Another important application of covector fields is to make coordinate independent sense of the notion of a line integral.

Suppose $[a,b]\subseteq \mathbb{R}$ is a compact interval and $\omega$ is a smooth covector field on $[a,b]$. If we let $t$ denote the standard coordinate on $\mathbb{R}$, $\omega$ can be written as ${\omega }_t=f(t)dt$ for some smooth function $f:[a,b]\to \mathbb{R}$. We define the integral of $\omega$ over $[a,b]$ to be $$\int_{[a,b]} \omega := \int_a^b f(t),dt.$$

Diffeomorphism Invariance of the Integral: Let $\omega$ be a smooth covector field on the compact interval $[a,b]\subseteq \mathbb{R}$. If $\phi :[c,d]\to [a,b]$ is an increasing diffeomorphism, then $$\int_{[c,d]} \varphi^\omega = \int_{[a,b]} \omega,$$and if $\phi$ is a decreasing diffeomorphism then $$\int_{[c,d]} \varphi^\omega =- \int_{[a,b]} \omega.$$

Now let $M$ be a smooth manifold. By a curve segment in $M$ we mean a continuous curve $\gamma :[a,b]\to M$ whose domain is a compact interval. It is a smooth curve segment if it has a smooth extension to an open interval containing $[a,b]$. A piecewise smooth curve segment is a curve segment $\gamma :[a,b]\to M$ with the property that there exists a finite partition $a=a_0<a_1<\cdots <a_k=b$ of $[a,b]$ such that $\gamma {|}_{[a_{i-1},a_i]}$ is smooth for each $i$.

Lemma: If $M$ is a connected smooth manifold, any two points of $M$ can be joined by a piecewise smooth curve segment.

If $\gamma :[a,b]\to M$ is a smooth curve segment and $\omega$ is a smooth covector field on $M$, we define the line integral of $\omega$ over $\gamma$ to be the real number$$\int_\gamma \omega:= \int_{[a,b]}\gamma^\omega. $$More generally, if $\gamma$ is piecewise smooth, we define $$\int_\gamma \omega := \sum_{i = 1}^k \int_{[a_{i-1},a_i]}\gamma^\omega,$$where $[a_{i-1},a_i]$, $i\in \{1,\dots ,k\}$, are the intervals on which $\gamma$ is smooth.

Properties of Line Integrals: Let $M$ be a smooth manifold. Suppose $\gamma :[a,b]\to M$ is a piecewise smooth curve segment and $\omega ,{\omega }_1,{\omega }_2\in {\mathrm{\Omega }}^1(M)$.

• For any $c_1,c_2\in \mathbb{R}$, $$\int_\gamma (c_1\omega_1+c_2\omega_2) = c_1\int_\gamma \omega_1 + c_2\int_\gamma \omega_2. $$
• It $\gamma$ is a constant map, then ${\int }_{\gamma }\omega =0.$
• If $a<c<b$, then$$\int_\gamma\omega = \int_{\gamma_1}\omega+ \int_{\gamma_2}\omega, $$where ${\gamma }_1:=\gamma {|}_{[a,c]}$ and ${\gamma }_2:=\gamma {|}_{[c,b]}$.

Prop: Suppose $F:M\to N$ is any smooth map, $\omega \in {\mathfrak{X}}^{\ast }(N)$, and $\gamma$ is a piecewise smooth curve segment in $M$, then $$\int_\gamma F^*\omega = \int_{F \circ \gamma}\omega. $$

Prop: If $\gamma :[a,b]\to M$ is a piecewise smooth curve segment, the line integral of $\omega$ over $\gamma$ can also be expressed as the ordinary integral $$\int_\gamma \omega = \int_a^b \omega_{\gamma(t)}(\gamma'(t)), dt. $$
Parameter Independence of Line Integrals: Suppose $M$ is a smooth manifold, $\omega$ is a smooth covector field on $M$, and $\gamma$ is a piecewise smooth curve in $M$. For any reparametrization $\tilde{\gamma }$ of $\gamma$ we have $$\int_{\widetilde\gamma} \omega = \begin{dcases}
\int_\gamma\omega & \text{if $\tilde{\gamma }$ is a forward reparametrization,} \
-\int_\gamma\omega & \text{if $\tilde{\gamma }$ is a backward reparametrization.}
\end

Conservative Covector Field

We say that a smooth covector field $\omega$ on a manifold $M$ is exact, or an exact differential on $M$ if there is a function $f\in {\mathcal{C}}^{\mathrm{\infty }}(M)$ such that $\omega =df$. In this case, the function $f$ is called a potential for $\omega$. The potential is not uniquely determined, but the difference between two potentials for $\omega$ must be a constant on each component of $M$.

We say that $\gamma$ is a closed curve segment if $\gamma (a)=\gamma (b)$.

We say that a smooth covector field $\omega$ is conservative if the line integral of $\omega$ over any closed piecewise smooth curve segment is zero.

Lemma: A smooth covector field $\omega$ is conservative iff the line integral of $\omega$ depends only on the endpoints of the curve, i.e., ${\int }_{\gamma }\omega ={\int }_{\tilde{\gamma }}\omega$ whenever $\gamma$ and $\tilde{\gamma }$ are piecewise smooth curve segments with the same starting and ending points.

Prop: If $M$ is a compact manifold,, then every exact covector field on $M$ vanishes at least at two points.

Th: Let $M$ be a smooth manifold with or without boundary. A smooth covector field on $M$ is conservative iff it is exact.

Let $f$ be any potential function for $\omega$, and let $(U,(x^i))$ be any smooth chart of $M$. Because $f$ is smooth, it satisfies the following identity on $U$:$$\frac{\partial^2 f}{\partial x^i\partial x^j} = \frac{\partial^2 f}{\partial x^j\partial x^i}.$$Writing $\omega ={\omega }_{i}d{x}^i$ in coordinates, the fact that $\omega =df$, is equivalent to ${\omega }_i=\frac{\partial f}{\partial x_i}$. Substituting this, we get that$$\frac{\partial \omega_i}{\partial x^j} = \frac{\partial \omega_j}{\partial x^i}.$$We say that a smooth covector field $\omega$ is closed if its components in every smooth chart satisfy the equality above.

Lemma: Every exact covector field is closed.

Prop: Let $\omega$ be a smooth covector field on a smooth manifold $M$ with or without boundary. The following are equivalent:

• $\omega$ is closed.
• $\omega$ satisfies $\frac{\partial {\omega }_i}{\partial x^j}=\frac{\partial {\omega }_j}{\partial x^i}$ in some smooth chart around every point.
• For any open subset $U\subseteq M$ and smooth vector fields $X,Y\in \mathfrak{X}(U)$, $$X(\omega(Y)) - Y(\omega(X)) = \omega[X, Y]. $$

Cor: If $G:M\to N$ is a local diffeomorphism, then the pullback $G^{\ast }:{\mathfrak{X}}^{\ast }(N)\to {\mathfrak{X}}^{\ast }(M)$ takes closed covector fields to closed covector fields, and exact one to exact ones.

The question of whether a particular closed covector field is exact is a global one, depending on the shape of the domain in question. This observation is the starting point for de Rham Cohomology, which expresses a deep relationship between smooth structures and topology.

Prop: If $U$ is a star-shaped open subset of ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$, then every closed covector field on $U$ is exact.

Local Exactness of Closed Covector Fields: Let $\omega$ be a closed covector field on a smooth manifold $M$ with or without boundary. Then every $p\in M$ has a neighbourhood on which $\omega$ is exact.