I am studying Stokes' Theorem and its applications, and dealing with the following two problems:
- Consider the surface $ S ={(x, y, z) \mid z=1-x^{2}-y^{2}>0}$. Find a smooth parametrisation $H$ of $S$ and then calculate the integral of the 2 -form $\omega=x \mathrm{~d} y \wedge \mathrm{d} z+$ $y \mathrm{~d} z \wedge \mathrm{d} x+z \mathrm{~d} x \wedge \mathrm{d} y$ over this surface.
- Use Stokes' Theorem to calculate the integral of the 2 -form
$$ \omega=y \mathrm{~d} y \wedge \mathrm{d} z+z \mathrm{~d} z \wedge \mathrm{d} x+\mathrm{d} x \wedge \mathrm{d} y $$ over the hemisphere $(x, y, z) \mid x^{2}+y^{2}+z^{2}=1, z>0$ oriented so that the normal vector points up.
For 1), I considered to use the polar coordinates: $H(r,\theta) = (rcos\theta, rsin\theta, 1-r^2)$, where H is defined on $U = \{ (r,\theta) : 0<\text{r}<1, 0<\theta<2\pi \}$. Then using definition of integrals of 2-forms, I can get
$\int_{H(U)} \omega = \int_U H^*(\omega)$
Then I computed $H^*(\omega)$ and calculated the resulting double integral as equal to $\frac{3\pi}{2}$ (I am adding my calculations below). The problem is I am not even sure that my change of variables function $H$ is correct.
For 2), I do not know really how to proceed. I am especially confused about the part "so that the normal vector points up." What should I understand from this? Note: both problems are from Joseph Taylor, "Foundations of Analysis". My level is not above of that book.
