Laplacian Definition and 138 Threads

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...

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}...

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...

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...

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...

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...

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...

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 +...

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...

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)) +...

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...

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}...

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…...

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...

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...

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}...

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...

Whoops, I figured it out!

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...

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 )...

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...

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...

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?

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

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...

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...

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...

(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...

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...

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...

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...

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...

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...

[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...

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...

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...

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...

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.

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...

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

This might sound kinda dumb, but what is the Inverse Laplace transform of a number? So L^{-1}(8) for example.

Hi, During the description of vector spherical harmonics, where N = curl of M , I came across the following : Laplacian of N = Laplacian of (Curl of M) = Curl of (Laplacian of M) How do we know that these operators can be interchanged ? What is the general rule for such interchanges...

Homework Statement I'm trying to show that the wave equation is not invariant under Galilean transform. To do that I need to figure out how the Laplacian transforms from S to S'. I seem to have trouble understanding why the laplacian actually changes. Homework Equations x'=x-vt, t'=t The...

In my derivation of one of Maxwell's Equations, I needed the fact that the Laplacian of 1/r vanishes everywhere except at r=0, where r is the norm of a radial vector. I cannot see how this is? I like to be solid in the math I use for a derivation, so this would really help if someone could...

Homework Statement \Delta f = \frac{1}{{r^2 }}\frac{\partial }{{\partial r}}\left( {r^2 \frac{{\partial f}}{{\partial r}}} \right) + \frac{1}{{r^2 \sin \phi }}\frac{\partial }{{\partial \phi }}\left( {\sin \phi \frac{{\partial f}}{{\partial \phi }}} \right) + \frac{1}{{r^2 \sin ^2 \phi...

Homework Statement Find the Laplacian of F = sin(k_x x)sin(k_y y)sin(k_z z) Homework Equations \nabla^2 f = \left( \frac{\partial}{\partial x} +\frac{\partial}{\partial y} + \frac{\partial}{\partial z} \right) \cdot \left( \frac{\partial}{\partial x} + \frac{\partial}{\partial y} +...

Tell me I'm not going mad. If I have a vector field of the form \mathbf{A}=(0,A(x,y,z),0) and I want to take the Laplacian of it, \nabla^{2}\mathbf{A}, can I take the Laplacian of the co-ordinate function A(x,y,z)? Will this be the same for the case of cylindrical co-ordinates? Mat

Hi, According to eg Nakahara's conventions the Laplacian on a form K is given by \Delta K = (dd^{\dagger} + d^{\dagger}d)K In my case K is a two form living in R^3. I've calculated the Laplacian and arrive at \Delta K = \Bigl( \frac{1}{3!}\epsilon^{klm}\epsilon^n_{\...

The http://en.wikipedia.org/wiki/Fine_structure#Darwin_term" contains a (3D-)delta function as a result of taking the Laplacian of the Coulomb potential. I'm trying to find out why. I've been searching, and I've so far come across different views of the Laplacian of 1/r at the origin. Either...

Why does the laplacian vanish for harmonic functions? Can someone explain this in physical terms?