Integration of Forms and the General Stokes Theorem — Examples

Independence of atlas, partition and refinement: The compact-support integral on an oriented manifold is independent of the chart cover, coordinate maps, subordinate partition, and refinement. If $U\subset M$ is open and contains $\mathrm{supp}\omega$, with its restricted orientation, then ${\int }_U\omega {\mathrm{\mid }}_U={\int }_M\omega$.

Chart integral with its orientation sign: Let $M^n$ be oriented and $\omega$ a smooth top form with compact support contained in a connected chart $(U,\varphi )$. For $n\ge 1$ write $({\varphi }^{-1}{)}^{\ast }\omega =fd{x}^1\wedge \cdots \wedge d{x}^n\mathrm{.}$ Let ${\sigma }_{\varphi }\in \{1,-1\}$ be the sign of its coordinate frame relative to the chosen orientation. Define the chart integral by $I_{\varphi }(\omega )={\sigma }_{\varphi }{\int }_{{\mathbb{R}}^n}\tilde{f}(x)dx\mathrm{.}$ Here $\tilde{f}$ is the Riemann-integrable zero extension, including across a genuine half-space face, as in lem-chart-supported-coefficients-have-well-defined-riemann-integrable-half-space-extensions. For $n=0$, a connected chart is a point $p$, and set $I_p(\omega )=\epsilon (p)\omega (p)$ using its determinant-line sign. Empty support gives zero. Negative charts are allowed: the upper-half-line chart $u=b-t$ at the right endpoint of an increasing interval has sign $-1$.

Change of variables on oriented manifolds: Let $F:M\to N$ be a diffeomorphism of oriented smooth $n$-manifolds and $\omega \in {\mathrm{\Omega }}_c^n(N)$. If $F$ preserves orientation everywhere, ${\int }_M{F}^{\ast }\omega ={\int }_N\omega$; if it reverses orientation everywhere, ${\int }_M{F}^{\ast }\omega =-{\int }_N\omega$. If the sign varies between components, apply the appropriate signed equality on each component and add.

Orientation-free density integration and its properties: Compactly supported smooth density integration is independent of charts and partition, linear, local, nonnegative on nonnegative densities and strictly positive for a nonzero nonnegative density. It is invariant under every diffeomorphism, without choosing an orientation. The finite-parametrization formula holds under the hypotheses of prop-integration-of-top-forms-by-finite-parametrizations, with orientation preservation omitted and absolute Jacobians used.

Orientation-free density integration and its properties: Compactly supported smooth density integration is independent of charts and partition, linear, local, nonnegative on nonnegative densities and strictly positive for a nonzero nonnegative density. It is invariant under every diffeomorphism, without choosing an orientation. The finite-parametrization formula holds under the hypotheses of prop-integration-of-top-forms-by-finite-parametrizations, with orientation preservation omitted and absolute Jacobians used.

Pullback of densities by local diffeomorphisms: For a local diffeomorphism $F:M^n\to N^n$, pullback of smooth densities is smooth and in coordinates satisfies $F^{\ast }(f\mathrm{\mid }dy\mathrm{\mid })=(f\circ F)\mathrm{\mid }\det DF\mathrm{\mid }\mathrm{\mid }dx\mathrm{\mid }\mathrm{.}$ It is real-linear, obeys $F^{\ast }(a\delta )=(a\circ F)F^{\ast }\delta$ for smooth functions $a$ on $N$, and $(F\circ G{)}^{\ast }=G^{\ast }{F}^{\ast }$ for composable local diffeomorphisms.

Stokes agrees with the fundamental theorem of calculus: For $a<b$, orient $[a,b]$ increasingly. Every smooth $f$ on this interval satisfies ${\int }_{[a,b]}df=f(b)-f(a),$ where the boundary point signs are $-1$ at $a$ and $+1$ at $b$. This agrees with the Riemann fundamental theorem of calculus.

Compact-support Stokes on the upper half-space: Give $H^n=\{x_n\ge 0\}$ the standard orientation, $n\ge 1$, and its face the outward-normal-first orientation. If $\eta \in {\mathrm{\Omega }}_c^{n-1}(H^n)$ and $j:\mathrm{\partial }{H}^n\hookrightarrow H^n$, then ${\int }_{H^n}d\eta ={\int }_{\mathrm{\partial }{H}^n}{j}^{\ast }\eta \mathrm{.}$ With $\eta ={\sum }_i{a}_{i}d{x}^1\wedge \cdots \wedge \widehat{d{x}^i}\wedge \cdots \wedge d{x}^n$, both sides are $(-1{)}^n{\int }_{{\mathbb{R}}^{n-1}}{a}_n(x^{\mathrm{\prime }},0)d{x}^{\mathrm{\prime }}$ for $n>1$, and $-a_1(0)$ for $n=1$.

General Stokes agrees with both planar Green formulas: For a compact smooth planar region $D$ oriented by $dx\wedge dy$ and smooth $P,Q$ on a neighborhood, general Stokes gives ${\int }_{\mathrm{\partial }D}(Pdx+Qdy)={\int }_D(Q_x-P_y)dxdy,$ ${\int }_{\mathrm{\partial }D}(Pdy-Qdx)={\int }_D(P_x+Q_y)dxdy\mathrm{.}$ When $D$ also has the supplied finite elementary Green decomposition required by the classical results, these are exactly their circulation and outward-flux formulas. Outer boundary curves run counterclockwise and holes clockwise.

Computing form integrals by finite parametrizations: Let $n\ge 1$, let $M^n$ be oriented, and let $\omega \in {\mathrm{\Omega }}_c^n(M)$. For $1\le i\le m$ let $D_i\subset {\mathbb{R}}^n$ be bounded open Jordan domains and $F_i:{\bar{D}}_i\to M$ continuous and smooth up to the boundary in target coordinates: near each parameter point, a target coordinate representative extends smoothly to a Euclidean neighborhood. Suppose $F_i{\mathrm{\mid }}_{D_i}$ is an orientation-preserving diffeomorphism onto an open $W_i\subset M$, the $W_i$ are pairwise disjoint, and $\mathrm{supp}\omega \subset {\bigcup }_i{\bar{W}}_i$. Then ${\int }_M\omega ={\sum }_{i=1}^m{\int }_{D_i}{F}_i^{\ast }\omega \mathrm{.}$ An empty family is allowed when the support is empty. No nonsingularity of $D{F}_i$ on $\mathrm{\partial }{D}_i$, and no $M$-valued extension across a genuine target boundary, is assumed.

Agreement of general and classical surface Stokes: Let $S\subset {\mathbb{R}}^3$ be a compact oriented smooth embedded surface with boundary, and let $F$ be smooth on an open neighborhood of $S$. Set $\alpha =F_{x}dx+F_{y}dy+F_{z}dz$ and $\mu =dx\wedge dy\wedge dz$. Then $d\alpha ={\iota }_{\mathrm{curl}F}\mu ,{\int }_{\mathrm{\partial }S}\alpha ={\int }_S{\iota }_{\mathrm{curl}F}\mu \mathrm{.}$ On an oriented parametrization $r(u,v)$ the latter integrand is $(\mathrm{curl}F)(r)\cdot (r_u\times r_v)du\wedge dv$; on a boundary curve it is $F(r)\cdot r^{\mathrm{\prime }}dt$. On the common smooth patch scope this is the published classical Stokes theorem, using the standard Euclidean metric identification.

Computing form integrals by finite parametrizations: Let $n\ge 1$, let $M^n$ be oriented, and let $\omega \in {\mathrm{\Omega }}_c^n(M)$. For $1\le i\le m$ let $D_i\subset {\mathbb{R}}^n$ be bounded open Jordan domains and $F_i:{\bar{D}}_i\to M$ continuous and smooth up to the boundary in target coordinates: near each parameter point, a target coordinate representative extends smoothly to a Euclidean neighborhood. Suppose $F_i{\mathrm{\mid }}_{D_i}$ is an orientation-preserving diffeomorphism onto an open $W_i\subset M$, the $W_i$ are pairwise disjoint, and $\mathrm{supp}\omega \subset {\bigcup }_i{\bar{W}}_i$. Then ${\int }_M\omega ={\sum }_{i=1}^m{\int }_{D_i}{F}_i^{\ast }\omega \mathrm{.}$ An empty family is allowed when the support is empty. No nonsingularity of $D{F}_i$ on $\mathrm{\partial }{D}_i$, and no $M$-valued extension across a genuine target boundary, is assumed.

Agreement with classical Gauss flux in Euclidean space: For $\mu =dx\wedge dy\wedge dz$ and a smooth Euclidean field $F$, the volume-form divergence is ${\mathrm{\partial }}_x{F}_x+{\mathrm{\partial }}_y{F}_y+{\mathrm{\partial }}_z{F}_z$. For a surface parametrization $r(u,v)$, $r^{\ast }({\iota }_F\mu )=(F(r)\cdot (r_u\times r_v))du\wedge dv\mathrm{.}$ Consequently the volume-form divergence theorem agrees with the classical Gauss flux theorem on compact smooth regions that also admit the supplied elementary-solid presentation required by that classical theorem.

Computing form integrals by finite parametrizations: Let $n\ge 1$, let $M^n$ be oriented, and let $\omega \in {\mathrm{\Omega }}_c^n(M)$. For $1\le i\le m$ let $D_i\subset {\mathbb{R}}^n$ be bounded open Jordan domains and $F_i:{\bar{D}}_i\to M$ continuous and smooth up to the boundary in target coordinates: near each parameter point, a target coordinate representative extends smoothly to a Euclidean neighborhood. Suppose $F_i{\mathrm{\mid }}_{D_i}$ is an orientation-preserving diffeomorphism onto an open $W_i\subset M$, the $W_i$ are pairwise disjoint, and $\mathrm{supp}\omega \subset {\bigcup }_i{\bar{W}}_i$. Then ${\int }_M\omega ={\sum }_{i=1}^m{\int }_{D_i}{F}_i^{\ast }\omega \mathrm{.}$ An empty family is allowed when the support is empty. No nonsingularity of $D{F}_i$ on $\mathrm{\partial }{D}_i$, and no $M$-valued extension across a genuine target boundary, is assumed.

A nonzero period obstructs exactness and bounding: Let $S\hookrightarrow M$ be an oriented compact boundaryless embedded $k$-submanifold, $k\ge 1$, and let $\omega$ be a closed smooth $k$-form on $M$. If ${\int }_S\omega \mathrm{\ne }0$, then $\omega$ is not exact on $M$, and $S$ cannot be the induced oriented boundary of a compact embedded $(k+1)$-submanifold of $M$.

Computing form integrals by finite parametrizations: Let $n\ge 1$, let $M^n$ be oriented, and let $\omega \in {\mathrm{\Omega }}_c^n(M)$. For $1\le i\le m$ let $D_i\subset {\mathbb{R}}^n$ be bounded open Jordan domains and $F_i:{\bar{D}}_i\to M$ continuous and smooth up to the boundary in target coordinates: near each parameter point, a target coordinate representative extends smoothly to a Euclidean neighborhood. Suppose $F_i{\mathrm{\mid }}_{D_i}$ is an orientation-preserving diffeomorphism onto an open $W_i\subset M$, the $W_i$ are pairwise disjoint, and $\mathrm{supp}\omega \subset {\bigcup }_i{\bar{W}}_i$. Then ${\int }_M\omega ={\sum }_{i=1}^m{\int }_{D_i}{F}_i^{\ast }\omega \mathrm{.}$ An empty family is allowed when the support is empty. No nonsingularity of $D{F}_i$ on $\mathrm{\partial }{D}_i$, and no $M$-valued extension across a genuine target boundary, is assumed.

The general Stokes theorem: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M$ be an oriented smooth $n$-manifold with boundary, $n\ge 1$, and let $\eta \in {\mathrm{\Omega }}_c^{n-1}(M)$. With $j:\mathrm{\partial }M\hookrightarrow M$ and the outward-normal-first orientation, ${\int }_{M}d\eta ={\int }_{\mathrm{\partial }M}{j}^{\ast }\eta \mathrm{.}$ An empty boundary contributes zero; in dimension one its integral is a finite signed sum of point values.

Computing form integrals by finite parametrizations: Let $n\ge 1$, let $M^n$ be oriented, and let $\omega \in {\mathrm{\Omega }}_c^n(M)$. For $1\le i\le m$ let $D_i\subset {\mathbb{R}}^n$ be bounded open Jordan domains and $F_i:{\bar{D}}_i\to M$ continuous and smooth up to the boundary in target coordinates: near each parameter point, a target coordinate representative extends smoothly to a Euclidean neighborhood. Suppose $F_i{\mathrm{\mid }}_{D_i}$ is an orientation-preserving diffeomorphism onto an open $W_i\subset M$, the $W_i$ are pairwise disjoint, and $\mathrm{supp}\omega \subset {\bigcup }_i{\bar{W}}_i$. Then ${\int }_M\omega ={\sum }_{i=1}^m{\int }_{D_i}{F}_i^{\ast }\omega \mathrm{.}$ An empty family is allowed when the support is empty. No nonsingularity of $D{F}_i$ on $\mathrm{\partial }{D}_i$, and no $M$-valued extension across a genuine target boundary, is assumed.

Chart integral with its orientation sign: Let $M^n$ be oriented and $\omega$ a smooth top form with compact support contained in a connected chart $(U,\varphi )$. For $n\ge 1$ write $({\varphi }^{-1}{)}^{\ast }\omega =fd{x}^1\wedge \cdots \wedge d{x}^n\mathrm{.}$ Let ${\sigma }_{\varphi }\in \{1,-1\}$ be the sign of its coordinate frame relative to the chosen orientation. Define the chart integral by $I_{\varphi }(\omega )={\sigma }_{\varphi }{\int }_{{\mathbb{R}}^n}\tilde{f}(x)dx\mathrm{.}$ Here $\tilde{f}$ is the Riemann-integrable zero extension, including across a genuine half-space face, as in lem-chart-supported-coefficients-have-well-defined-riemann-integrable-half-space-extensions. For $n=0$, a connected chart is a point $p$, and set $I_p(\omega )=\epsilon (p)\omega (p)$ using its determinant-line sign. Empty support gives zero. Negative charts are allowed: the upper-half-line chart $u=b-t$ at the right endpoint of an increasing interval has sign $-1$.

Compact-support Stokes on the upper half-space: Give $H^n=\{x_n\ge 0\}$ the standard orientation, $n\ge 1$, and its face the outward-normal-first orientation. If $\eta \in {\mathrm{\Omega }}_c^{n-1}(H^n)$ and $j:\mathrm{\partial }{H}^n\hookrightarrow H^n$, then ${\int }_{H^n}d\eta ={\int }_{\mathrm{\partial }{H}^n}{j}^{\ast }\eta \mathrm{.}$ With $\eta ={\sum }_i{a}_{i}d{x}^1\wedge \cdots \wedge \widehat{d{x}^i}\wedge \cdots \wedge d{x}^n$, both sides are $(-1{)}^n{\int }_{{\mathbb{R}}^{n-1}}{a}_n(x^{\mathrm{\prime }},0)d{x}^{\mathrm{\prime }}$ for $n>1$, and $-a_1(0)$ for $n=1$.

Change of variables on oriented manifolds: Let $F:M\to N$ be a diffeomorphism of oriented smooth $n$-manifolds and $\omega \in {\mathrm{\Omega }}_c^n(N)$. If $F$ preserves orientation everywhere, ${\int }_M{F}^{\ast }\omega ={\int }_N\omega$; if it reverses orientation everywhere, ${\int }_M{F}^{\ast }\omega =-{\int }_N\omega$. If the sign varies between components, apply the appropriate signed equality on each component and add.

Computing form integrals by finite parametrizations: Let $n\ge 1$, let $M^n$ be oriented, and let $\omega \in {\mathrm{\Omega }}_c^n(M)$. For $1\le i\le m$ let $D_i\subset {\mathbb{R}}^n$ be bounded open Jordan domains and $F_i:{\bar{D}}_i\to M$ continuous and smooth up to the boundary in target coordinates: near each parameter point, a target coordinate representative extends smoothly to a Euclidean neighborhood. Suppose $F_i{\mathrm{\mid }}_{D_i}$ is an orientation-preserving diffeomorphism onto an open $W_i\subset M$, the $W_i$ are pairwise disjoint, and $\mathrm{supp}\omega \subset {\bigcup }_i{\bar{W}}_i$. Then ${\int }_M\omega ={\sum }_{i=1}^m{\int }_{D_i}{F}_i^{\ast }\omega \mathrm{.}$ An empty family is allowed when the support is empty. No nonsingularity of $D{F}_i$ on $\mathrm{\partial }{D}_i$, and no $M$-valued extension across a genuine target boundary, is assumed.