Exterior Algebra and Calculus| 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, 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(ω ∧ θ):=
dω ∧ θ +
(−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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 15 apr 2021