Two form surface integral over sphere

Question

I'm trying to compute $\int_M \omega$ with \begin{align*} \omega &= x^4 dy \wedge dz + y^4 dz \wedge dx + z^4 dx \wedge dy, \\ M &: x^2 + y^2 + z^2 = R^2. \end{align*} I have done this in two ways: Stokes' theorem and direct computation of the wedge product via a spherical coordinate transformation. Using $\phi$ as azimuthal and $\theta$ as polar angle and applying Stokes I obtain \begin{align*} 4 \int_0^{2 \pi}\int_0^{\pi} (x^3 + y^3 + z^3) R^2 \sin \theta d\theta d\phi = 0, \end{align*} where I used the standard \begin{align*} x &= R \sin \theta \cos \phi \\ y &= R \sin \theta \sin \phi \\ z &= R \cos \theta. \end{align*} By directly computing the wedge products and plugging them in I obtain \begin{align*} dy \wedge dz &= R^2 \sin^2 \theta \cos \phi d\theta \wedge d\phi \\ dz \wedge dx &= R^2 \sin^2 \theta \sin \phi d\theta \wedge d\phi \\ dx \wedge dy &= R^2 \cos \theta \sin \theta d\theta \wedge d\phi \\ \int_M x^4 dy &\wedge dz + y^4 dz \wedge dx + z^4 dx \wedge dy = 0. \end{align*} The solution given by my professor was $\frac{12}{5} \pi^2 R^5$, which neither method agrees with. Then I repeat this procedure but instead of taking a sphere, I take a hemisphere. Changing the limits in the $\theta$ integrals to $[0, \frac{\pi}{2}]$, the Stokes' method yields $2 \pi R^5$ and the direct wedge computation yields $\frac{\pi R^6}{3}$. The solution given then is just $\frac{6}{5} \pi^2 R^5$, which also does not agree with my solutions. Any help with my mistakes would be greatly appreciated.

Answer 1

In case of sphere, the surface integral is indeed zero.

Please note in below working of mine, I use $\theta$ as azimuthal angle and $\phi$ as polar. I think you have the other way round. So want to call out to avoid confusion.

In case of hemisphere above $z = 0$,

Parametrization of sphere is $r(\phi, \theta) = (R \cos \theta \sin \phi, R \sin\theta \sin\phi, R \cos\phi)$

For outward normal vector, $\displaystyle r'_{\phi} \times r'_\theta = R^2 \sin\phi(\cos\theta \sin\phi, \sin\theta \sin\phi, \cos\phi)$

The integral becomes,

$\displaystyle \small \int_0^{2\pi}\int_0^{\pi/2} R^2 \sin\phi \ (R^4 \cos^4\theta\sin^4\phi, R^4 \sin^4\theta\sin^4\phi, R^4 \cos^4\phi) \cdot (\cos\theta \sin\phi, \sin\theta \sin\phi, \cos\phi) \ d\phi \ d\theta$

$= \displaystyle \frac{\pi R^6}{3} \ $, which is same as your working.

In case of applying divergence theorem, we close the surface with a disk at $z = 0$.

$\small \nabla \cdot (x^4, y^4, z^4) = 4(x^3+y^3+z^3) = 4r^3(\cos^3\theta \sin^3\phi+ \sin^3\theta \sin^3\phi + \cos^3\phi)$

where $0 \leq r \leq R$.

So volume integral is,

$\displaystyle \small \int_0^{2\pi} \int_0^{\pi/2} \int_0^R 4r^5 \sin\phi (\cos^3\theta \sin^3\phi+ \sin^3\theta \sin^3\phi + \cos^3\phi) \ dr \ d\phi \ d\theta = \frac{\pi R^6}{3}$.

Now note the flux through the bottom disk is zero so flux through spherical surface is $\frac{\pi R^6}{3}$, the same answer we received through direct surface integral.