Does the Jacobian in delta function change in non-Cartesian coordinates?

[rotemblu]

when using delta function $$\delta(r)$$ in cordinates othe then cartesian when does it needs to be divided by the jacobian for example in spherical coordinates

$$\delta(x)=\frac{1}{r^2sin(\vartheta)}\delta(r-r_{0})\delta(\vartheta-\vartheta_{0})\delta(\phi-\phi_{0})$$
but if you want a delta function that involves only the radial coordinate doy you still need to divide all the jacobian like this $$\delta(r)=\frac{1}{r^2sin(\vartheta)}\delta(r-r_{0})$$
or only the radial part like this
$$\delta(r)=\frac{1}{r^2}\delta(r-r_{0})$$
thanks

 

[lippyka]

in advanceThe latter is correct: \delta(r)=\frac{1}{r^2}\delta(r-r_{0}). You only need to divide by the radial component of the Jacobian.