What is Differential form; line integral? — Mathematics

Definition 20.1 University Mathematics — Year 2 · Chapter 20 — Line Integrals and Multiple Integrals

Let $U \subseteq \R^2$ be open. A differential form of degree $1$ and class $\mathcal{C}^0$ on $U$ is an expression $\omega = P\,\dd x + Q\,\dd y$ with $P, Q \colon U \to \R$ continuous — formally, a continuous map from $U$ into the dual of $\R^2$ , $\omega(M) = P(M)\,e_1^* + Q(M)\,e_2^*$ . For a $\mathcal{C}^1$ arc $\gamma \colon [a, b] \to U$ , $\gamma(t) = (x(t), y(t))$ , the line integral of $\omega$ along $\gamma$ is

\int_\gamma \omega = \int_a^b \Bigl(P(\gamma(t))\,x'(t) + Q(\gamma(t))\,y'(t)\Bigr)\,\dd t .

The definitions extend verbatim to $\R^3$ (forms $P\,\dd x + Q\,\dd y + R\,\dd z$ ) and to piecewise $\mathcal{C}^1$ arcs (sum over the pieces).

Examples

Example 20.4 (One integral, two parametrizations, one sign trap)

Compute $\int_\gamma x\,\dd y$ along the upper unit semicircle from $(1, 0)$ to $(-1, 0)$ . With $\gamma(t) = (\cos t, \sin t)$ , $t \in \intcc0\pi$ :

\int_0^\pi\cos t\cdot\cos t\,\dd t = \frac\pi2 .

With the graph parametrization $x \mapsto (x, \sqrt{1 - x^2})$ , $x$ from $1$ to $-1$ (note the direction!):

\int_1^{-1}x\cdot\frac{-x}{\sqrt{1 - x^2}}\,\dd x = \int_{-1}^{1}\frac{x^2}{\sqrt{1 - x^2}}\,\dd x = \frac\pi2

( $x = \sin u$ reduces it to a Wallis integral). Same value, as Proposition 20.2 guarantees — but only because both runs go from $(1,0)$ to $(-1,0)$ ; reversing the travel flips the sign. Closing the path along the $x$ -axis (where $\dd y = 0$ ) adds nothing, and the total $\frac\pi2$ is the area of the half-disk: the first instance of the boundary-area formulas of Green–Riemann below.

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