and the third is from Sean Carroll. They are: A Wikipedia formula in ##n## dimensions: \begin{align} \nabla^2=\frac{1}{\sqrt{\left|g\right|}}\frac{\partial}{\partial...
14. MHB Hankel transform on the polar form of the Laplacian.

Why first terms equal to zero ? ? ?
15. The Laplacian of the potential q*exp(-r)/r

Hello, I have a problem where I'm supposed to calculate the charge distribution ρ. I need to calculate it by applying the Laplacian operator to the potential Θ. The potential is the function: q*exp(-αr)/r I found on the internet that for this type of potentials I cannot just apply the...
16. A Replacing a non-harmonic function with a harmonic function

I am solving a problem of the boundary condition of Dirichlet type, in order to solve the problem, the functions within the differential equations suppose to be harmonic. I have a function f(x,y,z) (the function attached) which is not harmonic. I must find an equivalent function g(x,y,z) which...
17. I Derivative of f() as a function of a Laplacian

I need a little help with understanding a differential relationship between functions. If g and f are vector fields and f(g(x,y),q(x,y))=∇2g(x,y) How could you, if possible, express ∂f/∂g explicitly? Please help a bit confused.
18. Divergence operator for multi-dimensional neutron diffusion

Homework Statement [1] is the one-speed steady-state neutron diffusion equation, where D is the diffusion coefficient, Φ is the neutron flux, Σa is the neutron absorption cross-section, and S is an external neutron source. Solving this equation using a 'homogeneous' material allows D to be...
19. Laplacian in spherical coordinates

Homework Statement Hello at all! I have to calculate total energy for a nucleons system by equation: ##E_{tot}=\frac{1}{2}\sum_j(t_{jj}+\epsilon_j)## with ##\epsilon_j## eigenvalues and: ##t_{jj}=\int \psi_j^*(\frac{\hbar^2}{2m}\triangledown^2)\psi_j dr## My question is: if I'm in...
20. Laplacian for hyperbolic plates

Homework Statement Find a complete set of conditions on the constants a, b, c, n such that, for Cartesian coordinates (x, y, z), V = axn + byn + czn is a solution of Laplace’s equation ##∇^2V = 0##. A mass filter for charged particles consists of 4 electrodes extended along the z direction...
21. I How does the Vector Laplacian come about?

So the Laplacian of a scalar is divergence of the gradient of a scalar field, and it comes out to the double derivative of the field in X, Y, and Z. My book says the Laplacian of a vector field is the double derivative of the X component of the field with respect to X, the double derivative of...
22. S

What is the gradient in polar coordinates?

Hi, on this page: https://en.wikipedia.org/wiki/Laplace_operator#Two_dimensions the Laplacian is given for polar coordinates, however this is only for the second order derivative, also described as \delta f . Can someone point me to how to represent the first-order Laplacian operator in polar...
23. S

I Is the Laplacian Operator Different in Radial Coordinates?

Hi, I have that the Laplacian operator for three dimensions of two orders, \nabla ^2 is: 1/r* d^2/dr^2 (r) + 1/r^2( 1/sin phi d/d phi sin phi d/d phi + 1/sin^2 phi * d^2/d theta^2) Can this operator be used for a radial system, where r and phi are still valid, but theta absent, by setting...

27. I Derivation of the Laplacian in Spherical Coordinates

Hi all, Sorry if this is the wrong section to post this. For some time, I have wanted to derive the Laplacian in spherical coordinates for myself using what some people call the "brute force" method. I knew it would take several sheets of paper and could quickly become disorganized, so I...
28. I 2D Laplacian in polar coordinates

The 2D Laplacian in polar coordinates has the form of $$\frac{1}{r}(ru_r)_r +\frac{1}{r^2}u_{\theta \theta} =0$$ By separation of variables, we can write the ## \theta## part as $$\Theta'' (\theta) = \lambda \Theta (\theta)$$ Now, the book said because we need to satisfy the condition ##...
29. I Norm of Laplacian Let: Formula for | ∇X|² in Coordinates

Let ##(M,g)## a manifold with a Levi-Civita connection ## \nabla ## and ##X## is a vector field. What is the formula of ## | \nabla X|^2 ## in coordinates-form? I know that ##|X|^2= g(X,X)## is equivalent to ## X^2= g_{ij} X^iX^j## and ##\nabla X## to ##\nabla_i X^j = \partial_i X^j +...
30. Laplacian of polar coordinates

Homework Statement I am trying to calculate the laplacian in polar coordinates but I failed.Please see the attached Homework Equations The Attempt at a Solution My solution to this was uploaded in the attached.I was wondering what's wrong with the purple brackets since they shouldn't exist(...
31. I Laplacian in integration by parts in Jackson

I am reviewing Jackson's "Classical Electromagnetism" and it seems that I need to review vector calculus too. In section 1.11 the equation ##W=-\frac{\epsilon_0}{2}\int \Phi\mathbf \nabla^2\Phi d^3x## through an integration by parts leads to equation 1.54 ##W=\frac{\epsilon_0}{2}\int |\mathbf...
32. A Laplacian cylindrical coordinates

Laplacian in cylindrical coordinates is defined by \Delta=\frac{\partial^2}{\partial \rho^2}+\frac{1}{\rho}\frac{\partial}{\partial \rho}+\frac{1}{\rho^2}\frac{\partial^2}{\partial \varphi^2}+\frac{\partial^2}{\partial z^2} I am confused. I I have spherical symmetric function f(r) then \Delta...
33. Converting Laplacian to polar coordinates

Homework Statement $$U_{tt}=\alpha^2\bigtriangledown^2U$$ in polar coordinates if solution depends only on R, t. Homework EquationsThe Attempt at a Solution So, the books solution is $$U_{tt}=\alpha^2[U_{rr}+\frac{1}{r}U_r]$$. I am getting stuck along the way can't figure out this last step I...
34. Prove Laplacian of Dyad: Indicial Notation

Homework Statement [/B] Given the dyad formed by two arbitrary position vector fields, u and v, use indicial notation in Cartesian coordinates to prove: $$\nabla^2 ({\vec u \vec v}) = \vec v \nabla^2 {\vec u} + \vec u \nabla^2 {\vec v} + 2\nabla {\vec u} \cdot {(\nabla \vec v)}^T$$Homework...
35. M

Understanding the Position Vector in Calculus Problems

Homework Statement With ##\vec{r}## the position vector and ##r## its norm, we define $$\vec{f} = \frac{\vec{r}}{r^n}.$$ Show that $$\nabla^2\vec{f} = n(n-3)\frac{\vec{r}}{r^{n+2}}.$$ Homework Equations Basic rules of calculus. The Attempt at a Solution From the definition of...
36. I L'Hospital's rule for Laplacian

In the above expression for the Laplacian, how exactly does the author apply l'Hospital's rule? And is this transformation only valid for ## \rho = 0##?
37. I Vector Laplacian: different results in different coordinates

Hello, I calculated the Vector Laplacian of a uniform vector field in Cartesian and in Cylindrical coordinates. I found different results. I can't see why. In Cartesian coordinates the vector field is: (vx,vy,vz)=(1,0,0). Its Laplacian is: (0,0,0) . That's the result I expected. In...
38. Laplacian of the value function

Homework Statement Laplacian of the function V(x,t)=-1/2* x' D x + h' *x + D Homework EquationsThe Attempt at a Solution is equals D.
39. I Chladni plate with Neumann conditions

Hi there, I'm trying to simulate a vibrating plate with free edges. If i consider a consider a plate with fixed edges, the eigenvectors of the matrix bellow (which repesents the Laplacien operator) with S as a nxn tridiagonal matrix with -4 on the diagonal and 1s on either side (making the...
40. I Vector Laplacian: Scalar or Vector?

according to this page https://en.wikipedia.org/wiki/Vector_Laplacian value of Vector_Laplacian is vector, but according to this page https://en.wikipedia.org/wiki/D'Alembertian value of Vector_Laplacian is scalar Is on of these pages wrong or I misunderstand it? I am asking because I want to...
41. I Question about solution to Laplacian in Spherical Polars

I was following this derivation of the solution to the Laplacian in spherical polars. I was wondering where the two equations ##\lambda_{1} + \lambda_{2} = -1## and ##\lambda_{1}\lambda_{2} = -\lambda## come from? Thanks.
42. Question about a boundary-value problem (electrostatics)

Laplacian for polars: $$\frac{1}{r}\frac{\partial}{\partial r}\left( r\frac{\partial \phi}{\partial r}\right) + \frac{1}{r^{2}}\frac{\partial^{2} \phi}{\partial \theta^{2}} = 0$$ This is in relation to a problem relating to a potential determined by the presence of a wedge shaped metallic...
43. Prove the Laplacian of Function g is equal to g

Homework Statement Homework Equations the gradient of g is (d/dx,d/dy,d/dz) the divergence of g is d/dx+d/dy+d/dz The Attempt at a Solution When I run through even using only a few terms to see if I can get the final result of it equaling g I end up with u^2 terms as coefficients and this...
44. Deriving the Wave Equation from Laplacian and Partial Derivatives

Homework Statement derive the following wave equation ∇2H = 1/c2 (∂2H/∂t2) Homework EquationsThe Attempt at a Solution I'm not sure how to derive it. I suppose I can break it into a whole bunch of partial derivatives because of the del squared operator and then just lump the three partial...
45. Trouble understanding ##g^{jk}\Gamma^{i}{}_{jk}##

Hi friends, I'm learning Riemannian geometry. I'm in trouble with understanding the meaning of ##g^{jk}\Gamma^{i}{}_{jk}##. I know it is a contracting relation on the Christoffel symbols and one can show that ##g^{jk}\Gamma^i{}_{jk}=\frac{-1}{\sqrt{g}}\partial_j(\sqrt{g}g^{ij})## using the...
46. Integrate Laplacian operator by parts

This is the key step to transform from position space Schrodinger equation to its counterpart in momentum space. How is the first equation transformed into 3.21? To be more specific, how to integral Laplacian term by parts?
47. Delta Function Identity in Modern Electrodynamics, Zangwill

I am currently reading Modern Electrodynamics by Andrew Zangwill and came across a section listing some delta function identities (Section 1.5.5 page 15 equation 1.122 for those interested), and there is one identity that really confused me. He states: \begin{align*} \frac{\partial}{\partial...
48. Obtain the magnetic field from this experimental setup?

Homework Statement Hi all, I would appreciate some help with the following problem; I need to obtain and visualize the current flow and magnetic field profile of an elliptic cylinder (made from ferromagnetic material) which has a left section set at 0 volts and a right section set at 5 volts...
49. What Are Common Mistakes When Calculating the Laplacian of |r|?

Homework Statement Given: |r|=√(x^2+y^2+z^2) r=xi+yj+zk (i)Find the partial derivative with respect to x of |r|. (ii) Find the Laplacian of |r|. Homework EquationsThe Attempt at a Solution For (i) I got x/|r| but then for (ii) I got 2/r which I don't think is correct
50. Calculating Mean Curvature of a Scalar Field | Homework Solution

Homework Statement Consider the scalar field φ=x2+y2-z2-1. Let H be the scalar field defined by H = -0.5∇.(∇φ/ abs(∇φ)), where abs(∇φ) is the magnitude of ∇φ. Which makes that some sort of unit quantity. When H is evaluated for φ=0 it is the mean curvature of the level surface φ=0. Calculate...