Delta function for \nabla(log s), where s = \sqrt{x^2+y^2}

[zhuang382]

Homework Statement

What is ∇^2ln s in two dimensions where s =√x2+y2

Relevant Equations

delta function homework problem

My intuition for this problem is to use divergence theorem:

$\int_V \nabla^2 u dV = \int_S \nabla u \cdot \vec{n} dS$

But note that $\vec{n}$ is perpendicular to x-y plane, and makes $\int_S \nabla \ln s \cdot \vec{n} dS = 0$

If we take laplacian in polar coordinate directly, then $\nabla^2 \ln s = 0$

Can someone helps me on this?

 

[Antarres]

If $\nabla$ is defined as $\frac{\partial}{\partial x}\vec{i} + \frac{\partial}{\partial y}\vec{j}$ in two dimensions, why wouldn't you just compute the partial derivatives and see what you get? Also...I don't see a delta function from the title mentioned anywhere. Maybe you meant 'del', meaning $\nabla$?

 

[vela]

My intuition for this problem is to use divergence theorem:

$\int_V \nabla^2 u dV = \int_S \nabla u \cdot \vec{n} dS$

But note that $\vec{n}$ is perpendicular to x-y plane, and makes $\int_S \nabla \ln s \cdot \vec{n} dS = 0$

In two dimensions, $V$ would be an area in the $xy$ plane, and $S$ would be the boundary of that area.

If we take laplacian in polar coordinate directly, then $\nabla^2 \ln s = 0$

This result is only valid in the region $s>0$. You do need to keep in mind that you've excluded the origin.