Analysis Theorems

11 Line Integrals

Lemma 11.1. If \(\psi :[a,b]\to \RR ^n\) is a rectifiable curve and equivalent to \(\phi :[c,d]\to \RR ^n\), then \(\phi \) is rectifiable.

Proof. Let \(h:[a,b]\to [c,d]\) be the map of equivalence. Then for any partition \(\cP \) of \([a,b]\), \(h(\cP )\) is a partition of \([c,d]\) yielding the same lengths. □

Lemma 11.2. If \(\psi :[a,b]\to \RR ^n\) is a rectifiable curve, then \(L(\psi ) = L(\psi |_{[a,c]})+L(\psi |_{[c,b]})\) for any \(c\in [a,b]\), and \(L(\psi )\geq \Vert \psi (b)-\psi (a)\Vert _2\).

Proof. For the first one, just add in the point \(c\) to any partition of \([a,b]\). For the second, look at the trivial (initial) partition. □

Lemma 11.3. If \(\psi :[0,1]\to \RR ^n\) is a rectifiable curve, then \(s:[0,1]\to [0,L(\psi )]\), \(s(t) = L(\psi |_{[0,t]})\) is continuously monotonically increasing, and is constant on a subinterval \([a,b]\) iff \(\psi \) is.

Proof. Since \(L\) is non-negative, by Lemma 11.2 \(s\) is monotonic. If \(\psi \) is constant, certainly \(s\) is by the same Lemma, and conversely if \(\psi \) is not constant, then since there is a nonzero partition, \(s\) cannot be constant.

For continuity, for any \(\ee >0\) and \(t_0 \in [0,1]\) we have \(|s(t)-s(t_0)| = L(\psi |_{[t_0,t]}) = (L(\psi |_{[t_0,t]})-L(\psi |_{[t_0,t]},\cP ))+L(\psi |_{[t_0,t]},\cP )\) and now we choose a \(\cP \) to bound this. In particular, we have our partition of \([0,1]\) be less than \(\frac {\ee }{2}\) away from \(L(\psi )\) and we assume that \(f\) varies at most \(\frac {\ee }{2}\) in each subinterval, which is at most \(\delta \) in length. As long as \(t \in (t_0-\delta ,t_0+\delta )\), we have then \((L(\psi |_{[t_0,t]})-L(\psi |_{[t_0,t]},\cP ))+L(\psi |_{[t_0,t]},\cP ) \leq \ee \), giving continuity. □

Proposition 11.4. A \(\cC ^1\) parameterized curve \(\psi :[0,1]\to \RR ^n\) is rectifiable, its arc length given by \(L(\psi ) = \int _{[0,1]}\Vert \psi '\Vert _2\).

Proof. For any partition \(\cP \) we have

\[L(\psi ,\cP ) = \sum _{[a,b]\in \cP }\Vert \psi (b)-\psi (a)\Vert _2= \sum _{[a,b]\in \cP }\bigg \Vert \int _{[a,b]}\psi '\bigg \Vert _2 \leq \sum _{[a,b]\in \cP }\int _{[a,b]}\Vert \psi '\Vert _2=\int _{[0,1]}\Vert \psi '\Vert _2\]

so \(\psi \) is rectifiable. For the second part, if \(h>0\) is small, \(u\) is the unit vector in the direction of \(\psi '(t)\) and \(\tau \) comes from Theorem 4.2 we have from Cauchy-Schwarz inequality and Lemma 11.2:

\[|u\cdot \psi '(\tau )|=\frac {1}{h}|u\cdot (\psi (t+h)-\psi (t))| \leq \frac {1}{h}(s(t+h)-s(t)) \leq \frac {1}{h} \int _{[t,t+h]}\Vert \psi '\Vert _2\]

Letting \(h\to 0\) on the left we get \(\Vert \phi '(t)\Vert _2\), and on the right we also get \(\Vert \phi '(t)\Vert _2\) by Theorem 10.3, so the middle, which is \(s'(t)\), must be that (we also treat \(h<0\) similarly). □

The arc length gives a natural parameterization of a curve for integration of a function, but this is unnecessary for line integration over a vector field. A vector field should be thought of as a section of the tangent bundle.

Theorem 11.5. If \(f:\RR ^n\to \RR ^n\) is a vector field continuous near \(\phi ([a,b])\) where \(\phi \) is a \(\cC ^1\) parameterized curve with \(\phi '\neq 0\), then if \(\tau \) is the unit tangent vector and \(\psi \) is arc length parameterization, \(\int _\psi (f\cdot \tau )ds = \int _{[a,b]}(f\circ \phi )\phi '\).

Proof. If \(\psi \) denotes the parameterization by arc length, and \(\phi '\neq 0\), by Proposition 11.4 \(s\) is a \(\cC ^1\) diffeomorphism with \(\psi \). Moreover, by the chain rule, \(\phi ' = (\psi \circ s^{-1})' = (\psi '\circ s^{-1})(s^{-1})'\) so we have by Theorem 10.15 \(\int _{[a,b]} (f\circ \phi )\cdot \phi ' = \int _{[0,L(\phi )]}(f\circ \psi )\cdot \psi ' = \int _\psi (f\cdot \tau )ds\). □

Definition 11.6. A conservative vector field \(\RR ^n\to \RR ^n\) is one which is the gradient of a function \(\RR ^n\to \RR \), which is called its potential.

Theorem 11.7. If \(f:\RR ^n\to \RR ^n\) is a conservative vector field in a connected open set \(U\) with \(\cC ^1\) potential \(h\), and \(\gamma \) is a \(\cC ^1\) parameterized curve with unit tangent vector \(\tau \) from \(a\) to \(b\) with \(\gamma '\neq 0\), then \(\int _\gamma f\cdot \tau ds= h(b)-h(a)\). That \(f\) is \(\cC ^1\) satisfies this property is also sufficient for it to be conservative.

Proof. By Theorem 11.5 we may assume \(\gamma \) is arc length and by the chain rule,

\[\frac {d}{ds}h = (h'\circ \gamma )\gamma ' = (\nabla h\circ \gamma )\cdot \gamma ' = (f\circ \gamma )\cdot \gamma ' = (f\circ \gamma )\cdot \tau \]

so this follows from Theorem 10.3. For the converse, define \(h(a) = \int _{\gamma _a}f\cdot \tau ds\) where \(\gamma _a\) is any path to \(a\) from a fixed point \(b\). Now we can find the partial derivatives of \(h\) at a point \(a\) by integrating along a small path \(a\to a+\delta \) in one component and taking the derivative. If \(u\) is the unit vector in the \(x_i\) direction which \(\delta \) is in, this yields \(h(a+\delta )-h(a) = \int _{\gamma _{a,a+\delta }}f\cdot \tau ds = \int _{[0,\delta ]}(f(a+\delta )-f(a))\cdot u\) which by Theorem 10.3 shows \(f_i = \partial _i(h)\). □