If I am integrating a function such as:
$X=\iiint f(a,b,c)\delta(a,b,c)\textrm{ d}a\textrm{ d}b\textrm{ d}c$
where $f$ is a multidimensional (in this example, 3-dimensional) smooth function that is symmetric, i.e. the result is independent of the 3-tuple permutations of its arguments (e.g., f(a,b,c) = f(b,a,c) = f(b,c,a), etc.), $\delta$ is discontinuous and numerically equal to:
$x$ to the power of the number of arguments sharing a common value
Since the 'region' when a=b or a=c or b=c is infinitesimal, I was thinking if I can ignore $\delta$ - the answer I think is no. Is it correct if I simplify it to:
$X= \iiint f(a,b,c)\textrm{ d}a\textrm{ d}b\textrm{ d}c + 3x^2\iint f(a,a,b)\textrm{ d}a\textrm{ d}b + x^3\int f(a,a,a)\textrm{ d}a$ ?
Thanks!