Integration on manifolds

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Let 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 be the standard -cube. A singular cube in is a smooth mapping . Let be a -form on (cf. Differential form). Then the integral of over a singular -cube is defined as


where is the unique smooth function such that on and where on the right-hand side the ordinary Lebesgue integral is taken. A singular -chain is a formal finite sum of singular -cubes with coefficients in . One defines


Now let be oriented and let and be two singular -chains such that for all and such that all the are orientation preserving. Then . In particular, if the fit together to define a piecewise-smooth -dimensional submanifold of , then the integral is well-defined.

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


where is a -form and is a singular -chain. This is the analogue of the fundamental theorem of calculus.

A particular consequence is Green's theorem: Let be a compact -dimensional manifold with boundary and let be differentiable. Then


Let now be an oriented -dimensional Riemannian manifold, i.e. for each an orientation has been given on . The volume form on is now defined by requiring that for one (and hence each) orthonormal basis of in the given orientation class of . Another consequence of the general Stokes' theorem (a3) is the divergence theorem:


Here is a vector field on , is a three-dimensional oriented manifold in , if , is an outward normal to , and and are, respectively, the volume and area elements of and . The inner product is induced from the standard one in .

Finally there is the classical Stokes' formula: Let be an oriented two-dimensional submanifold with boundary . Give an orientation such that together with the outward normal it gives back the orientation of . Let parametrize and let be the vector field on such that everywhere. One then has the formula


where the curl of a vector field on is defined by:


All these theorems have higher-dimensional analogues.


[a1] M. Spivak, "Calculus on manifolds" , Benjamin (1965)
[a2] M. Hazewinkel, "A tutorial introduction to differentiable manifolds and calculus on manifolds" W. Schiehlen (ed.) W. Wedig (ed.) , Analysis and estimation of stochastic mechanical systems , Springer (Wien) (1988) pp. 316–340
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