I am interested in applying Leibniz integral rule to the following general contour integral.
$$ g(r)=\int_{f(x,y)=r} h(x,y) d\Gamma(x,y) $$ Note that the dependence on r exists only through the contour $f(x,y)=r$, and that $h(x,y)$ is independent of r.
Thus, I need some expression for $$ \frac{d }{dr}g(r) = ??? $$ It seems it might be something like $$ \frac{d }{dr}g(r) = [ h(x,y) ]_{\partial(f(x,y)-r)} $$ but I'm not confident and unable to find any resources for this subject. Also, $\partial(f(x,y)-r)$ refers to the boundary of the curve $(f(x,y)=r)$ which may be a null set.
Thank you for your attention.