Line integrals: 1. Exact, conservative and closed forms

Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis

The line integral is a useful tool for working with vector fields on \(\mathbb{R}^n\), (co)vector fields on manifolds, and complex differentiable functions. However, it is often unclear how these different versions of the line integral are related to each other. In the next few posts, I will be presenting a very general form of the line integral along with the standard theorems and some basic applications. Familiarity with the Fréchet derivative is assumed.

Let \(E,F\) be (real or complex) Banach spaces and let \(U\subseteq E\) be an open set. We define a form on \(U\) to be a continuous map \(\omega:U\to L(E,F)\), where \(L(E,F)\) is the space of continuous linear maps from \(E\) to \(F\). A path in \(U\) is a continuous map \(\gamma:[a,b]\to U\). If \(\gamma(a)=\gamma(b)\) then we say that \(\gamma\) is a closed path. We say that a continuous map \(\gamma:[a,b]\to U\) is a curve if there is a partition \(a=a_0 < \cdots < a_k=b\) of \([a,b]\) such that \(\gamma|_{[a_{i-1},a_i]}\) is a continuously differentiable path for each \(i\). If \(\gamma(a)=\gamma(b)\) then we say that \(\gamma\) is a closed curve.

If \(\gamma:[a,b]\to U\) is a \(C^1\) path and \(\omega\) is a form on \(U\), we define the (line) integral of \(\omega\) along \(\gamma\) by $$
\int_\gamma \omega = \int_\gamma \omega(x)\,dx = \int_a^b \omega(\gamma(t))\gamma'(t)\,dt.
$$ (Note that \(\omega(\gamma(t))\) is a linear map from \(E\) to \(F\), and \(\gamma'(t)\in E\) since we are identifying \(\gamma'(t)\) with \(D\gamma(t)(1)\).) If \(\gamma\) is a curve with partition \(\{a_0,\dots,a_k\}\), we define the integral of \(\omega\) along \(\gamma\) by $$
\int_\gamma \omega = \int_\gamma \omega(x)\,dx = \sum_{i=1}^k \int_{\gamma|_{[a_{i-1},a_i]}} \omega.

Theorem 1 (Properties of line integrals). Let \(\gamma:[a,b]\to U\) be a curve and let \(\omega,\eta\) be forms on \(U\).

  1. For any scalars \(a,b\), $$\int_\gamma (a\omega+b\eta)=a\int_\gamma \omega+b\int_\gamma \eta.$$
  2. For any Banach space \(G\) and any continuous linear map \(f:F\to G\), $$\int_\gamma f\omega = f\left(\int_\gamma \omega\right),$$ where \(f\omega:U\to L(E,G)\) is the form defined by \((f\omega)(x)u=f(\omega(x)u)\).
  3. If \(\gamma\) is constant then $$\int_\gamma \omega = 0.$$
  4. If \(\gamma_1=\gamma|_{[a,c]}\) and \(\gamma_2=\gamma|_{[c,b]}\) where \(a < c < b\) then $$\int_\gamma \omega = \int_{\gamma_1} \omega + \int_{\gamma_2} \omega.$$
  5. If \(\varphi:[c,d]\to [a,b]\) is a continuously differentiable function with \(\varphi(c)=a\) and \(\varphi(d)=b\) then $$\int_{\gamma\circ\varphi} \omega = \int_\gamma \omega.$$ If \(\varphi(c)=b\) and \(\varphi(d)=a\) (i.e. \(\varphi\) is decreasing) then $$\int_{\gamma\circ\varphi} \omega = -\int_\gamma \omega.$$
  6. We have $$\left\vert\int_\gamma \omega\right\vert \le L(\gamma) \sup_{t\in[a,b]} |\omega(\gamma(t))|,$$ where \(L(\gamma)\) is the length of \(\gamma\) defined by $$L(\gamma)=\sum_{i=1}^k\int_{a_{i-1}}^{a_i} |\gamma'(t)|\,dt.$$
  7. If \(\{\omega_n\}\) is a sequence of forms on \(U\) converging uniformly to a form \(\omega\), then $$\int_\gamma \omega = \lim_{n\to\infty}\int_\gamma \omega_n.$$

Proof. By linearity, we can assume that \(\gamma\) is a \(C^1\) path. For (2), we have \begin{align}
\int_\gamma f\omega &= \int_a^b f(\omega(\gamma(t))\gamma'(t))\,dt \\
&= f\left(\int_a^b \omega(\gamma(t))\gamma'(t)\,dt\right) \\
&= f\left(\int_\gamma \omega\right)
\end{align} using one of the basic properties of the regulated integral or Bochner integral. For (4), we have \begin{align}
\int_{\gamma\circ\varphi} \omega &= \int_c^d \omega((\gamma\circ\varphi)(t))(\gamma\circ\varphi)'(t)\,dt \\
&= \int_c^d \omega(\gamma(\varphi(t)))\gamma'(\varphi(t))\varphi'(t)\,dt \\
&= \int_a^b \omega(\gamma(t))\gamma'(t)\,dt \\
&= \int_\gamma \omega
\end{align} by the change of variables formula. For (5), we have \begin{align}
\left\vert\int_\gamma \omega\right\vert &= \left\vert\int_a^b \omega(\gamma(t))\gamma'(t)\,dt\right\vert \\
&\le \int_a^b |\omega(\gamma(t))||\gamma'(t)|\,dt \\
&\le L(\gamma) \sup_{t\in[a,b]} |\omega(\gamma(t))|.
\end{align} For (6), we have \begin{align}
\left\vert\int_\gamma \omega_n – \int_\gamma \omega\right\vert &= \left\vert\int_\gamma (\omega_n-\omega)\right\vert \\
&\le L(\gamma) \sup_{t\in[a,b]} |(\omega_n-\omega)(\gamma(t))| \\
&\to 0
\end{align} as \(n\to\infty\). \(\square\)

Example (Line integrals in \(\mathbb{R}^n\)). If \(U\subseteq\mathbb{R}^n\) is an open set and \(F:U\to\mathbb{R}^n\) is a vector field, then its associated form \(\omega_F:U\to L(\mathbb{R}^n,\mathbb{R})\) is given by $$\omega_F(x)v=F(x)\cdot v,$$ where \(\cdot\) is the usual dot product on \(\mathbb{R}^n\). We can then define $$\int_\gamma F \cdot dr = \int_\gamma \omega_F = \int_a^b F(\gamma(t)) \cdot \gamma'(t)\,dt$$ for any curve \(\gamma:[a,b]\to U\).

Example (Complex line integrals). If \(U\subseteq\mathbb{C}\) is an open set and \(f:U\to\mathbb{C}\) is continuous, then its associated form \(\omega_f:U\to L(\mathbb{C},\mathbb{C})\) is given by $$\omega_f(z)w=wf(z).$$ We can then define $$\int_\gamma f(z)\,dz = \int_\gamma \omega_f = \int_a^b f(\gamma(t))\gamma'(t)\,dt$$ for any curve \(\gamma:[a,b]\to U\). Later, we will examine this type of line integral more closely.

Exact, conservative and closed forms

We have an important generalization of the fundamental theorem of calculus to line integrals.

Theorem 2 (Fundamental theorem for line integrals). Let \(f:U\to F\) be continuously differentiable and let \(\gamma:[a,b]\to U\) be a curve. Then $$\int_\gamma Df = f(\gamma(b))-f(\gamma(a)).$$

Proof. First assume that \(\gamma\) is a \(C^1\) path. Then \begin{align}
\int_\gamma Df &= \int_a^b Df(\gamma(t))\gamma'(t)\,dt \\
&= \int_a^b (f\circ\gamma)'(t)\,dt \\
&= f(\gamma(b))-f(\gamma(a))
\end{align} by the fundamental theorem of calculus. If \(\gamma\) is a curve with partition \(\{a_0,\dots,a_k\}\) then $$
\int_\gamma Df = \sum_{i=1}^k [f(\gamma(a_k))-f(\gamma(a_{k-1}))] = f(\gamma(b))-f(\gamma(a)).
$$ \(\square\)

Note that in particular we have $$\int_\gamma Df = 0$$ for any closed curve \(\gamma\) in \(U\). If \(\omega\) is a form on \(U\), a function \(f:U\to F\) satisfying \(\omega=Df\) is called a potential for \(\omega\). We say that a form \(\omega\) is exact if it has a potential function. Note that if \(U\) is connected then this result implies that \(f-g\) is constant if \(f,g\) are any two potentials for \(\omega\). If the integral of \(\omega\) along any closed curve is zero, then we say that \(\omega\) is conservative. It is easy to see that a form is conservative if and only if it is path-independent, in the sense that $$\int_\gamma \omega=\int_{\widetilde{\gamma}} \omega$$ whenever \(\gamma,\widetilde{\gamma}\) are curves with the same starting and ending points.

Theorem 3. A form is conservative if and only if it is exact.

Proof. Theorem 2 shows that every exact form is conservative, so it remains to show that every conservative form is exact. Let \(\omega\) be a conservative form on \(U\). We can assume that \(U\) is connected, for otherwise we can obtain a potential function \(f_\alpha\) on each component \(U_\alpha\) of \(U\) and define a potential \(f:U\to F\) for \(\omega\) by setting \(f|_{U_\alpha}=f_\alpha\). Since \(\omega\) is path-independent, for any two points \(x,y\in U\) we can define $$\int_x^y \omega=\int_\gamma \omega$$ where \(\gamma\) is any curve from \(x\) to \(y\). Choose some \(x_0\in U\) and let $$f(x)=\int_{x_0}^x \omega;$$ we want to show that \(\omega=Df\). Let \(x\in U\) and choose \(r > 0\) so that the open ball of radius \(r\) around \(x\) is contained in \(U\). For all \(|h| < r\) the straight line from \(x\) to \(x+h\) is contained in \(U\), so \begin{align} \frac{1}{|h|} |f(x+h)-f(x)-\omega(x)h| &= \frac{1}{|h|}\left\vert\int_{x_0}^{x+h} \omega - \int_{x_0}^x \omega - \omega(x)h\right\vert \\ &= \frac{1}{|h|}\left\vert\int_x^{x+h} \omega - \omega(x)h\right\vert \\ &= \frac{1}{|h|}\left\vert\int_0^1 \omega(x+th)h\,dt-\int_0^1 \omega(x)h\,dt\right\vert \\ &= \frac{1}{|h|}\left\vert\left(\int_0^1 [\omega(x+th)-\omega(x)]\,dt\right)h\right\vert \\ &\le \sup_{t\in[0,1]} |\omega(x+th)-\omega(x)| \\ &\to 0 \end{align} as \(h\to 0\) since \(\omega\) is continuous. \(\square\) If \(\omega\) is a differentiable form on \(U\), we say that \(\omega\) is closed if \(D\omega(x)\in L(E,E;F)\) is symmetric for every \(x\in U\). If \(\omega=Df\) for some \(C^2\) map \(f:U\to F\) then \(D\omega(x)=D^2 f(x)\) is always symmetric, so we have the following result:

Theorem 4. Every exact \(C^1\) form is closed. \(\square\)

The converse of Theorem 4 holds for certain kinds of sets. A set \(A\) in a vector space is star-shaped with respect to \(x_0\in A\) if the line segment from \(x_0\) to any \(x\in A\) is contained in \(A\).

Theorem 5 (Poincaré lemma). Let \(U\subseteq E\) be a star-shaped open set. Every closed \(C^1\) form on \(U\) is exact.

Proof. Suppose that \(U\) is star-shaped with respect to some \(x_0\in U\). By translating \(U\), we can assume that \(x_0=0\). Let \(\omega\) be a closed form on \(U\). Define $$f(x)=\int_0^1 \omega(tx)x\,dt,$$ which is simply the integral of \(\omega\) along the straight line segment from \(0\) to \(x\). Using differentiation under the integral sign and the symmetry of \(D\omega(tx)\), \begin{align}
Df(x)u &= \int_0^1 [tD\omega(tx)(u,x)+\omega(tx)u]\,dt \\
&= \int_0^1 [tD\omega(tx)(x,u)+\omega(tx)u]\,dt \\
&= \left(\int_0^1 [tD\omega(tx)x+\omega(tx)]\,dt\right)u \\
&= \left(\int_0^1 \frac{d}{dt}(t\omega(tx))\,dt\right)u \\
&= \omega(x)u.
\end{align} \(\square\)

Lemma 6. Let \(R=[a,b]\times[c,d]\) be a rectangle and let \(\omega\) be a closed (differentiable) form defined on an open subset of \(\mathbb{R}^2\) containing \(R\). Then $$\int_{\partial R} \omega=0,$$ where the integral is taken counterclockwise along the boundary of \(R\).

Proof. First note that \(D(D\omega(x))(y)=D\omega(x)\) is symmetric for all \(y\in\mathbb{R}^2\), so the Poincaré lemma shows that \(D\omega(x)\) is exact for all \(x\in R\). Decompose \(R\) into the four rectangles \begin{align}
R_1 &= [a,\tfrac{b-a}{2}]\times[c,\tfrac{d-c}{2}], \\
R_2 &= [\tfrac{b-a}{2},b]\times[c,\tfrac{d-c}{2}], \\
R_3 &= [a,\tfrac{b-a}{2}]\times[\tfrac{d-c}{2},d], \\
R_4 &= [\tfrac{b-a}{2},b]\times[\tfrac{d-c}{2},d].
\end{align} Due to the orientations of \(\partial R_1,\dots,\partial R_4\), the inside boundaries cancel and we have $$
\int_{\partial R} \omega = \sum_{i=1}^4 \int_{\partial R_i} \omega \quad\mathrm{and}\quad \left\vert\int_{\partial R} \omega\right\vert \le \sum_{i=1}^4 \left\vert\int_{\partial R_i} \omega\right\vert,$$ so there is a rectangle \(R^{(1)}\) among \(R_1,\dots,R_4\) for which $$\left\vert\int_{\partial R^{(1)}} \omega\right\vert \ge \frac{1}{4}\left\vert\int_{\partial R} \omega\right\vert.$$ Replacing \(R\) with \(R^{(1)}\) in the above, we have a rectangle \(R^{(2)}\subseteq R^{(1)}\) such that $$\left\vert\int_{\partial R^{(2)}} \omega\right\vert \ge \frac{1}{4}\left\vert\int_{\partial R^{(1)}} \omega\right\vert.$$ Repeating this process, we obtain a sequence of rectangles $$R^{(1)}\supseteq R^{(2)}\supseteq \cdots$$ such that $$\left\vert\int_{\partial R^{(n+1)}} \omega\right\vert \ge \frac{1}{4}\left\vert\int_{\partial R^{(n)}} \omega\right\vert$$ for all \(n\). Therefore $$\left\vert\int_{\partial R^{(n)}} \omega\right\vert \ge \frac{1}{4^n}\left\vert\int_{\partial R} \omega\right\vert.$$ If \(L_0\) is the length of \(\partial R\) and \(L_n\) is the length of \(\partial R^{(n)}\), then \(L_n=L_0/2^n\), and if \(\operatorname{diam} R\) is the diameter of \(R\), then \(\operatorname{diam} R^{(n)} = (\operatorname{diam} R)/2^n\). Since every \(R^{(n)}\) is compact and \(\operatorname{diam}R^{(n)} \to 0\) as \(n\to\infty\), there is exactly one point $$x_0\in\bigcap_{n=1}^\infty R^{(n)}.$$ Since \(\omega\) is differentiable at \(x_0\), there exists a neighborhood \(U\) of \(x_0\) such that $$\omega(x)=\omega(x_0)+D\omega(x_0)(x-x_0)+\theta(x-x_0)$$ for every \(x\in U\), where \(\theta\) is a continuous function into \(L(E,F)\) satisfying $$\lim_{x\to x_0} \frac{\theta(x-x_0)}{|x-x_0|} = 0.\tag{*}$$ (This follows directly from the definition of the derivative.) For sufficiently large \(n\) we have \(R^{(n)}\subseteq U\) and \begin{align}
\int_{\partial R^{(n)}} \omega &= \int_{\partial R^{(n)}} \omega(x_0)\,dx + \int_{\partial R^{(n)}} D\omega(x_0)(x-x_0)\,dx + \int_{\partial R^{(n)}} \theta(x-x_0)\,dx \\
&= \int_{\partial R^{(n)}} [\omega(x_0)-D\omega(x_0)x_0]\,dx + \int_{\partial R^{(n)}} D\omega(x_0) + \int_{\partial R^{(n)}} \theta(x-x_0)\,dx \\
&= \int_{\partial R^{(n)}} \theta(x-x_0)\,dx
\end{align} since \(x\mapsto [\omega(x_0)-D\omega(x_0)x_0]x\) is a primitive for the constant form \(\omega(x_0)-D\omega(x_0)x_0\) and \(D\omega(x_0)\) is exact. Therefore \begin{align}
\left\vert\int_{\partial R} \omega\right\vert &\le 4^n \left\vert\int_{\partial R^{(n)}} \omega\right\vert \\
&\le 4^n L_n \sup_{x\in R^{(n)}} |\theta(x-x_0)| \\
&\le 4^n L_n (\operatorname{diam} R^{(n)}) \sup_{x\in R^{(n)}\setminus\{x_0\}} \frac{|\theta(x-x_0)|}{|x-x_0|} \\
&\le L_0 (\operatorname{diam} R) \sup_{x\in R^{(n)}\setminus\{x_0\}} \frac{|\theta(x-x_0)|}{|x-x_0|} \\
&\to 0
\end{align} as \(n\to\infty\) by (*). \(\square\)

Theorem 7 (Morera’s theorem). If \(\omega\) is a form on a disc $$U=\{x\in\mathbb{R}^2:|x-x_0| < r\}$$ and $$\int_{\partial R} \omega = 0$$ for every rectangle \(R\) contained in \(U\), then \(\omega\) is exact.

Proof. Define $$f(x)=\int_{x_0}^x \omega,$$ where the integral is taken along the sides of a rectangle whose opposite vertices are \(x_0\) are \(x\). By considering an appropriate rectangle we have $$f(x+h)-f(x)=\int_x^{x+h} \omega,$$ so we can use the argument in Theorem 3 to show that \(\omega=Df\). \(\square\)

Corollary 8 (Goursat’s theorem). If \(\omega\) is a closed (differentiable) form on a disc $$U=\{x\in\mathbb{R}^2:|x-x_0| < r\},$$ then \(\omega\) is exact.

Proof. Apply Lemma 6 and Theorem 7. \(\square\)

A closed form \(\omega\) on \(U\) is said to be locally exact if for every \(x\in U\) there is a neighborhood \(V\subseteq U\) of \(x\) on which \(\omega\) is exact. The Poincaré lemma shows that every closed \(C^1\) form on an open subset of \(E\) is locally exact, and Corollary 8 shows that every closed form (not necessarily of class \(C^1\)) on an open subset of \(\mathbb{R}^2\) is locally exact. This latter result is fundamental to complex analysis.

Next time we will see why locally exact forms are important, and how we can interpret line integrals in the framework of singular homology.

Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis

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