曲线与曲面上的微分形式积分

The pullback does all the work

A worked pullback

Pulling back two-forms

Manifolds, the general objects

In pulling $\omega = x\, dy$ back through $x = \cos t$, $y = \sin t$, what replaces $dy$?

The pullback of $\omega = -y\, dx + x\, dy$ over the counterclockwise unit circle gave $2\pi$. What does the clockwise traversal give?

What does the wedge algebra produce from $dx \wedge dy$ under a parametrization $x(u,v), y(u,v)$?

Under the substitution $x = t^2$, the differential transforms as $dx = c \cdot t\, dt$. Give $c$, as a plain number.

For $x = u + v$, $y = u - v$, give the Jacobian $\partial(x, y)/\partial(u, v)$, as a plain number.