Exterior Algebra and Calculus  

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

\[ (\omega\wedge\theta)_{a...bc...d} = {(p+q)!\over p!\,q!}\,\omega_{[a...b}\,\theta_{c...d]}\;.\]

* Properties:
Under permutation, ωθ = (−1)pq θω.
Contraction with a vector field, v · (ωθ) = (v · ω) ∧ θ + (−1)p ω ∧ (v · θ).
> Online resources: see MathWorld page; Wikipedia page.

Exterior Calculus / Derivatives > s.a. cohomology; differential forms; lie derivatives.
$ Def: An operator d: Ω p(M) → Ω 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(αω + βη) = α dω + β dη, for all p-forms ω, η and numbers α, β;
(3) Relation with exterior product, d(ωθ):= dωθ + (−1)p ω ∧ dθ, for all p-forms ω and q-forms θ;
(4) Square, d2ω = d(dω) = 0 for all p-forms ω.
* 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 ω = ωi...j dx i ∧ ... ∧ dx j,

(dω)ma... b = (p+1) ∇[m ωa... b] ,      dω = ∂k ωi... j dxk ∧ dx i ∧ ... ∧ dx j .

* Properties: It commutes with taking the Lie derivative with respect to some vector field va, d(\(\cal L\)v ω) = \(\cal L\)v(dω).
@ General references: Colombaro et al Math(19)-a2002 [introduction].
@ Discrete: Harrison mp/06 [unified with continuum]; Arnold et al BAMS(10) [finite-element exterior calculus, cohomology and Hodge theory]; Boom et al a2104 [applied to linear elasticity].
@ Other generalized: Okumura PTP(96)ht [in non-commutative geometry]; Gozzi & Reuter IJMPA(94)ht/03 [quantum deformed, on phase space]; Tarasov JPA(05) [of fractional order]; Yang a1507-wd [in non-commutative geometry, nilpotent matrix representation].
> Online resources: see Wikipedia page.

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