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}* f*_{a...
b}:=
∫_{U}* f*(*x*)
d^{n}*x* , sometimes
written ∫_{U }*f*_{a...
b}
d*v*^{a... b} ,

where *U* has to be covered by one (right-handed) coordinate chart (otherwise
we
generalize the definition using partitions of unity) and

*f*_{a... b} = *f*(*x*)
(d*x*^{1} ∧ ...
∧ d*x*^{n})_{a...
b} .

* __For functions__: Choose a volume element or measure (which could be
defined by a metric), and define

∫_{U}* f* (d^{n}*v*):=
∫_{U}* f* *ε*_{a...
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.

**On Submanifolds**

* __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}* A*_{a...
b}
d*v*^{a... b} , or ∫_{S}* B*^{m...
n}
d*S*_{m... 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
Loop.

* __In 3D space__:

∫_{}_{S }∇ ×
*A* · d*s* = ∫_{}_{∂(S)} * A* · d*l* ; 2 ∫_{}_{S }∇_{a}* A*_{b} d*v*^{ab}
= ∫_{}_{∂(S)}
*A*_{a} d*v*^{a} ; ∫_{}_{V }∇ ·
*A* d^{3}*v* = ∫_{}_{∂(V)} *A* ·
d*s* .

* __In general__:

∫_{}_{U}
d*ω* =
∫_{}_{∂(U)} *ω* , from
which ∫_{}_{U}∇_{a} *v*^{a}
= ∫_{}_{∂(U)} *v*^{a }*ε*_{ab...c}
= ∫_{}_{∂(U)} *v*^{a} *n*_{a}
;

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]} d*S*_{b} = –\(1\over2\)∫_{}_{∂(S)}* T*^{ [ab]} d*S*_{ab}
.

@ __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].

**Generalizations**

@ __Generalized integrals__: Yekutieli 15 [non-abelian multiplicative integration on surfaces].

> __Generalized manifolds__: see differential
geometry; fractals; grassmann; operator theory; Supermanifolds.

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nov 2016