# 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 $\Delta _ {n} = [ 0, 1] ^ {n} \subset \mathbf R ^ {n}$ be the standard $n$- cube. A singular cube in $M$ is a smooth mapping $s: \Delta _ {k} \rightarrow M$. 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

$$\tag{a1 } \int\limits _ { s } \omega = \ \int\limits _ {\Delta _ {k} } f ,$$

where $f$ is the unique smooth function such that $s ^ {*} \omega = f dx _ {1} \wedge \dots \wedge dx _ {k}$ on $\Delta _ {k}$ 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

$$\tag{a2 } \int\limits _ { c } \omega = \ \sum _ { i } n _ {i} \int\limits _ {s _ {i} } \omega .$$

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 $s _ {i} ( \Delta _ {k} ) = s _ {i} ^ \prime ( \Delta _ {k} )$ 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

$$\tag{a3 } \int\limits _ { c } d \omega = \ \int\limits _ {\partial c } \omega ,$$

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 $f , g: M \rightarrow \mathbf R$ be differentiable. Then

$$\tag{a4 } \int\limits _ {\partial M } ( f dx + g dy) = \ {\int\limits \int\limits } _ { M } \left ( \frac{\partial g }{\partial x } - \frac{\partial f }{\partial y } \right ) dx dy .$$

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:

$$\tag{a5 } \int\limits _ { M } \mathop{\rm div} \psi dV = \ \int\limits _ {\partial M } \langle \psi , n \rangle dA .$$

Here $\psi$ is a vector field on $\mathbf R ^ {3}$, $M$ is a three-dimensional oriented manifold in $\mathbf R ^ {3}$, $\mathop{\rm div} \psi = {\partial \psi _ {i} } / {\partial x _ {i} }$ 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 $\phi$ be the vector field on $\partial M$ such that $ds ( \phi ) = 1$ everywhere. One then has the formula

$$\tag{a6 } \int\limits _ { M } \langle \mathop{\rm curl} \psi , n \rangle dA = \ \int\limits _ {\partial M } \langle \psi , \phi \rangle ds ,$$

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

$$\tag{a7 } \mathop{\rm curl} \psi = \ \left ( \frac{\partial \psi _ {3} }{\partial x _ {2} } - \frac{\partial \psi _ {2} }{\partial x _ {3} } \right ) { \frac \partial {\partial x _ {1} } } +$$

$$+ \left ( \frac{\partial \psi _ {1} }{\partial x _ {3} } - \frac{\partial \psi _ {3} }{\partial x _ {1} } \right ) { \frac \partial {\partial x _ {2} } } + \left ( \frac{\partial \psi _ {2} }{\partial x _ {1} } - \frac{\partial \psi _ {1} }{\partial x _ {2} } \right ) { \frac \partial {\partial x _ {3} } } .$$

All these theorems have higher-dimensional analogues.

How to Cite This Entry:
Integration on manifolds. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integration_on_manifolds&oldid=47384