Multivariable Dirac Delta Functions

[freechus9]

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

\int\intF(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).