Multivariable Dirac Delta Functions

[freechus9]

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

$$\int$$$$\int$$F(x,y)$$\delta$$[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 $$(0,\pi)$$? Depending on the values of $$a$$ and $$b$$ (e.g. if one of them is negative) the function $$g(x,y)=a(cos(x)-1)+b(cos(y)+1)$$ could have a number of roots between the limits of integration.

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