I am trying to solidify my understanding of differential volume form by integrating the same section of the unit sphere using two separate parameterizations. With the parameterization $\varphi^{-1}_1(\phi, \theta)=(\sin\phi\sin\theta, \sin\phi\cos\theta, \cos\phi)'$, $A_1 := \{ (\phi, \theta): \phi\in (0, \pi/4), \theta \in (0,\pi)\}$, I calculate the integral according to Amann (2009):
$$\text{Vol}_{g, U}(A) := \int_{\varphi(A)} \sqrt{\det(D {\varphi^{-1}}'D \varphi^{-1})} da$$
which here is just
$$\text{Vol}_{g, U}(A_1) := \int_{0}^\pi \int_0^{\pi/4} |\sin \phi| ~d\phi~ d\theta \approx 0.920151.$$
Next, I use the parameterization $\varphi^{-1}_2(x,y)=(x, y, \sqrt{1-x^2-y^2})'$, $A_1 := \{ (x,y): x\in \left(-\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right), y \in (0,\sqrt{\frac{1}{\sqrt{2}} – x^2})\}$ (which I think should be the same section of the sphere) to get
$$\text{Vol}_{g, U}(A_2) := \int_{-\frac{1}{\sqrt{2}}}^\frac{1}{\sqrt{2}} \int_0^{\sqrt{\frac{1}{\sqrt{2}} – x^2}} \left|\frac{1}{\sqrt{1-x^2-y^2}}\right| ~dy~ dx \approx 1.30593.$$
So, two questions:
- Where am I going wrong here? I think that the spherical coordinates one is okay. I would really like an example where it matches the direct parameterization.
- I believe that $|\sin \phi|d\phi\theta$ in its entirety is the Riemannian volume form — is this correct?
Thanks!
Edit Oct1: I've added a photo of the region.
Best Answer
@TedShifrin was kind enough to lead me to my silly mistake. An updated integral would be: $$\text{Vol}_{g, U}(A_2) := \int_{-\frac{1}{\sqrt{2}}}^\frac{1}{\sqrt{2}} \int_0^{\sqrt{\frac{1}{2} - x^2}} \left|\frac{1}{\sqrt{1-x^2-y^2}}\right| ~dy~ dx \approx 0.920151.$$
I'll leave this posted for anyone trying to better understand volume form!