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.
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}##?
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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.
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...
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...
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...
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...
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...
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...
Homework Statement
I have to solve the following problem
$$
\left\{
\begin{array}{ll}
\dfrac{ \partial^{2} u }{ \partial x^{2} } + \dfrac{ \partial^{2} u }{ \partial y^{2} } =0 & \qquad \forall x \in (0, L), y > 0 \\
& \\
\dfrac{ \partial u }{ \partial x } (0,y) =0, & \qquad \forall y > 0 \\
&...
Homework Statement
Problem 2 http://math.mit.edu/~jspeck/18.152_Spring%202017/Exams/Practice%20Midterm%20Exam.pdf
"Let ##u## such that ##Laplacian( u)=0##
Show if ##u \le \sqrt{x}##, then ##u=0##
Homework Equations
At the solution...
Dear friends,
I have found a derivation of the fact that, under the assumptions made in physics on ##\rho## (to which we can give the physical interpretation of charge density) the function defined by
$$V(\mathbf{x},t):=\frac{1}{4\pi\varepsilon_0}\int_{\mathbb{R}^3}...
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...
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 ##...
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 +...
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(...
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...
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...
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...
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...
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...
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##?
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...
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...
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...
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.
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...
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...
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...
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...
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?
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...
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...
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
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...