外导数与广义斯托克斯定理

Differentiating a form

d squared is zero

One theorem for everything

Reading the unification

What is $d\omega$ for $\omega = x^2 y\, dx + \sin y\, dy$?

The generalized theorem with $k = 3$ becomes which classical statement?

A computation yields $d(d\omega) = xy\, dx \wedge dy \wedge dz$ for some form $\omega$. What follows?

Give the coefficient of $dx \wedge dy$ in $d(y\, dx)$, as a plain number.

By the generalized theorem, $\int_a^b f'(x)\, dx$ equals $f(b) - f(a)$. The boundary of $[a, b]$ consists of how many points? Give the count, as a plain number.