Laplacian Definition and 138 Threads

I am trying to show that the laplacian: L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} can also be represented as: L = \frac{1}{2}(\vec{E}^{2}-\vec{B}^{2}) where F^{\mu\nu} = \partial{}^{\mu}A^{\nu} - \partial{}^{\nu}A^{\mu} and F_{\mu\nu} = g_{\mu\alpha}F^{\alpha\beta}g_{\beta\nu} A is the...

Homework Statement Two concentric spherical shells. The outer shell is split into two hemispheres at potentials +Vo for the upper half and -Vo for the lower half. The inner shell is at zero potential (see attachment). " what is the potential in the region; r > R' " (the potential in the...

Homework Statement What is a laplacian of a laplacian? Homework Equations laplacian = \Delta = \nabla^2 = \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2} The Attempt at a Solution Is the follow true? \nabla^2\nabla^2=\nabla^4 Also...

Stupid thing I noticed today: \nabla^2 U=tr(H(U)) Or, in other words, the Laplacian of a function is just the trace of its Hessian matrix. Whoop-de-frickin do, right? Is this useful knowledge or should I forget it immediately? N!

Can all vector fields be described as the vector Laplacian of another vector field?

Something occurred to me just now. A question about the scalar potential. First I will do some calculations of the laplacian of the scalar potential in different electrostatic situations to give myself a basis for my question. Point charge: \phi =\frac{1}{4\pi\epsilon_0} \frac{q}{r}...

Hey All, In my vector calculus class my lecturer was showing that the laplacian of 1/r is zero. He further said that since 1/r and its derivatives are not defined at the origin we state that the Laplacian of 1/r is zero for all values of r not equal to zero. He then says that this caveat is...

I wish to solve a 2D steady state heat equation analytically. The boundary is a square. The top side is maintained at 100 C, while the other sides are maintained at 0 C. The differential equation governing the temperature distribution will be the laplacian equation. To solve the equation...

Lets consider the equation: \nabla^2 f=0 I know that in spherical coordinates this equation may be decomposed into two equations, first which depends only on r, and the second one which has the form of spherical harmonics equation except that the l(l+1) is an arbitrary constant, let's say C...

Has anyone ever heard about a Complex Laplace Operator? I would like to build one from first principles as in differential geometry ∆=d*d, where d is the exterior derivative, but I don't know where to start. Actually, I was even unsure in which forum to post the question. If one defines d to...

Homework Statement I want to calculate \nabla\times[\vec{F}(r)] and \nabla^2[\vec{F}(r)] where F if a function that depends of r, and r = \sqrt{x^2+y^2+z^2} Homework Equations 1)\nabla \times \vec A = \left|\begin{matrix} \mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}} \\ \\...

Homework Statement I have $\int \nabla^2 \vec{u} \cdot \vec{v} dV$ where u and v are velocities integrated over a volume. I want to perform integration by parts so that the derivative orders are the same. This is the Galerkin method. Homework Equations The Attempt at a Solution I have...

Hi; I am trying to understand the rytov approximation... and when I was studying that, I could not understand a manipilation... ΔeØ + k2eØ = 0 ▼[▼ØeØ] + k2eØ = 0 ▼2ØeØ + (▼Ø)2eØ+k2eØ = 0 I can not understand these manipilations... for a long time, I have searched the properties of...

http://buyanik.wordpress.com/2009/05/02/laplacian-in-spherical-coordinates/"

∇(∇ * q) does this equal the laplacian or something else?

http://img243.imageshack.us/img243/1816/laplaceds2.jpg Right I have a fair few questions on this, it's relating to question 7 only, although you need to refer back to the equation derived from question 6. 1) I used the equation from q6. as a Fourier series substituting r=a. I end up with an...

What is the physical significance of the laplacian operator? The laplacian operator is the divergence of the gradient. I understand the intuitive meanings of both. The gradient when dotted against a unit vector gives the rate of change in that direction. The divergence is the flow in or out...

Strange Laplacian of 1/|x-x’| !? Please read the file first (http://www.pa.msu.edu/courses/2007fall/PHY481/lectures/lecture08.pdf" ) .. and look into page 8 there is a sentence like this "Evaluate right side with sphere, radius R around origin" Now there comes up against a question ...

Hi. In http://www-personal.umich.edu/~pran/jackson/P505/p1s.pdf" solution(page 2) to Jackson 1.5 it is stated that \nabla^2 \left(\frac{1}{r}\right)=-4\pi\delta^3(\mathbf r). But why is this true? \nabla^2\left(\frac{1}{r}\right)=\frac{1}{r^2}\frac{d}{d...

Hey! I'm self-studying a bit of quantum chemistry this summer. My introductory P.chem book (David Ball) doesn't specifically show the conversion of the laplacian operator from Cartesian to spherical coordinates. I don't really feel satisfied until I've actually derived it myself... So...

Hi, Someone has some suggestion about self-study book about "Laplacian" and "Hamilton Operator". Thanks

Homework Statement If anyone could clarify this statement for me, I'm having a bit of difficulty interpreting what the heck I'm supposed to do: "For a given potential difference V0 between the inner and outer conductors and for a given fixed value of b, determine the inner radius a for which...

I've been reading up on these three recently, and wondered if anyone could confirm what I think they do. I'm not 100% I understand these. del (\bigtriangleup), when applied to a scalar, creates a vector with that scalar as each of the XYZ values. eg \bigtriangleup . x = (x,x,x)...

how do you write the laplacian operator in spherical coordinates and cylindrical coordinates from a cartesian basis?

Can the laplacian of a scalar field be throught of as its curvature (either approximately or exactly)?

They say that vector laplacian is defined as the following: \nabla^2 \vec{A} = \nabla(\nabla\cdot\vec{A}) - \nabla\times(\nabla \times\vec{A}) Is the above definition true for all coordinate systems or just for cartesian coordinate system? --- --- --- Also, wikipedia say the following...

I need to know the steps involved in solving this laplacian. E[SIZE="1"]x(r,z) = E[SIZE="1"]o*e^[-(r/r[SIZE="1"]o)^2]*e^[-ibz] the laplacian \/^2*E[SIZE="1"]o = ? Eo is a vector \/ is laplacian symbol any help would be appreciated. Thanks in advance.

Hey... Could someone help me out with deriving the LaPlacian in spherical coordinates? I tried using the chain rule but it just isn't working out well.. any sort of hint would be appriciated. :) \nabla^2 = \frac{1}{r^2} [ \frac{\partial}{\partial r} ( r^2 \frac{\partial}{\partial r} ) +...

Given: F(s) = (s-1)/(s+1)^3 Find: f(t) Solution: Using the equation that when F(s) = n!/(s-a)^(n=1), L^(-1){F(s)} = t^n*e^(at) So far I find that f(t) = e^(-t)*(-t^2+__) The book says that f(t) = e^(-t)*(t-t^2) How did they get the t?

we are given the laplacian: (d^2)u/(dx^2) + (d^2)u/(dy^2) = 0 where the derivatives are partial. we have the B.C's u=0 for (-1<y<1) on x=0 u=0 on the lines y=plus or minus 1 for x>0 u tends to zero as x tends to infinity. Using separation of variable I get the general solution u =...

I'm supposed to prove the laplacian in cylindrical coord. is what it is. I tried tackling the problem in two ways and none work! I have no idea what's the matter. The first way is to calculate d²f/dr² , d²f/dO² and d²f/dz² and isolate d²f/dx² , d²f/dy² and d²f/dz². In cylindrical coord...

Why does the laplacian of u=0 when u is a solution to a certain boundary problem? Is this always the case?

Does anybody out there know what the Laplacian is for two dimensions?

According to the definition, an operator T that commutes with all components of the angular momentum operator is a scalar, or rank zero, operator. What is the mathematical definition to that statement? How can I prove that the four dimensional Laplacian is a scalar operator? Regards, :biggrin:

I need to take the \nabla^2 of x^2+y^2+z^2. This is how far I got \begin{gather*} \nabla^2 = \frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr} + \frac{1}{r^2}(\frac{1}{sin^2\theta}\frac{d^2}{d\Phi^2} + \frac{1}{sin\theta}\frac{d}{d\theta} sin\theta\frac{d}{d\theta})\\ \nabla^2(r^2sin^2\theta...

Consider a function U(x,y) where x, and y are spatial variables (have units of length) Assume that the symbol V^2 corresponds to the Laplacian operator. Then V^2U= Uxx + Uyy where the subscript indicates partial differentiation. Consider now a function F(x,t) where x is spatial...

I have a problem on my homework that is really confusing. I need to solve the partial differential equation in a spherical shell with inner radius = a and outer radius=b: (Laplacian u)=1 in spherical coordinates. The boundary conditions are u=0 on the inner radius r=a, and du/dr=0 on outer...

Here's the problem: Find the Laplacian of sqrt(x^-y^2) and ln(r^2). Will i just take the gradient of each one of these twice?