14 KiB
14 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Exterior calculus identities | 4/7 | https://en.wikipedia.org/wiki/Exterior_calculus_identities | reference | science, encyclopedia | 2026-05-05T08:14:13.652508+00:00 | kb-cron |
=== Cohomology === The
k
{\displaystyle k}
-th cohomology of a manifold
M
{\displaystyle M}
and its exterior derivative operators
d
0
,
…
,
d
n
−
1
{\displaystyle d_{0},\ldots ,d_{n-1}}
is given by
H
k
(
M
)
:=
ker
(
d
k
)
im
(
d
k
−
1
)
{\displaystyle H^{k}(M):={\frac {{\text{ker}}(d_{k})}{{\text{im}}(d_{k-1})}}}
Two closed
k
{\displaystyle k}
-forms
α
,
β
∈
Ω
k
(
M
)
{\displaystyle \alpha ,\beta \in \Omega ^{k}(M)}
are in the same cohomology class if their difference is an exact form i.e.
[
α
]
=
[
β
]
⟺
α
−
β
=
d
η
for some
η
∈
Ω
k
−
1
(
M
)
{\displaystyle [\alpha ]=[\beta ]\ \ \Longleftrightarrow \ \ \alpha {-}\beta =d\eta \ {\text{ for some }}\eta \in \Omega ^{k-1}(M)}
A closed surface of genus
g
{\displaystyle g}
will have
2
g
{\displaystyle 2g}
generators which are harmonic.
=== Dirichlet energy === Given
α
∈
Ω
k
(
M
)
{\displaystyle \alpha \in \Omega ^{k}(M)}
, its Dirichlet energy is
E
D
(
α
)
:=
1
2
⟨
⟨
d
α
,
d
α
⟩
⟩
+
1
2
⟨
⟨
δ
α
,
δ
α
⟩
⟩
{\displaystyle {\mathcal {E}}_{\text{D}}(\alpha ):={\dfrac {1}{2}}\langle \!\langle d\alpha ,d\alpha \rangle \!\rangle +{\dfrac {1}{2}}\langle \!\langle \delta \alpha ,\delta \alpha \rangle \!\rangle }
== Properties ==
=== Exterior derivative properties ===
∫
Σ
d
α
=
∫
∂
Σ
α
{\displaystyle \int _{\Sigma }d\alpha =\int _{\partial \Sigma }\alpha }
( Stokes' theorem )
d
∘
d
=
0
{\displaystyle d\circ d=0}
( cochain complex )
d
(
α
∧
β
)
=
d
α
∧
β
+
(
−
1
)
k
α
∧
d
β
{\displaystyle d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1)^{k}\alpha \wedge d\beta }
for
α
∈
Ω
k
(
M
)
,
β
∈
Ω
l
(
M
)
{\displaystyle \alpha \in \Omega ^{k}(M),\ \beta \in \Omega ^{l}(M)}
( Leibniz rule )
d
f
(
X
)
=
∂
X
f
{\displaystyle df(X)=\partial _{X}f}
for
f
∈
Ω
0
(
M
)
,
X
∈
Γ
(
T
M
)
{\displaystyle f\in \Omega ^{0}(M),\ X\in \Gamma (TM)}
( directional derivative )
d
α
=
0
{\displaystyle d\alpha =0}
for
α
∈
Ω
n
(
M
)
,
dim
(
M
)
=
n
{\displaystyle \alpha \in \Omega ^{n}(M),\ {\text{dim}}(M)=n}
=== Exterior product properties ===
α
∧
β
=
(
−
1
)
k
l
β
∧
α
{\displaystyle \alpha \wedge \beta =(-1)^{kl}\beta \wedge \alpha }
for
α
∈
Ω
k
(
M
)
,
β
∈
Ω
l
(
M
)
{\displaystyle \alpha \in \Omega ^{k}(M),\ \beta \in \Omega ^{l}(M)}
( alternating )
(
α
∧
β
)
∧
γ
=
α
∧
(
β
∧
γ
)
{\displaystyle (\alpha \wedge \beta )\wedge \gamma =\alpha \wedge (\beta \wedge \gamma )}
( associativity )
(
λ
α
)
∧
β
=
λ
(
α
∧
β
)
{\displaystyle (\lambda \alpha )\wedge \beta =\lambda (\alpha \wedge \beta )}
for
λ
∈
R
{\displaystyle \lambda \in \mathbb {R} }
( compatibility of scalar multiplication )
α
∧
(
β
1
+
β
2
)
=
α
∧
β
1
+
α
∧
β
2
{\displaystyle \alpha \wedge (\beta _{1}+\beta _{2})=\alpha \wedge \beta _{1}+\alpha \wedge \beta _{2}}
( distributivity over addition )
α
∧
α
=
0
{\displaystyle \alpha \wedge \alpha =0}
for
α
∈
Ω
k
(
M
)
{\displaystyle \alpha \in \Omega ^{k}(M)}
when
k
{\displaystyle k}
is odd or
rank
α
≤
1
{\displaystyle \operatorname {rank} \alpha \leq 1}
. The rank of a
k
{\displaystyle k}
-form
α
{\displaystyle \alpha }
means the minimum number of monomial terms (exterior products of one-forms) that must be summed to produce
α
{\displaystyle \alpha }
.
=== Pull-back properties ===
d
(
ϕ
∗
α
)
=
ϕ
∗
(
d
α
)
{\displaystyle d(\phi ^{*}\alpha )=\phi ^{*}(d\alpha )}
( commutative with
d
{\displaystyle d}
)
ϕ
∗
(
α
∧
β
)
=
(
ϕ
∗
α
)
∧
(
ϕ
∗
β
)
{\displaystyle \phi ^{*}(\alpha \wedge \beta )=(\phi ^{*}\alpha )\wedge (\phi ^{*}\beta )}
( distributes over
∧
{\displaystyle \wedge }
)
(
ϕ
1
∘
ϕ
2
)
∗
=
ϕ
2
∗
ϕ
1
∗
{\displaystyle (\phi _{1}\circ \phi _{2})^{*}=\phi _{2}^{*}\phi _{1}^{*}}
( contravariant )
ϕ
∗
f
=
f
∘
ϕ
{\displaystyle \phi ^{*}f=f\circ \phi }
for
f
∈
Ω
0
(
N
)
{\displaystyle f\in \Omega ^{0}(N)}
( function composition )
=== Musical isomorphism properties ===
(
X
♭
)
♯
=
X
{\displaystyle (X^{\flat })^{\sharp }=X}
(
α
♯
)
♭
=
α
{\displaystyle (\alpha ^{\sharp })^{\flat }=\alpha }
=== Interior product properties ===
ι
X
∘
ι
X
=
0
{\displaystyle \iota _{X}\circ \iota _{X}=0}
( nilpotent )
ι
X
∘
ι
Y
=
−
ι
Y
∘
ι
X
{\displaystyle \iota _{X}\circ \iota _{Y}=-\iota _{Y}\circ \iota _{X}}
ι
X
(
α
∧
β
)
=
(
ι
X
α
)
∧
β
+
(
−
1
)
k
α
∧
(
ι
X
β
)
{\displaystyle \iota _{X}(\alpha \wedge \beta )=(\iota _{X}\alpha )\wedge \beta +(-1)^{k}\alpha \wedge (\iota _{X}\beta )}
for
α
∈
Ω
k
(
M
)
,
β
∈
Ω
l
(
M
)
{\displaystyle \alpha \in \Omega ^{k}(M),\ \beta \in \Omega ^{l}(M)}
( Leibniz rule )
ι
X
α
=
α
(
X
)
{\displaystyle \iota _{X}\alpha =\alpha (X)}
for
α
∈
Ω
1
(
M
)
{\displaystyle \alpha \in \Omega ^{1}(M)}
ι
X
f
=
0
{\displaystyle \iota _{X}f=0}
for
f
∈
Ω
0
(
M
)
{\displaystyle f\in \Omega ^{0}(M)}
ι
X
(
f
α
)
=
f
ι
X
α
{\displaystyle \iota _{X}(f\alpha )=f\iota _{X}\alpha }
for
f
∈
Ω
0
(
M
)
{\displaystyle f\in \Omega ^{0}(M)}
=== Hodge star properties ===