Integration on manifolds

Let $M$ be a finite-dimensional smooth manifold. Tangent spaces and such provide the global analogues of differential calculus. There is also an "integral calculus on manifolds" . Let ${\mathrm{\Delta }}_n=[0,1{]}^n\subset {\mathbf{R}}^n$ be the standard $n$- cube. A singular cube in $M$ is a smooth mapping $s:{\mathrm{\Delta }}_k\to M$. Let $\omega$ be a $k$- form on $M$( cf. Differential form). Then the integral of $\omega$ over a singular $k$- cube $s$ is defined as

$$\begin{array}{}\text{(a1)}& \underset{s}{\int }\omega =\underset{{\mathrm{\Delta }}_k}{\int }f,\end{array}$$

where $f$ is the unique smooth function such that $s^{\ast }\omega =fd{x}_1\wedge \cdots \wedge d{x}_k$ on ${\mathrm{\Delta }}_k$ and where on the right-hand side the ordinary Lebesgue integral is taken. A singular $k$- chain is a formal finite sum $c=\sum n_i{s}_i$ of singular $k$- cubes with coefficients in $\mathbf{Z}$. One defines

$$\begin{array}{}\text{(a2)}& \underset{c}{\int }\omega =\sum _i{n}_i\underset{s_i}{\int }\omega .\end{array}$$

Now let $M$ be oriented and let $c=\sum n_i{s}_i$ and $c^{\prime }=\sum n_i{s}_i^{\prime }$ be two singular $k$- chains such that $s_i({\mathrm{\Delta }}_k)=s_i^{\prime }({\mathrm{\Delta }}_k)$ for all $i$ and such that all the $s_i,s_i^{\prime }$ are orientation preserving. Then ${\int }_c\omega ={\int }_{c^{\prime }}\omega$. In particular, if the $s_i$ fit together to define a piecewise-smooth $k$- dimensional submanifold $N$ of $M$, then the integral ${\int }_N\omega$ is well-defined.

Let $d$ denote the exterior derivative on exterior forms (cf. Exterior form) and $\partial$ the (obvious) boundary operator on oriented (singular) chains. Then one has Stokes' theorem

$$\begin{array}{}\text{(a3)}& \underset{c}{\int }d\omega =\underset{\partial c}{\int }\omega ,\end{array}$$

where $\omega$ is a $(k-1)$- form and $c$ is a singular $k$- chain. This is the analogue of the fundamental theorem of calculus.

A particular consequence is Green's theorem: Let $M\subset {\mathbf{R}}^2$ be a compact $2$- dimensional manifold with boundary and let $f,g:M\to \mathbf{R}$ be differentiable. Then

$$\begin{array}{}\text{(a4)}& \underset{\partial M}{\int }(fdx+gdy)={\int \int }_M(\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y})dxdy.\end{array}$$

Let $M$ now be an oriented $n$- dimensional Riemannian manifold, i.e. for each $x\in M$ an orientation has been given on $T_{x}M$. The volume form ${\omega }_M$ on $M$ is now defined by requiring that ${\omega }_M(x)(v_1\dots v_n)=1$ for one (and hence each) orthonormal basis of $T_{x}M$ in the given orientation class of $T_{x}M$. Another consequence of the general Stokes' theorem (a3) is the divergence theorem:

$$\begin{array}{}\text{(a5)}& \underset{M}{\int }\mathrm{div}\psi dV=\underset{\partial M}{\int }⟨\psi ,n⟩dA.\end{array}$$

Here $\psi$ is a vector field on ${\mathbf{R}}^3$, $M$ is a three-dimensional oriented manifold in ${\mathbf{R}}^3$, $\mathrm{div}\psi =\partial {\psi }_i/\partial x_i$ if $\psi =\sum {\psi }_i\partial /\partial x_i$, $n$ is an outward normal to $\partial M$, and $dM$ and $dA$ are, respectively, the volume and area elements of $M$ and $\partial M$. The inner product is induced from the standard one in ${\mathbf{R}}^3$.

Finally there is the classical Stokes' formula: Let $M\subset {\mathbf{R}}^3$ be an oriented two-dimensional submanifold with boundary $\partial M$. Give $\partial M$ an orientation such that together with the outward normal it gives back the orientation of $M$. Let $s$ parametrize $\partial M$ and let $\varphi$ be the vector field on $\partial M$ such that $ds(\varphi )=1$ everywhere. One then has the formula

$$\begin{array}{}\text{(a6)}& \underset{M}{\int }⟨\mathrm{curl}\psi ,n⟩dA=\underset{\partial M}{\int }⟨\psi ,\varphi ⟩ds,\end{array}$$

where the curl of a vector field $\psi$ on ${\mathbf{R}}^3$ is defined by:

$$\begin{array}{}\text{(a7)}& \mathrm{curl}\psi =(\frac{\partial {\psi }_3}{\partial x_2}-\frac{\partial {\psi }_2}{\partial x_3})\frac{\partial }{\partial x_1}+\end{array}$$

$$+(\frac{\partial {\psi }_1}{\partial x_3}-\frac{\partial {\psi }_3}{\partial x_1})\frac{\partial }{\partial x_2}+(\frac{\partial {\psi }_2}{\partial x_1}-\frac{\partial {\psi }_1}{\partial x_2})\frac{\partial }{\partial x_3}.$$

All these theorems have higher-dimensional analogues.

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