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, *ω* ∧ *θ* =
(–1)^{pq }*θ* ∧ *ω*.

– Contraction with a vector
field, *v* · (*ω* ∧ *θ*)
= (*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, d*f*(*X*):= *X*(*f*), for all 0-forms *f* and vector fields *X*;

(2) Linearity, d(*αω* + *βη*)
= *α* d*ω* + *β* d*η*, for all *p*-forms *ω*,
*η* and numbers *α*, *β*;

(3) Relation with exterior product, d(*ω* ∧ *θ*):=
d*ω* ∧ *θ* +
(–1)^{p} *ω* ∧ d*θ*,
for all *p*-forms *ω* and *q*-forms *θ*;

(4) Square, d^{2}*ω* = d(d*ω*)
= 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* = *ω*_{i...j} d*x*^{ i} ∧ ...
∧ d*x*^{ j},

(d*ω*)_{ma... b} =
(*p*+1) ∇_{[m }*ω*_{a...
b]}
, d*ω* = ∂_{k }*ω*_{i...
j} d*x*^{k}
∧ dx^{ i} ∧ ...
∧ d*x*^{ j} .

* __Properties__:
It commutes with taking the Lie derivative with respect to some vector field
*v*^{a}, d(\(\cal L\)_{v} *ω*)
= \(\cal L\)_{}_{v}(d*ω*).

@ __Discrete__: Harrison mp/06 [unified
with continuum]; Arnold et al BAMS(10)
[finite-element exterior calculus, cohomology and Hodge theory].

@ __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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