I'm preparing for an upcoming exam and here's a practice problem:
Let $N = \{(w, x, y, z) \in \mathbb R^4 | w^2 + x^2 + y^2 + z^2 = 1 | z \geq 0\}$ be the northern hemisphere of the 3-sphere. Calculate:
$$\int_{N} e^xy^2 dx \wedge dw \wedge dz + 2y e^x dx \wedge dy \wedge dz + z dz \wedge dy \wedge dz$$
I'm thinking of using Stokes' theorem for this and then I would just integrate $d\eta$ over the boundary $w^2 + w^2 + y^2 = 1$ where $\eta$ is the expression within the integral. But before that, we notice that $dz = 0$ on this boundary. So would the whole integral just be $0$ or am I missing something?