question about differential form on manifold

Question

1. If I define a 1-form on $\mathbf R^{2}$, for example, $\omega_1=-y\,dx+x\,dy$, and integrate $\omega_1$ over a circle $C$ in $\mathbf R^{2}$,can I say that I define an 1-form $\omega_2=-y\,dx+x\,dy$ only on the circle $C$ and integrate $\omega_2$ over this circle? Are these two integrals same?

2. What is $d\omega_1$ and $d\omega_2$, and where is the difference between these two forms?

Answer 1

Any $2$-form on a $1$-dimensional manifold must be $0$. The key point is this: If $\iota\colon C\to\Bbb R^2$ is the inclusion map of the circle into the plane, then the restriction of a form $\omega$ on the plane to the circle is given by $\iota^*\omega$. Since $d$ commutes with pullback, we always have $$d(\iota^*\omega) = \iota^*(d\omega).$$

This holds when $\iota$ is the inclusion map of any submanifold $X\subset M$ and $\omega$ is a form on $M$. Moreover, if $\omega$ is a $k$-form on $M$ and $X$ is an oriented $k$-dimensional submanifold, then we typically write $$\int_X \omega \quad\text{for}\quad \int_X \iota^*\omega.$$