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, \\
\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