Integrating δ(1 - cosθ) over the entire solid angle in spherical coordinates.

Question

I don't understand why the following is wrong: link here

Should dθ sinθ be converted to -dcosθ? If yes should dθ sinθ be converted to -dcosθ for every integral involving cosθ?

Answer 1

The answer is $0$. If you want to use $-d \cos(\theta)$ in place of $d\theta \sin \theta$, then you get $$ C \int_0^{2\pi} d\phi \int_{-1}^1 \delta_{1-t} dt, $$ which is $0$ if you define $\delta$ function appropriately.