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外微分形式的积分
定义
我们熟知有积分变量代换公式: 设$f\in C_c(\mathbb{R}^m),$
$h:\mathbb{R}^m\approx \mathbb{R}^m$为微分同胚, 则
$$
\int_{\mathbb{R}^m}f(x)dx^1\cdots dx^m=\int_{\mathbb{R}^m}(f\circ h)(y)|\det Dh|dy^1\cdots dy^m.
$$
若$h:\mathbb{R}^m\rightarrow \mathbb{R}^m$满足$\det Dh(y)>0,$
则处处有$\det Dh>0.$ 称它是保定向的微分同胚.
由上节课内容,
$\Omega={}dx^{1}\wedge\cdots\wedge{}dx^{ {}m}\in \mathcal{A}^m(\mathbb{R}^m)$给了$\mathbb{R}^m$一个自然的定向.
现$(\mathbb{R}^m,\mathrm{id};x),(\mathbb{R}^m,h^{-1};y)$为两个不同坐标系,
令$\omega=f(x)dx^1\wedge\cdots\wedge dx^m$为具紧支集连续$m$次外微分形式,
那么由于
$$
dx^1\wedge\cdots\wedge dx^m=\det(Dh)(y){}dy^{1}\wedge\cdots\wedge{}dy^{ {}m},
$$
我们知道$\omega=f\circ h(y)\det(Dh){}dy^{1}\wedge\cdots\wedge{}dy^{ {}m}$为另一表示.
上述公式提示:
$$
\int_{\mathbb{R}^m}\omega:=\int_{\mathbb{R}^m}f(x)dx^1\cdots dx^m.
$$
设$(M^m,\Omega)$是定向$C^\infty$-流形,
$\omega$为$M$上具紧支集$m$次外微分形式,
即$\operatorname{supp}\omega=\overline{\left\{q\mid\omega(q)\neq 0\right\}} \Subset M.$
下面来定义$\omega$在$M$上的积分.
当$\operatorname{supp}\omega\subset (U,\varphi;x^i),$
与定向$\Omega$相符的坐标系时, $\omega=f dx^1\wedge\cdots\wedge dx^m,$
$f\in C_c(U).$ 由前面的叙述, 定义
$$
\int_M \omega:=\int_U \omega=\int_{\varphi(U)}(f\circ\varphi^{-1})(x)dx^1\cdots dx^m.
$$
一般地, 取$\{(U_i,\varphi_i)\}$为与定向相符的坐标图册,
以及从属于它的单位分解. 那么$\omega=\sum_i f_i\omega,$
$\operatorname{supp}(f_i\omega)\subset U_i.$ 由$\omega$具紧支集,
$\{\operatorname{supp}f_i\}$局部有限, 和式总有意义. 于是我们定义
$$
\int_M \omega:=\sum_i \int_M f_i\omega,
$$
其中$\int_M f_i\omega$是已定义好的.
良定性
我们需要验证该定义的良定性: 是否与坐标系/图册以及单位分解选取无关?
若$\operatorname{supp}\omega \subset (U,\varphi;x)\cap (V,\psi;y),$
$f{}dx^{1}\wedge\cdots\wedge{}dx^{ {}m}=\omega=g{}dy^{1}\wedge\cdots\wedge{}dy^{ {}m}.$
需验证:
$$
\int_{\psi(U)}(g\circ\psi^{-1})(y)dy^1\cdots dy^m=\int_M\omega=\int_{\varphi(U)}(f\circ\varphi^{-1})(x)dx^1\cdots dx^m.
$$
令$h=\varphi\circ\psi^{-1},$ $h(y)=x.$ 那么由积分变量代换公式,
以及$g=f\det(Dh),$ 结合坐标系都是与定向$\Omega$相符的, 即有等式成立.
设有两种单位分解$\{f_i\},\{g_j\},$
那么$\{f_ig_j=g_jf_i\}$也是一个单位分解. 因此
$$
\sum_j\int_M g_j\omega=\sum_{i,j}\int_M f_ig_j\omega=\sum_i\int_M f_i\omega.
$$
其它性质
容易看出, $\int_{(M,\Omega)}\omega=-\int_{(M,-\Omega)}\omega.$
定义$\mathcal{K}$为全体具紧支集连续的$m$次形式, 构成向量空间,
且$\int_M:\mathcal{K}\rightarrow \mathbb{R}$为线性泛函.
设$\varphi:N^k\rightarrow M$为$k$维子流形.
若$\varphi^\ast \omega$在$N$中具紧支集, 则
$$
\int_N \varphi^\ast \omega:=\int_{\varphi(N)} \omega.
$$
Stokes公式
回忆$\mathbb{R}^m$上的多元微积分, 我们可将微积分基本定理, Green公式,
Gauss公式, Stokes公式统一为:
$$
\int_D d\omega=\int_{\partial D}\omega,\quad D\subset \mathbb{R}^m.
$$
设$M^m$为$m$维$C^\infty$流形. 如果$M$中子集$D$由下面两类点构成:
内点: $p\in D,$ $\,\exists\,V\ni p$为开邻域, $V\subset D.$
边界点: $p\in D,$ $\,\exists\,(U,\varphi;x^i)$使得$\varphi(p)=0,$
$\varphi(U\cap D)=\{q\in D|x^m(q)\ge 0\}.$
称这样的坐标系为适应坐标系.
则称$D$为$M$中带边区域. 用$\partial D$表示边界全体.
易见内点全体$D^\circ$为$M$中开子集,
$U\cap \partial D=\{x^m(q)=0\}$为$(m-1)$维切片.
从而$\partial D$为$M$中正则超曲面.
全体适应坐标系构成了$\partial D$的一个覆盖,
构成了$\partial D$的坐标图册. 它确定了$\partial D$的$C^\infty$结构.
现$M$是定向流形, 则可要求$\partial D$的每个适应坐标系与定向$\Omega$相符.
在每个$(U_\alpha\cap \partial D;x_{\alpha}^1,\cdots,x_{\alpha}^{m-1})$上,
局部定义$(-1)^m{}dx_{(\alpha)}^{1}\wedge\cdots\wedge{}dx_{(\alpha)}^{ {}m-1}.$
下面证明它与$(-1)^m{}dx_{(\beta)}^{1}\wedge\cdots\wedge{}dx_{(\beta)}^{ {}m-1}$定向相符,
因此$\partial D$上有一诱导定向.
定理 1. 设$D$是$C^\infty$-流形$M$中带边区域, 如果$(M,\Omega)$是可定向的, 则$\partial D$也是可定向的.
证: 取$(U,\varphi;x^i),(V,\psi;y^j)$为$\partial D$的适应坐标系,
与定向$\Omega$相符,
需考虑$\det\frac{D(y^1,\cdots,y^{m-1})}{D(x^1,\cdots,x^{m-1})}$的符号.
已知$\det D(\psi\circ \varphi^{-1})>0.$
由于$\frac{\partial {}y^m}{\partial {}x^{i} }=0,$ $\,\forall\,i\neq m;$
$\frac{\partial {}y^m}{\partial {}x^m}\ge 0,$
立即推得$\det\frac{D(y^1,\cdots,y^{m-1})}{D(x^1,\cdots,x^{m-1})}>0.$
因此$\partial D$是可定向的.
特别地,
我们选取$(-1)^m {}dx_{(\alpha)}^{1}\wedge\cdots\wedge{}dx_{(\alpha)}^{ {}m-1}$与$\partial D$定向相符,
使Stokes公式成立.
定理 2 (Stokes公式). 设$(M^m,\Omega)$是定向$C^\infty$流形, $D$是$M$中带边区域, 从$M$中自然诱导定向. $\omega$是$M$一个具紧支集$C^1$的$(m-1)$次外微分形式, 则$\int_D d\omega=\int_{\partial D}\omega.$ 如果$\partial D=\varnothing,$ 则右端为零.
文章最后更新于 2021-12-22 18:57:10
Integration of Differential Forms
definition
We know that there are Integral variable substitution formula: Set $f\in C_c(\mathbb{R}^m),$
$h:\mathbb{R}^m\approx \mathbb{R}^m$ is diffeomorphism, then
$$
\int_{\mathbb{R}^m}f(x)dx^1\cdots dx^m=\int_{\mathbb{R}^m}(f\circ h)(y)|\det Dh|dy^1\cdots dy^m.
$$
If $h:\mathbb{R}^m\rightarrow \mathbb{R}^m$ satisfies $\det Dh(y)>0,$
Then there is $\det Dh>0.$ everywhere and it is called Keep direction diffeomorphism of .
From the content of the previous lesson,
$\Omega={}dx^{1}\wedge\cdots\wedge{}dx^{ {}m}\in \mathcal{A}^m(\mathbb{R}^m)$ gives $\mathbb{R}^m$ a natural orientation.
Now $(\mathbb{R}^m,\mathrm{id};x),(\mathbb{R}^m,h^{-1};y)$ is two different coordinate systems,
Let $\omega=f(x)dx^1\wedge\cdots\wedge dx^m$ be the continuous exterior derivative form of $m$ with compact support,
Then due to
$$
dx^1\wedge\cdots\wedge dx^m=\det(Dh)(y){}dy^{1}\wedge\cdots\wedge{}dy^{ {}m},
$$
We know that $\omega=f\circ h(y)\det(Dh){}dy^{1}\wedge\cdots\wedge{}dy^{ {}m}$ is another representation.
Tips for the above formula:
$$
\int_{\mathbb{R}^m}\omega:=\int_{\mathbb{R}^m}f(x)dx^1\cdots dx^m.
$$
Let $(M^m,\Omega)$ be the oriented $C^\infty$-manifold,
$\omega$ is the exterior derivative form of the compact support $m$ on $M$,
That is $\operatorname{supp}\omega=\overline{\left\{q\mid\omega(q)\neq 0\right\}} \Subset M.$
Next, define the integral of $\omega$ on $M$.
When $\operatorname{supp}\omega\subset (U,\varphi;x^i),$
When the coordinate system matches the orientation $\Omega$, $\omega=f dx^1\wedge\cdots\wedge dx^m,$
$f\in C_c(U).$ From the previous description, defined
$$
\int_M \omega:=\int_U \omega=\int_{\varphi(U)}(f\circ\varphi^{-1})(x)dx^1\cdots dx^m.
$$
Generally, $\{(U_i,\varphi_i)\}$ is taken as the coordinate atlas consistent with the orientation,
And the partition of unity subordinate to it. Then $\omega=\sum_i f_i\omega,$
$\operatorname{supp}(f_i\omega)\subset U_i.$ has a compact support from $\omega$,
$\{\operatorname{supp}f_i\}$ is locally finite, and the sum always makes sense. So we define
$$
\int_M \omega:=\sum_i \int_M f_i\omega,
$$
Where $\int_M f_i\omega$ is defined.
good quality
We need to verify the soundness of this definition: Is it independent of the coordinate system/atlas and partition of unity selection?
If $\operatorname{supp}\omega \subset (U,\varphi;x)\cap (V,\psi;y),$
$f{}dx^{1}\wedge\cdots\wedge{}dx^{ {}m}=\omega=g{}dy^{1}\wedge\cdots\wedge{}dy^{ {}m}.$
Need to verify:
$$
\int_{\psi(U)}(g\circ\psi^{-1})(y)dy^1\cdots dy^m=\int_M\omega=\int_{\varphi(U)}(f\circ\varphi^{-1})(x)dx^1\cdots dx^m.
$$
Let $h=\varphi\circ\psi^{-1},$ $h(y)=x.$ then substitute the formula by the integral variable,
And the $g=f\det(Dh),$ combined coordinate system is consistent with the orientation $\Omega$, that is, the equation is established.
There are two partitions of unity $\{f_i\},\{g_j\},$
Then $\{f_ig_j=g_jf_i\}$ is also a partition of unity. Therefore
$$
\sum_j\int_M g_j\omega=\sum_{i,j}\int_M f_ig_j\omega=\sum_i\int_M f_i\omega.
$$
Other properties
It is easy to see, $\int_{(M,\Omega)}\omega=-\int_{(M,-\Omega)}\omega.$
Define $\mathcal{K}$ as all consecutive $m$-order forms with compact supports, forming a vector space,
And $\int_M:\mathcal{K}\rightarrow \mathbb{R}$ is a linear functional.
Let $\varphi:N^k\rightarrow M$ be the $k$-dimensional submanifold.
If $\varphi^\ast \omega$ has compact support in $N$, then
$$
\int_N \varphi^\ast \omega:=\int_{\varphi(N)} \omega.
$$
Stokes formula
Recalling the multivariable calculus on $\mathbb{R}^m$, we can convert the fundamental theorem of calculus, Green's formula,
Gauss formula and Stokes formula are unified as:
$$
\int_D d\omega=\int_{\partial D}\omega,\quad D\subset \mathbb{R}^m.
$$
Let $M^m$ be the $m$-dimensional $C^\infty$ manifold. If the subset $D$ in $M$ consists of the following two types of points:
Interior points: $p\in D,$ $\,\exists\,V\ni p$ is an open neighborhood, $V\subset D.$
Boundary points: $p\in D,$ $\,\exists\,(U,\varphi;x^i)$ makes $\varphi(p)=0,$
$\varphi(U\cap D)=\{q\in D|x^m(q)\ge 0\}.$
Call such a coordinate system Adapt to coordinate system.
Then $D$ is called $M$ band edge area. Use $\partial D$ to represent the entire boundary.
It is easy to see that all interior points $D^\circ$ are the open subset of $M$,
$U\cap \partial D=\{x^m(q)=0\}$ is a $(m-1)$-dimensional slice.
Therefore $\partial D$ is the regular hypersurface in $M$.
The overall adaptive coordinate system constitutes a coverage of $\partial D$,
It constitutes the coordinate atlas of $\partial D$. It determines the $C^\infty$ structure of $\partial D$.
Now $M$ is an oriented manifold, then each adaptive coordinate system of $\partial D$ can be required to be consistent with the orientation $\Omega$.
On each $(U_\alpha\cap \partial D;x_{\alpha}^1,\cdots,x_{\alpha}^{m-1})$,
Local definition $(-1)^m{}dx_{(\alpha)}^{1}\wedge\cdots\wedge{}dx_{(\alpha)}^{ {}m-1}.$
The following proves that it is consistent with the $(-1)^m{}dx_{(\beta)}^{1}\wedge\cdots\wedge{}dx_{(\beta)}^{ {}m-1}$ orientation,
Therefore there is an induced orientation on $\partial D$.
Theorem 1. Let $D$ be the edge region in $C^\infty$-manifold $M$. If $(M,\Omega)$ is orientable, then $\partial D$ is also orientable.
Certificate: Take $(U,\varphi;x^i),(V,\psi;y^j)$ as the adaptive coordinate system of $\partial D$,
Consistent with orientation $\Omega$,
The symbol $\det\frac{D(y^1,\cdots,y^{m-1})}{D(x^1,\cdots,x^{m-1})}$ needs to be considered.
Known $\det D(\psi\circ \varphi^{-1})>0.$
Due to $\frac{\partial {}y^m}{\partial {}x^{i} }=0,$ $\,\forall\,i\neq m;$
$\frac{\partial {}y^m}{\partial {}x^m}\ge 0,$
Push immediately $\det\frac{D(y^1,\cdots,y^{m-1})}{D(x^1,\cdots,x^{m-1})}>0.$
Therefore $\partial D$ is orientable.
In particular,
We select $(-1)^m {}dx_{(\alpha)}^{1}\wedge\cdots\wedge{}dx_{(\alpha)}^{ {}m-1}$ to match $\partial D$ orientation,
Make Stokes formula hold.
Theorem 2 (Stokes formula). Assume $(M^m,\Omega)$ is the oriented $C^\infty$ manifold, $D$ is the edge region in $M$, and the orientation is naturally induced from $M$. $\omega$ is $M$ a $(m-1)$ sub-exterior derivative form with a compact support $C^1$, then $\int_D d\omega=\int_{\partial D}\omega.$ If $\partial D=\varnothing,$, the right end is zero.
The article was last updated on 2021-12-22 18:57:10
