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Hello.I was reading recently barton's book.I reached the part corresponding to dirac-delta functions in spherical polar coordinates.

he let :##(\theta,\phi)=\Omega## such that ##f(\mathbf {r})=f(r,\Omega)##

##\int d^3r...=\int_0^{\infty}drr^2\int_0^{2π }dφ\int_0^π dθsinθ...##

define

##\int_0^{2π}dφ\int_0^πdθsinθ=\int_0^{2π}dφ\int_{-1}^{1}dcosθ=\int d\Omega##

then## δ(\mathbf {r-r^{'}})=\frac {1}{r^2} δ(r-r')δ(\Omega-\Omega^{'})##>>>(1)

where ##δ(\Omega-\Omega^{'})=δ(φ-φ')δ(cosθ-cosθ')##>>>(2)

My problem is that I really didn't get how he switched from ##\int_0^{2π }dφ\int_0^π dθsinθ## into ##\int_{-1}^{1}dcosθ##

same thing corresponding to relations (1) and (2), I didn't get how he obtained them?

If somebody can give me a hint for obtaining them? thanks.

Relevant equations

##\int d^3rf(\mathbf{r})δ(\mathbf{r})=f(0)##

where##δ(r)=δ(x)δ(y)δ(z)##

such that ##δ(r)=1/(2π)^3\int d^3kexp(i\mathbf{k}.\mathbf{r})##

3. The attempt at a solution

I need hints to know where yo start.

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# Dirac-delta function in spherical polar coordinates

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