Definition 6.2.1 (Integrating a $k$-form field over a parametrized domain). Let $U\subset \R^k$ be a bounded open set with $\text{vol}_k \partial U = 0$. Let $V\subset \R^n$ be open, and let $[\gamma(U)]$ be a parametrized domain in $V$. Let $\partial$ be a $k$-form field on $V$.

Then the integral of $\varphi$ over $[\gamma(U)]$ is

$$ \int_{[\gamma(U)}\varphi = \int_U \varphi(P_{\gamma(u)}(D_1\gamma(u), \dots, D_k\gamma(u)))|d^k u|. $$

Example 6.2.2 (Integrating a 1-form field over a parametrized curve). Consider a case where $k=1, n=2$ and $\gamma(u) = \begin{pmatrix} R\cos u\\ R\sin u\end{pmatrix}$. We will take $U$ to be the interval $[0, a]$ for some $a > 0$. If we integrate $xdy - ydx$ over $[\gamma(u)]$, then

$$ \int_{[\gamma(U)]}(xdy - ydx) = \int_{[0, a]}(xdy - ydx) P_{\begin{pmatrix}R\cos u\\R\sin u\end{pmatrix}}\begin{bmatrix} -R \sin u\\ R \cos u\end{bmatrix}|du|\\ = \int_{[0, a]} (R\cos u R\cos u - (R\sin u) (-R\sin u))|du| = \int_{[0, a]}R^2|du| = \int_{0}^a R^2du = R^2a. $$