I have been looking everywhere but I am unable to prove $\delta(\vec{x}-\vec{a}) = \frac{1}{fgh}\delta(x_u-a_u)\delta(x_v-a_v) \delta(x_w-a_w)$
Where $f,g,h$ are scale factors for an orthogonal system $u,v,w$. If $\vec{a}$ lies on a degenerate coordinate then $\delta(\vec{x}-\vec{a}) = \frac{1}{fg\int hdw}\delta(x_u-a_u)\delta(x_v-a_v) \delta(x_w-a_w)$
I know that the delta function is a generalized function, and is generally used in the form $\int_{r_0\in V} f(\vec{r})\delta (\vec{r}-\vec{r_0})dV = f(\vec{r_0})$
But I am unsuccesful in using this to prove the above expressions.