I need to compute the following integral
$\int\int_{\infty}^{+\infty}{f(\vec r)\delta(\vec r) d\vec r}$
where I defined $\vec r=m_1\vec a_1+m_2\vec a_2$ with $m_1, m_2\in \mathbb{R}$
Does the integral results just $f(0)$?
In the case of single variable, let's say $x$, the integral should be:
$\int\limits_{\infty}^{+\infty}{f(x)\delta(g(x)) dx}=\frac{f(x_0)}{|g'(x_0)|}$
where $x_0$ any of the values of $x$ that makes the function $g$ be zero.
When $g(x)=x$ the result is obviously $f(x_0)$. If now I try to apply this definition to the first integral, that is setting $g(\vec r)=\vec r$, here is what I end up with:
$\int\int_{\infty}^{+\infty}{f(\vec r)\delta(\vec r) d\vec r}=\frac{f(0)}{|\vec\nabla\cdot\vec r(0)|}$
Now I think I might have a bit of confusion. Is the following expression right?
$\vec\nabla\cdot\vec r=\frac{d}{dm_1}m_1+\frac{d}{dm_2}m_2=1+1=2$
In case it was correct, the result of the first integral would be $\frac{f(0)}{2}$.
Can you help me to understand where my mistakes are?
Thank you