|Integration on Manifolds|
In General > s.a. differential forms; lie groups.
* Idea: Integration on an n-manifold (without a metric, with just an orientation) is defined only for n-forms, by
∫U fa... b:= ∫U f(x) dnx , sometimes written ∫U fa... b dva... b ,
where U has to be covered by one (right-handed) coordinate chart (otherwise we generalize the definition using partitions of unity) and
fa... b = f(x) (dx1 ∧ ... ∧ dxn)a... b .
* For functions: Choose a volume element or measure (which could be defined by a metric), and define
∫U f (dnv):= ∫U f εa... b .
* For other objects: Need even more structure, except for some integrals
over lower-dimensional submanifolds.
@ In curved spacetime, for general relativity: DeBenedictis phy/98.
* Idea: Given a p-dimensional submanifold Σ in an n-dimensional manifold M, the natural things to integrate there are p-forms, but we could integrate on Σ an n-form f defined on M by defining the result to be the integral of the p-form φ*f, where φ is the embedding φ: Σ → M.
* Notation: An integral over a p-submanifold S can be indicated in one of two ways,
∫S Aa... b dva... b , or ∫S Bm... n dSm... n ,
where A is a p-form on S and B an (n–p)-th rank tensor with indices normal to S.
Stokes' Theorem > s.a. Gauss' Theorem; Wilson
* In 3D space:
∫S ∇ × A · ds = ∫∂(S) A · dl ; 2 ∫S ∇a Ab dvab = ∫∂(S) Aa dva ; ∫V ∇ · A d3v = ∫∂(V) A · ds .
* In general:
∫U dω = ∫∂(U) ω , from which ∫U∇a va = ∫∂(U) va εab...c = ∫∂(U) va na ;
Special cases are the 3D version, Gauss' theorem, and the Green identities
(> see vector calculus).
* For a rank-two (antisymmetric) tensor:
∫S ∇a T [ab] dSb = –\(1\over2\)∫∂(S) T [ab] dSab .
@ General references: in Flanders 63; Saslow EJP(07)
[computationally based proof].
@ Generalized: Bralic ht/93 ["surfaceless"]; Mensky PLA(04)gq [in terms of path group in Minkowski space]; Meunier EJC(08) [combinatorial].
@ Non-abelian: Diakonov & Petrov hl/00, JETP(01)ht/00 [including Yang-Mills and gravity]; Kondo ht/00-proc [Wilson loops]; Broda in(01)mp/00; Broda & Duniec mp/01 [Wilson loops]; Mensky gq/02-conf [in terms of path group in Minkowski]; Matsudo & Kondo PRD-a1509 [and implications for quark confinement; > s.a. loops [gauge theories and loop variables]; path integrals for gauge theories.
> Online resources: see Wikipedia page.
Specific Results and Types of Manifolds > s.a. lie
groups and examples; Reynolds Theorem.
@ References: Hannay & Nye JPA(04) [2-sphere, Fibonacci numerical integration]; Somogyi JMP(11)-a1101 [angular integrals in d dimensions]; Felder & Kazhdan a1608 [regularization of divergent integrals].
@ Generalized integrals: Yekutieli 15 [non-abelian multiplicative integration on surfaces].
> Generalized manifolds: see differential geometry; fractals; grassmann; operator theory; Supermanifolds.
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