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.

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