Subjects: Differential Geometry
Links: Line Integral over a Vector Field, Vector Surface Integral, Riemann Integral in Rn, Orientations of Smooth Manifolds, Change of Variable Theorem in Rn, Smooth Partitions of Unity for Manifolds, Differential Forms on Smooth Manifolds

Recall that a domain of integration in ${\mathbb{R}}^n$ is a bounded subset whose boundary has measure zero.

Def: Let $D\subseteq \mathbb{R}$ be a domain of integration, and let $\omega$ be a continuous $n$-form on $\bar{D}$. Any such form can be written as $\omega =fd{x}^1\wedge \cdots \wedge d{x}^n$ for some continuous function $f:\bar{D}\to \mathbb{R}$. We define the integral of $\omega$ over $D$ to be $$\int_D \omega := \int_D f.$$This can be written more suggestively as $$\int_D f ; dx^1 \wedge \dots \wedge dx^n := \int_D f; dx^1\cdots dx^n.$$
Somewhat more generally, let $U$ be an open subset of ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$, and suppose $\omega$ is compactly supported $n$-form on $U$. We define $$\int_U \omega = \int_D \omega,$$where $D\subseteq {\mathbb{R}}^n$ or ${\mathbb{H}}^n$ is any domain of integration containing $\text{supp }\omega$, and $\omega$ is extended to be zero on the complement of its support.

Prop: Suppose $D$ and $E$ are open domain of integration in ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$, and $G:\bar{D}\to \bar{E}$ is a smooth map that restricts to an orientation-preserving or orientation-reversing diffeomorphism from $D$ to $E$. If $\omega$ is an $n$-form on $\bar{E}$, then $$\int_D G^* \omega = \begin{dcases}\int_E \omega & \text{if }G \text{ is orientation-preserving,} \ -\int_E \omega & \text{if }G\text{ is orientation-reversing.}\end{dcases} $$

Lemma: Suppose $U$ is an open subset of ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$, and $K$ is a compact subset of $U$. Then there is a domain of integration $K\subseteq D\subseteq \bar{D}\subseteq U$.

Prop: Suppose $U$, $V$ are open subsets of ${\mathbb{R}}^n$ or ${\mathbb{H}}^n$, and $G:U\to V$ is an orientation-preserving or orientation-reversing diffeomorphism. If $\omega$ is a compactly supported $n$-form on $V$, then $$\int_V \omega = \pm \int_U G^*\omega,$$with the positive sign if $G$ is orientation-preserving, and the negative sign otherwise.

Integration on Manifolds

Def: Let $M$ be an oriented smooth $n$-manifold with or without boundary, and let $\omega$ be an $n$-form on $M$. Suppose first that $\omega$ is compactly supported in a domain of a single smooth chart $(U,\phi )$ that is either positively or negatively oriented. We define the integral of $\omega$ over $M$ to be $$\int_M \omega =: \pm \int_{\varphi[U]} \left(\varphi^{-1}\right)^* \omega, $$with the positive sign for positively oriented chart, and the negative sign otherwise.

Prop: With $\omega$ as above, ${\int }_M\omega$ does not depend on the choice of smooth chart whose domain contains $\text{supp }\omega$.

To integrate over an entire manifold, we combine this definition with a partition of unity.

Def: Suppose $M$ is an oriented smooth $n$-manifold with or without boundary, and $\omega$ is a compactly supported $n$-form on $M$. Let $\{U_i\}$ be a finite open cover of $\text{supp }\omega$ by domains of positively or negatively oriented smooth charts and let $\{{\psi }_i\}$ be the subordinates smooth partition of unity. Define the integral of $\omega$ over $M$ to be $$\int_M \omega := \sum_{i} \int_M \psi_i \omega. $$
Prop: The definition of ${\int }_M\omega$ given above doesn't depend on the choice of open cover or partition of unity.

Just we have for orientations, we have a special definition in the zero-dimensional case.

Def: The integral of a compactly supported $0$-form $f$ over an oriented $0$-manifold $M$ is to be defined to be the sum $$\int_M f := \sum_{p\in M}\pm f(p), $$where we take the positive sign where the orientation is positive sign and the negative sign at points where it is negative.

Def: If $S\subseteq M$ is an oriented immersed $k$-dimensional submanifold with or without boundary, and $\omega$ is a $k$-form on $M$ whose restriction to $S$ is compactly supported we interpret ${\int }_S\omega$ to mean ${\int }_S{\iota }_S^{\ast }\omega$, where ${\iota }_S:S\hookrightarrow M$ is inclusion. In particular, if $M$ is a compact, oriented, smooth $n$-manifold with boundary and $\omega$ is an $(n-1)$-form on $M$, we can interpret ${\int }_{\partial M}\omega$ unambiguously as the integral of ${\iota }_{\partial M}^{\ast }\omega$ over $\partial M$, where $\partial M$ is always understood to have the Stoke's orientation.

Properties of Integral of Forms: Suppose $M$ and $N$ are nonempty oriented smooth $n$-manifolds with or without boundary, and $\omega$, $\eta$ are compactly supported $n$-forms on $M$.

• Linearity: If $a,b\in \mathbb{R}$, then $$\int_M a\omega + b\eta = a \int_M \omega + b\int_M \eta. $$
• Orientation Reversal:: If $-M$ denotes $M$ with the opposite orientation, then $$\int_{-M}\omega = -\int_M \omega. $$
• Positivity: If $\omega$ is a positively oriented orientation form, the ${\int }_M\omega >0$.
• Diffeomorphism Invariance: If $F:N\to M$ is an orientation-preserving or orientation-reversing diffeomorphism, then $$\int_M \omega = \begin{dcases}\int_N F^\omega & \text{if }F \text{ is orientation-preserving,} \ -\int_N F^ \omega & \text{if }F\text{ is orientation-reversing.}\end{dcases} $$
  Cor: Suppose $E$ and $M$ are smooth $n$-manifolds with or without boundary, and $\pi :E\to M$ is a smooth $k$-sheeted covering map or generalised covering map. If $E$ and $M$ are oriented and $\pi$ is orientation-preserving $$\int_E \pi^*\omega = k \int_M \omega $$for any compactly supported $n$-form $\omega$ on $M$.

Integrations Over Parametrizations: Let $M$ be an oriented smooth $n$-manifold with or without boundary, and let $\omega$ be a compactly supported $n$-form on $M$. Suppose $D_1,\dots ,D_k$ are open domains of integration in ${\mathbb{R}}^n$, and for $i=1,\dots ,k$, we are given smooth maps $F_i:{\bar{D}}_i\to M$ satisfying:

• $F_i$ restricts to an orientation-preserving diffeomorphism from $D_i$ onto an open subset $W_i\subseteq M$;
• $W_i\cap W_j=\varnothing$ when $i\ne j$;
• $\text{supp }\omega \subseteq \bigcup _{i=1}^k{\bar{W}}_i$.
  Then $$\int_M \omega = \sum_{i = 1}^k \int_{D_i} F^*_i \omega. $$

Integration on Lie Groups

Def: Let $G$ be a Lie group. A covariant tensor field $A$ on $G$ is said to be left-invariant if $L_g^{\ast }A=A$ for all $g\in G$.

Prop: Let $G$ be a compact Lie group endowed with a left-invariant orientation. Then $G$ has a unique positively oriented left-invariant $n$-form ${\omega }_G$ with the property that $$\int_G \omega_G = 1.$$
The orientation form whose existence is asserted in this proposition is called the Haar volume form on $G$. Similarly, the map $f\mapsto {\int }_{G}f{\omega }_G$ is called the Haar integral. This I suspect is just a special case of a Haar Measures

We see that every Lie group has a left-invariant orientation form that is uniquely determined up to constant multiple. It is only in the compact case that we can use the volume normalisation to single out a unique one.