Subjects: Differential Geometry
Links: Line Integral over a Vector Field, Covector Fields on Smooth Manifolds, Conservative Fields, The Cotangent Bundle, Local and Global Sections of Vector Bundles, Riemann Integral in R, Vector Fields on Smooth Manifolds

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.