Analysis Theorems

13 Integrals of Differential Forms

If we have an orientation \(y_1\dots y_n\) on \(\RR ^n\), we define \(\int _Efdy_1\wedge \dots \wedge dy_n = \int _Ef\) for \(E\subset \RR ^n\).

Theorem 13.1. If \(\phi :D\to E\) is a \(\cC ^1\) diffeomorphism between two subsets of \(\RR ^n\), and \(\omega \) is a continuous differential \(n\)-form, then \(\int _E\omega = \ee \int _D\phi ^*(\omega )\) where \(\ee \) is the sign of the determinant of \(\phi '\).

Proof. If \(x_1\dots x_n\) is the orientation for \(D\), \(y_1\dots y_n\) is the orientation for \(E\), and \(\omega = fdy_I\) we have \(\int _D\phi ^*(\omega ) = \int _D(f\circ \phi )\det (\phi ')dx_I = \int _D(f\circ \phi )\det (\phi ') = \ee \int _Ef = \ee \int _E\omega \) by the change of variables formula. □

A singular \(n\) cell in \(\RR ^m\) is a map from an oriented cell \(\Delta _n\) to \(\RR ^m\). We will mostly consider \(\cC ^\infty \) singular \(n\) cells. Two \(\cC ^\infty \) singular \(n\) cells are equivalent if there is an orientation preserving \(\cC ^\infty \) diffeomorphism between their domains that commutes with them. If we have a differential form \(\omega \) in an open set containing a singular cell \(\phi :\Delta _n\to \RR ^m\), we define \(\int _\phi \omega = \int _{\Delta _n}\phi ^*(\omega )\). By Theorem 13.1 this only depends on the equivalence class. Indeed given a notion of integration, the pullback may be defined as the unique form satisfying \(\int _D\phi ^*(\omega ) = \int _{\phi (D)}\omega \) (it is dual to the pushforward of a chain).

Indeed if \(\phi \) is a singular \(1\) cell with \(\phi '\neq 0\), and \(\omega = \sum _1^nf_idx_i\), is a \(1\)-form, then if \(f\) is the vector field with components \(f_i\), then \(\int _\phi \omega = \int _{[0,1]}(f\circ \phi )\cdot \phi ' = \int _\phi f\cdot \tau ds\).

Now since we have an inner product structure on \(\RR ^n\), we can identify covector with vectors, as a covector corresponds to a unique vector such that dotting with that vector is that covector. Thus covector fields (\(1\)-forms) may be identified with vector fields. Similarly if \(\phi \) is a singular \(2\) cell with orientation \(t_1,t_2\), and codomain \(\RR ^3\) with orientation \(x_1,x_2,x_3\), we can integrate a \(2\)-form associated with a vector field \(f\) \(\omega = f_1dx_1+f_2dx_2+f_3dx_3\) over this cell, yielding \(\int _\phi \omega = \int _\Delta f(\phi (t))\cdot (\partial _1\phi \times \partial _2\phi )dt_1\wedge dt_2\) where \(\times \) denotes cross product. If \(\nu \) denotes the unit vector in the direction of \(\partial _1\phi \times \partial _2\phi \), then \(\nu \) is the normal unit vector, and we can write \(\partial _1\phi \times \partial _2\phi = \nu \Vert \partial _1\phi \times \partial _2\phi \Vert _2\). Now we can interpret \(\int _\Delta \Vert \partial _1\phi \times \partial _2\phi \Vert _2 = \int _\phi 1dS\) as the surface area, and write \(\int _\phi \omega = \int _\phi f\cdot \nu dS\).

We would like to integrate over chains rather than cells, so we define a singular \(n\)-chain as a finite formal sum of singular \(n\) cells. We can let \(\Delta (x_1,\dots ,x_n)\) be the unit cell in \(\RR ^n\) with that orientation, and then let its boundary be defined as the singular \(n-1\)-chain

\[\partial \Delta (x_1,\dots ,x_n) = \sum _{i=1}^n\sum _{\ee = 0}^1(-1)^{i+\ee }\Delta (x_1,\dots ,\ee ,\dots ,x_n) \]

where \(\ee \) denotes the part of the boundary restricting \(x_i\) to \(\ee \). For a singular \(n\)-chain \(\phi \), we define \(\partial \phi = \phi (\partial \Delta (x_1,\dots ,x_n))\). We then have our notions of boundary and cycle for chains.

Lemma 13.2. A boundary is a cycle.

Proof. It suffices to show this on the unit cell \(\Delta \). We have

\[ \partial \partial \Delta (x_1,\dots ,x_n) = \sum _{i=1}^n\sum _{\ee = 0}^1(-1)^{i+\ee }\partial \Delta (x_1,\dots ,\ee ,\dots ,x_n) \]

\[= \sum _{i=1}^n\sum _{\ee = 0}^1\sum _{i=1}^{n-1}\sum _{\ee '=0}^1(-1)^{i+\ee +j+\ee '}\Delta (x_1,\dots ,\ee ',\dots ,\ee ,\dots ,x_n) \]

Now for a fixed \(\ee ',\ee \) and unordered pair \((a,b)\), the term \(\Delta (x_1,\dots ,\ee ',\dots ,\ee ,\dots ,x_n)\) where \(\ee '\) and \(\ee \) take up the \(a\) and \(b\) slots respectively appear twice in this sum. WLOG, \(a<b\), and so we can have both \(i=a,j=b-1\) or \(i=b,j=a\). Then as \(i+j\) is a different parity for these, the terms cancel out, so this sum is \(0\). □

Theorem 13.3 (Stoke’s Theorem). If \(\cC \) is a \(\cC ^\infty \) singular \(n\)-chain and \(\omega \) is a \(n-1\)-form, then \(\int _\cC d\omega = \int _{\partial \cC } \omega \).

Proof. Since pullback commutes with exterior derivative and integration over chains is linear, it suffices to prove this on the unit cell \(\Delta (x_1,\dots ,x_n)\) for a \(n-1\)-form \(\omega = f dX_r\) where \(dX_r\) indicates \(dx_r\) is missing from \(dX = dx_1\wedge \dots \wedge dx_n\). This way \(d\omega = (-1)^{r-1}\partial _rfdX\). Similarly let \(\Delta \) denote the cell, \(\Delta _{r,e}\) denote the boundary on the \(r^{th}\) side with \(\ee \), \(\Delta _r\) denote the cell \(\Delta \) with the \(r\) dimension missing. Then from Theorem 10.11 and Theorem 10.3 we get

\[ \int _\Delta d\omega = (-1)^{r-1}\int _\Delta \partial _rfdX= (-1)^{r-1}\int _{\Delta _r}\int _{\Delta (x_r)}\partial _rf\]

\[= (-1)^{r+1}\int _{\Delta _{r,1}}f(x_1,\dots ,1,\dots ,x_n)+(-1)^{r}\int _{\Delta _{r,0}}f(x_1,\dots ,0,\dots ,x_n) = \int _{\partial \Delta }fdX_r \]

Where \(\int _{\partial \Delta }fdX_r\) vanishes on all other parts of the boundary as on \(\Delta _{i,\ee }\) with \(i\neq r\) the \(x_i\) part is constant, so pulling back yields the form \(0\). □

Indeed nothing deep is going on here, one may define the exterior derivative as the map so that Theorem 13.3 holds. Note that there is a pairing between chains and forms (homology and cohomology), and thus Theorem 13.3 says simply that \(d\) is adjoint to \(\partial \) for this pairing.

Corollary 13.4. If \(H_1(U,\ZZ ) = 0\) for an open set \(U \subset \RR ^n\), then any closed \(\cC ^\infty \) differential \(1\)-form is exact.

Proof. By Theorem 11.7 it suffices to show line integrals depend only on the start and end, but this follows from the hypothesis and Theorem 13.3. □

Corollary 13.5. If \(D\) is the image of a simple (orientation preserving diffeomorphism) singular \(\cC ^\infty \) \(n\)-chain \(\cC \) and \(\omega = \sum _if_idX_i\) is an \(n-1\)-form, then

\[\int _D\sum _i(-1)^{i-1}\partial _if_i = \int _{\partial \cC }\omega \]

Proof. This follows from Theorem 13.3 and Theorem 13.1. □

Corollary 13.6 (Green’s Theorem). If \(D\) is the image of a simple \(\cC ^\infty \) singular \(2\)-chain \(\cC \), then

\[\int _{\partial \cC }f_1dx_1+f_2dx_2 = \int _D\partial _1f_2-\partial _2f_1\]

Proof. This is a special case of Theorem 13.5. □

Green’s theorem can be used to calculate area by integrating on the boundary the form \(x_1dx_2\) or \(-x_2dx_1\) for example. For the case of \(\partial \cC \) being a curve \(\gamma \), we can get \(\int _\gamma f\cdot \nu ds = \int _D \partial _1f_1+\partial _2f_2\) where \(\nu \) is the outward pointing normal vector defined as the unit vector in the direction of \(\begin {pmatrix} \partial _1\gamma _2\\ -\partial _1\gamma _1 \end {pmatrix}\). Then the left side is interpreted as work done by a vector field pushing a particle, and the right is interpreted as total flow of a substance moving across the curve with velocity given by the vector field.

Corollary 13.7 (Gauss’s Theorem). If \(D\) is the image of a simple \(\cC ^\infty \) singular \(3\)-chain \(\cC \), then

\[\int _{\partial \cC }f_1dx_2\wedge dx_3+f_2dx_3\wedge dx_1 + f_3dx_1\wedge dx_2 = \int _D\sum _{i=1}^3\partial _if_i\]

Proof. This is a special case of Theorem 13.5. □

This can also be used to calculate volume, and can similarly be written as \(\int _\gamma f\cdot \nu dS = \int _D \nabla \cdot f\), which can be interpreted as flow and amount of substance created.

Corollary 13.8 (Kelvin-Stokes Theorem). If \(\cC \) is a \(\cC ^1\) singular \(2\)-chain, then

\[ \int _{\partial \cC }f\cdot \tau ds = \int _{\cC }(\nabla \times f)\cdot \nu dS \]

Proof. This is a special case of Theorem 13.3. □

As an interpretation of the curl, let \(f\) be a vector field in \(\RR ^3\), and we can integrate around a tiny circle \(\gamma _\ee \) around a point \(a\) which bounds a disk \(D_\ee \) which has its unit normal vector \(\nu \) pointing in the direction of \(\nabla \times f(a)\). Then by Kelvin-Stokes, \(\frac {1}{\pi \ee ^2}\int _{\gamma _\ee }f\cdot \tau ds = \frac {\int _{D_\ee }(\nabla \times f)\cdot \nu dS}{\int _{D_\ee }dS}\). But as the curl is continuous, we get

\[ \lim _{\ee \to 0}\frac {1}{\pi \ee ^2}\int _{\gamma _\ee }f\cdot \tau ds = \Vert \nabla \times f(a)\Vert _2 \]