Multivariable Dirac Delta Functions

freechus9

Hello all. So I am trying to integrate a function of this form:

[tex]\int[/tex][tex]\int[/tex]F(x,y)[tex]\delta[/tex][a(Cos[x]-1)+b(Cos[y]+1)]dxdy

The limits of integration for x and y are both [0,2Pi). I know that this integral is only nonzero for x=0, y=Pi. So this should really only sample one point of F(x,y), namely F(0,Pi). However, I am having trouble figuring out what I need to divide by due to the fact that the delta function argument is a function of x and y, not x and y themselves. Does anyone have any ideas? Thanks!

appelberry

Are you sure that the only nonzero point is [tex](0,\pi)[/tex]? Depending on the values of [tex]a[/tex] and [tex]b[/tex] (e.g. if one of them is negative) the function [tex]g(x,y)=a(cos(x)-1)+b(cos(y)+1)[/tex] could have a number of roots between the limits of integration.

Given the following identity for a single variable Dirac Delta function:
[tex]\delta[g(x)]=\sum_{i}\frac{\delta(x-x_{i})}{|g'(x_{i})|}[/tex]
where [tex]x_{i}[/tex] are the roots of [tex]g(x)[/tex], I think the multivariable analogue will be
[tex]\delta[g(x,y)]=\sum_{i}\frac{\delta(x-x_{i})\delta(y-y_{i})}{|\nabla g(x_{i},y_{i})|}[/tex]
where [tex](x_{i},y_{i})[/tex] are the roots of [tex]g(x,y)[/tex].