Line integrals: 2. Locally exact forms and singular homology

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

As in part 1, \(E,F\) are Banach spaces and \(U\subseteq E\) is an open set. Recall that a form \(\omega:U\to L(E,F)\) is exact if \(\omega=Df\) for some \(f:U\to F\), closed if it is differentiable and \(D\omega(x)\in L(E,E;F)\) is symmetric for every \(x\in U\), and locally exact if it is closed and for every \(x\in U\) there is a neighborhood \(V\subseteq U\) of \(x\) on which \(\omega\) is exact. Last time we showed that every closed \(C^1\) form is locally exact, and that every closed form on an open subset of \(\mathbb{R}^2\) is locally exact (even if is differentiable but not \(C^1\)).

So far, we have only defined line integrals along \(C^1\) paths and curves. It turns out that for locally exact forms, we can extend the definition to paths that are merely continuous (and not necessarily differentiable).

Let \(\omega\) be a locally exact form on \(U\) and let \(\gamma:[a,b]\to U\) be a path. Since \(\gamma([a,b])\) is compact, there exists a partition \(P=\{a_0,\dots,a_k\}\) of \([a,b]\) and open balls \(B_1,\dots,B_k\) such that \(\omega\) is exact on \(B_i\) and \(\gamma([a_{i-1},a_i])\subseteq B_i\) for each \(i=1,\dots,k\). We define the integral of \(\omega\) along \(\gamma\) by $$
\int_\gamma \omega = \sum_{i=1}^k [g_i(\gamma(a_i))-g_i(\gamma(a_{i-1}))],
$$ where \(g_i\) is any potential for \(\omega\) on \(B_i\).


Lemma 9. The above integral is well-defined.

Proof. Suppose we are given different open balls \(\widetilde{B}_1,\dots,\widetilde{B}_k\) with the above properties, as well as corresponding potential functions \(\widetilde{g}_i\). For each \(i\) the functions \(g_i\) and \(\widetilde{g}_i\) are both potentials for \(\omega\) on the connected set \(B_i\cap \widetilde{B}_i\), so \(g_i-\widetilde{g}_i\) is constant and $$
g_i(\gamma(a_i))-g_i(\gamma(a_{i-1})) = \widetilde{g}_i(\gamma(a_i))-\widetilde{g}_i(\gamma(a_{i-1})).
$$ Next, we show that choosing a refinement of \(P\) does not change the value of the integral. Since any refinement of \(P\) can be obtained by adding finitely many points to \(P\), it suffices to show that adding a single point \(c\) to \(P\) does not change the value of the integral. Let $$
P=\{a_0,\dots,a_{j-1},c,a_j,\dots,a_k\}
$$ where \(a_{j-1} < c < a_j\). We can use the same open balls and potential functions as before, with \(B_j\) being used for the intervals \([a_{j-1},c]\) and \([c,a_j]\). Then the term $$ g_j(\gamma(a_j))-g_j(\gamma(a_{j-1})) $$ is replaced by $$ g_j(\gamma(a_j))-g_j(\gamma(c))+g_j(\gamma(c))-g_j(\gamma(a_{j-1})), $$ which does not change the value of the integral. This completes the proof, for if \(Q\) is another partition of \([a,b]\) then taking the common refinement \(P\cup Q\) does not change the value of the integral. \(\square\) It is easy to check that the fundamental theorem for line integrals (Theorem 2) still holds when \(\gamma\) is a path.

Let \(X,Y\) be topological spaces and let \(f,g:X\to Y\) be continuous maps. A homotopy from \(f\) to \(g\) is a continuous map \(H:X\times[0,1]\to Y\) such that \(H(s,0)=f(s)\) and \(H(s,1)=g(s)\) for all \(s\in[a,b]\). Let \(\gamma_1,\gamma_2:[a,b]\to X\) be paths (i.e. continuous maps). If \(t\mapsto H(0,t)\) and \(t\mapsto H(1,t)\) are constant, we say that \(H\) is a path homotopy. If there is a path homotopy from \(\gamma_1\) to \(\gamma_2\), we say that \(\gamma_1\) is homotopic to \(\gamma_2\).


Theorem 10 (Line integrals along homotopic paths). Let \(\omega\) be a locally exact form on \(U\). If \(\gamma_1,\gamma_2\) are paths in \(U\) that are homotopic, then $$\int_{\gamma_1} \omega = \int_{\gamma_2} \omega.$$

Proof. Let \(H:[a,b]\times[0,1]\to U\) be a path homotopy from \(\gamma_1\) to \(\gamma_2\). Since \([a,b]\times[0,1]\) is compact, there are partitions \begin{align}
a&=s_0\le\cdots\le s_m=b, \\
0&=t_0\le\cdots\le t_n=1
\end{align} such that for each rectangle \(R_{ij}=[s_{i-1},s_i]\times[t_{j-1},t_j]\) there is an open ball \(B_{ij}\subseteq U\) with \(H(R_{ij})\subseteq B_{ij}\) on which \(\omega\) is exact. For each \(j=0,\dots,n\), let \(\gamma^{(j)}(s)=H(s,t_j)\). Since \(\gamma^{(0)}=\gamma_1\) and \(\gamma^{(n)}=\gamma_2\), it suffices to show that $$
\int_{\gamma^{(j)}} \omega = \int_{\gamma^{(j-1)}} \omega
$$ for each \(j=1,\dots,n\). Fix some \(j\). For each \(i=1,\dots,m\), let \(g_i\) be a potential for \(\omega\) on \(B_{ij}\). Since \(g_i\) and \(g_{i-1}\) are both potentials for \(\omega\) on \(B_{ij}\cap B_{(i-1)j}\), they differ by a constant on \(B_{ij}\cap B_{(i-1)j}\). Therefore $$
g_i(\gamma^{(j)}(s_{i-1}))-g_i(\gamma^{(j-1)}(s_{i-1}))=g_{i-1}(\gamma^{(j)}(s_{i-1}))-g_{i-1}(\gamma^{(j-1)}(s_{i-1}))
$$ for each \(i=1,\dots,m\). We have \begin{align}
& \int_{\gamma^{(j)}} \omega – \int_{\gamma^{(j-1)}} \omega \\
&= \sum_{i=1}^m [g_i(\gamma^{(j)}(s_{i}))-g_i(\gamma^{(j)}(s_{i-1}))-g_i(\gamma^{(j-1)}(s_{i}))-g_i(\gamma^{(j-1)}(s_{i-1}))] \\
&= \sum_{i=1}^m [g_i(\gamma^{(j)}(s_{i}))-g_i(\gamma^{(j-1)}(s_{i}))-(g_{i-1}(\gamma^{(j)}(s_{i-1}))-g_{i-1}(\gamma^{(j-1)}(s_{i-1})))] \\
&= g_m(\gamma^{(j)}(b))-g_m(\gamma^{(j-1)}(b))-(g_0(\gamma^{(j)}(a))-g_0(\gamma^{(j-1)}(a))) \\
&= 0
\end{align} since \(\gamma^{(j)}\) and \(\gamma^{(j-1)}\) have the same starting and ending points. \(\square\)

An open set \(U\) is said to be simply connected if it is connected and every closed path in \(U\) is homotopic to a point (i.e. homotopic to a constant path).

Corollary 11. Every locally exact form on a simply connected open set is exact.

Proof. Apply Theorem 10 and Theorem 3. \(\square\)

The next theorem will be used in part 3.


Theorem 12. Every path in \(U\) is homotopic to a curve in \(U\).

Proof. Let \(\gamma:[a,b]\to U\) be a path. Since \(\gamma([a,b])\) is compact, there exists a partition \(P=\{a_0,\dots,a_k\}\) of \([a,b]\) and open balls \(B_1,\dots,B_k\subseteq U\) such that \(\gamma([a_{i-1},a_i])\subseteq B_i\) for each \(i=1,\dots,k\). Define \(\gamma_i:[0,1]\to B_i\) by \(\gamma_i(s)=\gamma(a_{i-1}+s(a_i-a_{i-1}))\); then $$
\gamma|_{[a_{i-1},a_i]}(s)=\gamma_i\left(\frac{s-a_{i-1}}{a_i-a_{i-1}}\right).
$$ For each \(i\) there is a path homotopy \(H_i:[0,1]\times[0,1]\to B_i\) from \(\gamma_i\) to the straight line segment \(\eta_i\) from \(\gamma(a_{i-1})\) to \(\gamma(a_i)\), so we can define a path homotopy \(H:[a,b]\times[0,1]\to U\) by setting $$
H|_{[a_{i-1},a_i]\times[0,1]}(s,t)=H_i\left(\frac{s-a_{i-1}}{a_i-a_{i-1}},t\right).
$$ Therefore \(\gamma\) is homotopic to the curve \(s\mapsto H(s,1)\). \(\square\)

Singular homology

We now turn to singular homology, and extend the line integral to singular 1-chains.

Given \(p+1\) affinely independent points \(\{v_0,\dots,v_p\}\) in \(\mathbb{R}^n\), the geometric \(p\)-simplex with vertices \(v_0,\dots,v_p\) is the subset of \(\mathbb{R}^n\) defined by $$
[v_0,\dots,v_p]=\left\{\sum_{i=0}^p t_i v_i : 0\le t_1\le 1\:\mathrm{and}\:\sum_{i=0}^p t_i=1\right\}.
$$ The integer \(p\) is called the dimension of the simplex. The simplices whose vertices are nonempty subsets of \(\{v_0,\dots,v_p\}\) are called the faces of the simplex. The \((p-1)\)-dimensional faces are the boundary faces of the simplex. The standard \(p\)-simplex is $$
\triangle_p=[e_0,\dots,e_p]\subseteq\mathbb{R}^p,
$$ where \(e_0=0\) and \(e_i\) is the \(i\)th standard basis vector. For each \(i=0,\dots,p\), we define the \(i\)th face map in \(\triangle_p\) to be the unique affine map \(F_{i,p}:\triangle_{p-1}\to\triangle_p\) satisfying $$
F_{i,p}(e_0)=e_0,\dots,\quad F_{i-p}(e_{i-1})=e_{i-1},\quad F_{i,p}(e_i)=e_{i+1},\dots,\quad F_{i,p}(e_{p-1})=e_p.
$$ Let \(U\subseteq E\) be an open set. A continuous map \(\sigma:\triangle_p\to U\) is called a singular \(p\)-simplex in \(U\). The singular chain group of \(U\) in degree \(p\), denoted by \(C_p(U)\), is the free abelian group generated by all singular \(p\)-simplices in \(U\). An element of \(C_p(U)\) is called a singular \(p\)-chain. The boundary of a singular \(p\)-simplex \(\sigma\) is the singular \((p-1)\)-chain defined by $$
\partial\sigma=\sum_{i=0}^p (-1)^i \sigma\circ F_{i,p}.
$$ For example, if \(\sigma\) is the identity map on \(\triangle_2\) then \(\partial\sigma=\sigma\circ F_{0,2}-\sigma\circ F_{1,2}+\sigma\circ F_{2,2}\) where \begin{align}
(\sigma\circ F_{0,2})(e_0)=e_1,\quad & (\sigma\circ F_{0,2})(e_1)=e_2, \\
(\sigma\circ F_{1,2})(e_0)=e_0,\quad & (\sigma\circ F_{1,2})(e_1)=e_2, \\
(\sigma\circ F_{2,2})(e_0)=e_0,\quad & (\sigma\circ F_{2,2})(e_1)=e_1.
\end{align} The map \(\partial\) extends uniquely to a group homomorphism \(\partial_p:C_p(U)\to C_{p-1}(U)\), called the singular boundary operator. We write \(\partial=\partial_p\) when the dimension \(p\) is clear.

Theorem 13. For any \(c\in C_p(U)\) we have \(\partial(\partial c)=0\).

We say that a singular \(p\)-chain is a cycle if \(\partial c=0\), and we say that \(c\) is a boundary if \(c=\partial b\) for some \(b\in C_{p+1}(U)\). Let \(Z_p(U)\) be the set of all singular \(p\)-cycles, and let \(B_p(U)\) be the set of all singular \(p\)-boundaries. Then \(Z_p(U)=\ker\partial_p\), and \(B_p(U)=\operatorname{im}\partial_{p-1}\). Since \(\partial_{p-1}\circ\partial_p=0\), we have \(B_p(U)\subseteq Z_p(U)\). The \(p\)th singular homology group of \(U\) is the quotient group $$
H_p(U)=Z_p(U)/B_p(U).
$$ The equivalence class in \(H_p(U)\) of a singular \(p\)-cycle \(c\) is called its homology class, and is denoted by \([c]\). If \([c]=[c’]\), then we say that \(c\) and \(c’\) are homologous.

Since \(\triangle_1=[0,1]\), any singular 1-simplex \(\gamma\) is a path in \(U\). Conversely, any path \(\gamma:[a,b]\to U\) can be considered as a singular 1-simplex in \(U\) since we have the reparametrization \(\widehat{\gamma}:[0,1]\to U\) given by \(\widehat{\gamma}(t)=\gamma(a+t(b-a))\).

Lemma 14.

  1. If \(\gamma\) is a singular 1-simplex then \([-\gamma]=-[\gamma]\), where \(-\gamma\) is the singular 1-simplex defined by $$(-\gamma)(t)+\gamma(1-t).$$
  2. If \(\gamma_1,\gamma_2\) are singular 1-simplices with \(\gamma_1(1)=\gamma_2(0)\), then \([\gamma_1+\gamma_2]=[\gamma_1\cdot\gamma_2]\), where \(\gamma_1\cdot\gamma_2\) is the singular 1-simplex defined by $$
    \gamma(t)=\begin{cases}
    \gamma_1(2t), & 0\le t\le 1, \\
    \gamma_2(2t-1), & 1 < t \le 2. \end{cases}$$
  3. If \(\gamma,\eta\) are singular 1-simplices that are (path) homotopic, then \([\gamma]=[\eta]\).

Let us denote a singular 0-simplex \(\sigma\) with \(\sigma(0)=x\) by \(P(x)\). If \(c\) is a singular 1-simplex in \(U\) and \(\partial c=0\) then by definition we have \(P(c(1))-P(c(0))=0\), i.e. \(c\) is a closed path.


Theorem 15. Every 1-cycle \(c\in Z_1(U)\) can be written in the form $$
[c]=\sum_{i=1}^k c_i[\gamma_i],
$$ where each \(\gamma_i:\triangle_1\to U\) is a closed path.

Proof. Suppose not; then we can write $$
c=\partial b+\sum_{i=1}^k c_i\gamma_i+\sum_{i=1}^j \sigma_i \tag{*}
$$ where \(b\) is a singular 2-chain, \(c_i,c’_i\ne 0\) for all \(i\), each \(\gamma_i\) is a closed path, and each \(\sigma_i\) is a path that is not closed. We can assume that \((j,k)\) is the smallest pair for which \(c\) can be written in this form (where we take \((j,k) < (j',k')\) if \(j < j'\), or \(j=j'\) and \(k < k'\)), and that \(j\ge 1\). Then $$ 0=\partial c=\sum_{i=1}^j [P(\sigma_i(1))-P(\sigma_i(0))]. $$ Suppose \(P(\sigma_i(1))=P(\sigma_1(0))\) for some \(i\ne 1\); then \([\sigma_i]+[\sigma_1]=[\sigma_i\cdot\sigma_1]\), so we can reduce either \(j\) or \(k\) in (*). Similarly, if \(P(\sigma_i(0))=P(\sigma_1(0))\) for some \(i\ne 1\), then \([\sigma_i]+[\sigma_1]=[(-\sigma_i)\cdot\sigma_1]\) and we can reduce \(j\) or \(k\). Both of these cases contradict the minimality of \((j,k)\), so we must have \(P(\sigma_1(0))=P(\sigma_1(1))\) since the coefficient of \(P(\sigma_1(0))\) in the above sum is \(0\). This contradicts the fact that \(\sigma_i\) is not closed. \(\square\) Let \(\omega\) be a locally exact form on \(U\) and let \(c\in C_1(U)\) be a 1-chain in \(U\). Write \(c=\sum_{i=1}^k c_i\sigma_i\) where \(c_i\in\mathbb{Z}\) and each \(\sigma_i\) is a 1-simplex in \(U\). We define the integral of \(\omega\) over \(c\) by $$
\int_c \omega = \sum_{i=1}^k c_i \int_{\sigma_i} \omega.
$$

Lemma 16. Let \(\sigma:\triangle_2\to U\) be a 2-simplex. Then $$\int_{\partial\sigma} \omega=0$$ for every locally exact form \(\omega\) on \(U\).

Proof. It is clear that $$
\int_{\partial\sigma} \omega=\int_{\sigma\circ\gamma} \omega
$$ where \(\gamma:[a,b]\to\triangle_2\) is a path traversing the boundary of \(\triangle_2\) counterclockwise. Since \(\triangle_2\) is convex, there is a path homotopy \(H:[a,b]\times[0,1]\to\triangle_2\) from \(\gamma\) to a point in \(\triangle_2\), so \(\sigma\circ H\) is a path homotopy from \(\sigma\circ\gamma\) to a point in \(\sigma(\triangle_2)\). Therefore $$\int_{\sigma\circ\gamma} \omega=0$$ by Theorem 10. \(\square\)


Theorem 17. If \(c\) and \(c’\) are homologous 1-cycles in \(U\), then $$\int_c \omega=\int_{c’} \omega$$ for every locally exact form \(\omega\) on \(U\). If \(\omega\) and \(\widetilde{\omega}\) are locally exact forms on \(U\) that differ by an exact form, then $$\int_c \omega=\int_c \widetilde{\omega}$$ for every 1-cycle \(c\) in \(U\).

Proof. If \(c\) and \(c’\) are homologous, then \(c-c’=\partial b\) for some 2-chain \(b\). Write \(b=\sum_{i=1}^k b_i\sigma_i\) where each \(\sigma_i\) is a 2-simplex. Then $$
\int_c \omega – \int_{c’} \omega = \int_{\partial b} \omega = \sum_{i=1}^k b_i \int_{\partial\sigma_i} \omega = 0
$$ by Lemma 16. Suppose \(\omega\) and \(\widetilde{\omega}\) are locally exact forms with \(\omega-\widetilde{\omega}=Df\) for some function \(f:U\to F\), and let \(c\) be a 1-cycle. By Theorem 15, \(c\) is homologous to \(\sum_{i=1}^k c_i\gamma_i\) where each \(\gamma_i\) is a closed path. Then $$
\int_c \omega – \int_c \widetilde{\omega} = \int_c Df = \sum_{i=1}^k c_i \int_{\gamma_i} Df = 0.
$$ \(\square\)

Next, time we will look at how the line integral, in this general formulation, is used in complex analysis.

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

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