17.2 Integration of n-forms

中文

There is no natural way to define the integral of a scalar function $f:M\to \mathbb{R}$ over a compact region $D.$ . For, if $D\subset U$ where $U$ is the domain of a coordinate chart $(U,\varphi ;x^i)$ the multiple integral

$${\int }_{D}f={\int }_{\varphi (D)}f\circ {\varphi }^{-1}\left(x^1,\dots ,x^n\right)d{x}^1\dots d{x}^n$$

will have a different expression in a second chart $(V,\psi ;y^i)$ such tha $D\subset U\cap V$

$$\begin{array}{c}{\int }_{\psi (D)}f\circ {\psi }^{-1}(\mathbf{y})\mathrm{d}{y}^1\dots \mathrm{d}{y}^n={\int }_{\varphi (D)}f\circ {\varphi }^{-1}(\mathbf{x})\left|\det \right[\frac{\partial y^i}{\partial x^j}\left]\right|\mathrm{d}{x}^1\dots \mathrm{d}{x}^n\\ \ne {\int }_{\varphi (D)}f\circ {\varphi }^{-1}(\mathbf{x})\mathrm{d}{x}^1\dots \mathrm{d}{x}^n.\end{array}$$

As seen above, $n$-forms absorb a Jacobian determinant in their coordinate transformation, and it turns out that these are the ideal objects for integration. However, it is necessary that the manifold be orientable so that the absolute value of the Jacobian occurring in the integral transformation law can be omitted.

Let $(M,\omega )$ be an $n$-dimensional oriented differentiable manifold, and $(U,\varphi ;x^i)$ a positively oriented chart. On $U$ we can write $\omega =g(\mathbf{x})\mathrm{d}{x}^1\wedge \cdot \cdot \cdot \wedge \mathrm{d}{x}^n$ where $g>0$ . The support of an $n$-form $\alpha$ is defined as the closure of the set on which $\alpha \ne 0$ ,

$$\mathrm{supp}\alpha =\overline{\{p\in M\mid {\alpha }_p\ne 0\}}.$$

If$\alpha$ has compact support contained in $U_{:}$ , and $\alpha =f\mathrm{d}{x}^1\wedge \mathrm{d}{x}^2\wedge \cdot \cdot \cdot \wedge \mathrm{d}{x}^n$ on $U$ we define its integral over $M$ to be

$${\int }_M\alpha ={\int }_{\varphi (U)}f\left(x^1,\dots ,x^n\right)\mathrm{d}{x}^1\dots \mathrm{d}{x}^n={\int }_{\varphi (\mathrm{supp}\alpha )}f\mathrm{d}{x}^1\dots \mathrm{d}{x}^n$$

where $f(x^1,\dots ,x^n)$ is commonly written in place of $\cdot \hat{f}=f\circ {\varphi }^{-1}.\mathrm{If}(V,\psi ;x^{\prime i})$ is a second positively oriented chart also containing the support of $\alpha$ and $f^{\prime }(x^{\prime 1},\dots ,x^{\prime n})\equiv f\circ {\psi }^{-1}$ we have by the change of variable formula in multiple integration

$$\begin{array}{l}{\int }_{\psi (V)}{f}^{\prime }(x^{\prime 1},\dots ,x^{\prime n})\mathrm{d}{x}^{\prime 1}\dots \mathrm{d}{x}^{\prime n}={\int }_{\psi (\mathrm{supp}\alpha )}{f}^{\prime }({\mathbf{x}}^{\prime })\mathrm{d}{x}^{\prime 1}\dots \mathrm{d}{x}^{\prime n}\\ ={\int }_{\varphi (\mathrm{supp}\alpha )}{f}^{\prime }({\mathbf{x}}^{\prime })\left|\det \right[\frac{\partial x^{\prime i}}{\partial x^j}\left]\right|\mathrm{d}{x}^1\dots \mathrm{d}{x}^n\\ ={\int }_{\varphi (U)}f(\mathbf{x})\mathrm{d}{x}^1\dots \mathrm{d}{x}^n\end{array}$$

since

$$\alpha =f\mathrm{d}{x}^1\wedge \cdots \wedge \mathrm{d}{x}^n=f^{\prime }\mathrm{d}{x}^{\prime 1}\wedge \cdots {\wedge }^{\prime n}\text{ where }f=f^{\prime }\det \left[\frac{\partial x^{\prime i}}{\partial x^j}\right]$$

and the Jacobian determinant is everywhere positive. The definition of the integral is there fore independent of the coordinate chart, provided the support lies within the domain of the chart.

For an arbitrary $n$-form $\alpha$ with compact support and atlas $(U_a,{\varphi }_a)$ , assumed to be locally finite, let $g_a$ be a partition of unity subordinate to the open covering $\{U_a\}$ . Evidently

$$\alpha =\sum _a{g}_a\alpha$$

and each of the summands $g_a\alpha$ has compact support contained in $U_a$ . We define the integra of $\alpha$ over $M$ to be

$${\int }_M\alpha =\sum _a{\int }_M{g}_a\alpha .\text{\hspace{1em}(17.1)}$$

Exercise

Prove that ${\int }_M$ is a linear operator, $\begin{array}{r}{\int }_M\alpha +c\beta ={\int }_M\alpha +c{\int }_M\beta .\end{array}$

If $\alpha$ is a differential $k$-form with compact support on $M$ and $\phi :N\to M$ is a regula embedding of a $k$-dimensional manifold $N$ in $M$ (see Section 15.4), define the integral of $\alpha$ on $\phi (N)$ to be

$${\int }_{\phi (N)}\alpha ={\int }_N{\phi }^{\ast }\alpha .$$

The right-hand side is well-defined since ${\phi }^{\ast }\alpha$ is a differential $k$-form on $N$ with compact support, since $\phi :M\to N$ is a homeomorphism from $N$ to $\phi (N)$ in the relative topology with respect to $M$.

Problems

Problem 17.3

Show that the definition of the integral of an $n$-form over a manifold $M$ given in Eq. (17.1) is independent of the choice of partition of unity subordinate to $\{U_a\}$