Exterior Algebra and Calculus

Exterior Product > s.a. forms.
$ Def: The associative, bilinear composition law for differential forms on a manifold : ^p(M) × ^q(M) → ^p+q(M) given by

* Properties:
– Under permutation, = (–1)^pq .
– Contraction with a vector field, v · ( ) = (v · ) + (–1)^p (v · ).

Exterior Derivative > s.a. cohomology; differential forms; lie derivatives.
$ Def: An operator d: ^p(M) → ^p+1(M) on the graded algebra of differential forms on a manifold, defined by
(1) Action on scalars, df(X):= X(f), for all 0-forms f and vector fields X;
(2) Linearity, d( + ) = d + d, for all p-forms , and numbers , ;
(3) Relation with exterior product, d( ):= d + (–1)^p d, for all p-forms and q-forms ;
(4) Square, d² = d(d) = 0 for all p-forms .
* Remark: It does not need a metric to be defined (it is a concomitant).
* Notation: In abstract index and coordinate notation, respectively, for a p-form = _i...j dx ⁱ ... dx ^j,

(d)_{ma... b} = (p+1) _{[m a... b]} ,      d = _{k i... j} dx^k dx ⁱ ... dx ^j .

* Properties: It commutes with taking the Lie derivative with respect to some vector field v^a, d(_v ) = _v(d).
@ Generalized: Gozzi & Reuter IJMPA(94)ht/03 [quantum deformed, on phase space]; Tarasov JPA(05) [of fractional order]; Harrison mp/06 [unified discrete and continuum].

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