14 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Exterior calculus identities | 3/7 | https://en.wikipedia.org/wiki/Exterior_calculus_identities | reference | science, encyclopedia | 2026-05-05T08:14:13.652508+00:00 | kb-cron |
A one-form
α
∈
Ω
1
(
M
)
{\displaystyle \alpha \in \Omega ^{1}(M)}
corresponds to the unique vector field
α
♯
∈
Γ
(
T
M
)
{\displaystyle \alpha ^{\sharp }\in \Gamma (TM)}
such that for all
X
∈
Γ
(
T
M
)
{\displaystyle X\in \Gamma (TM)}
, we have:
α
(
X
)
=
g
(
α
♯
,
X
)
.
{\displaystyle \alpha (X)=g(\alpha ^{\sharp },X).}
These mappings extend via multilinearity to mappings from
k
{\displaystyle k}
-vector fields to
k
{\displaystyle k}
-forms and
k
{\displaystyle k}
-forms to
k
{\displaystyle k}
-vector fields through
(
A
1
∧
A
2
∧
⋯
∧
A
k
)
♭
=
A
1
♭
∧
A
2
♭
∧
⋯
∧
A
k
♭
{\displaystyle (A_{1}\wedge A_{2}\wedge \cdots \wedge A_{k})^{\flat }=A_{1}^{\flat }\wedge A_{2}^{\flat }\wedge \cdots \wedge A_{k}^{\flat }}
(
α
1
∧
α
2
∧
⋯
∧
α
k
)
♯
=
α
1
♯
∧
α
2
♯
∧
⋯
∧
α
k
♯
.
{\displaystyle (\alpha _{1}\wedge \alpha _{2}\wedge \cdots \wedge \alpha _{k})^{\sharp }=\alpha _{1}^{\sharp }\wedge \alpha _{2}^{\sharp }\wedge \cdots \wedge \alpha _{k}^{\sharp }.}
=== Hodge star === For an n-manifold M, the Hodge star operator
⋆
:
Ω
k
(
M
)
→
Ω
n
−
k
(
M
)
{\displaystyle {\star }:\Omega ^{k}(M)\rightarrow \Omega ^{n-k}(M)}
is a duality mapping taking a
k
{\displaystyle k}
-form
α
∈
Ω
k
(
M
)
{\displaystyle \alpha \in \Omega ^{k}(M)}
to an
(
n
−
k
)
{\displaystyle (n{-}k)}
-form
(
⋆
α
)
∈
Ω
n
−
k
(
M
)
{\displaystyle ({\star }\alpha )\in \Omega ^{n-k}(M)}
. It can be defined in terms of an oriented frame
(
X
1
,
…
,
X
n
)
{\displaystyle (X_{1},\ldots ,X_{n})}
for
T
M
{\displaystyle TM}
, orthonormal with respect to the given metric tensor
g
{\displaystyle g}
:
(
⋆
α
)
(
X
1
,
…
,
X
n
−
k
)
=
α
(
X
n
−
k
+
1
,
…
,
X
n
)
.
{\displaystyle ({\star }\alpha )(X_{1},\ldots ,X_{n-k})=\alpha (X_{n-k+1},\ldots ,X_{n}).}
=== Co-differential operator === The co-differential operator
δ
:
Ω
k
(
M
)
→
Ω
k
−
1
(
M
)
{\displaystyle \delta :\Omega ^{k}(M)\rightarrow \Omega ^{k-1}(M)}
on an
n
{\displaystyle n}
dimensional manifold
M
{\displaystyle M}
is defined by
δ
:=
(
−
1
)
k
⋆
−
1
d
⋆
=
(
−
1
)
n
k
+
n
+
1
⋆
d
⋆
.
{\displaystyle \delta :=(-1)^{k}{\star }^{-1}d{\star }=(-1)^{nk+n+1}{\star }d{\star }.}
The Hodge–Dirac operator,
d
+
δ
{\displaystyle d+\delta }
, is a Dirac operator studied in Clifford analysis.
=== Oriented manifold === An
n
{\displaystyle n}
-dimensional orientable manifold M is a manifold that can be equipped with a choice of an n-form
μ
∈
Ω
n
(
M
)
{\displaystyle \mu \in \Omega ^{n}(M)}
that is continuous and nonzero everywhere on M.
=== Volume form === On an orientable manifold
M
{\displaystyle M}
the canonical choice of a volume form given a metric tensor
g
{\displaystyle g}
and an orientation is
d
e
t
:=
|
det
g
|
d
X
1
♭
∧
…
∧
d
X
n
♭
{\displaystyle \mathbf {det} :={\sqrt {|\det g|}}\;dX_{1}^{\flat }\wedge \ldots \wedge dX_{n}^{\flat }}
for any basis
d
X
1
,
…
,
d
X
n
{\displaystyle dX_{1},\ldots ,dX_{n}}
ordered to match the orientation.
=== Area form === Given a volume form
d
e
t
{\displaystyle \mathbf {det} }
and a unit normal vector
N
{\displaystyle N}
we can also define an area form
σ
:=
ι
N
det
{\displaystyle \sigma :=\iota _{N}{\textbf {det}}}
on the boundary
∂
M
.
{\displaystyle \partial M.}
=== Bilinear form on k-forms === A generalization of the metric tensor, the symmetric bilinear form between two
k
{\displaystyle k}
-forms
α
,
β
∈
Ω
k
(
M
)
{\displaystyle \alpha ,\beta \in \Omega ^{k}(M)}
, is defined pointwise on
M
{\displaystyle M}
by
⟨
α
,
β
⟩
|
p
:=
⋆
(
α
∧
⋆
β
)
|
p
.
{\displaystyle \langle \alpha ,\beta \rangle |_{p}:={\star }(\alpha \wedge {\star }\beta )|_{p}.}
The
L
2
{\displaystyle L^{2}}
-bilinear form for the space of
k
{\displaystyle k}
-forms
Ω
k
(
M
)
{\displaystyle \Omega ^{k}(M)}
is defined by
⟨
⟨
α
,
β
⟩
⟩
:=
∫
M
α
∧
⋆
β
.
{\displaystyle \langle \!\langle \alpha ,\beta \rangle \!\rangle :=\int _{M}\alpha \wedge {\star }\beta .}
In the case of a Riemannian manifold, each is an inner product (i.e. is positive-definite).
=== Lie derivative === We define the Lie derivative
L
:
Ω
k
(
M
)
→
Ω
k
(
M
)
{\displaystyle {\mathcal {L}}:\Omega ^{k}(M)\rightarrow \Omega ^{k}(M)}
through Cartan's magic formula for a given section
X
∈
Γ
(
T
M
)
{\displaystyle X\in \Gamma (TM)}
as
L
X
=
d
∘
ι
X
+
ι
X
∘
d
.
{\displaystyle {\mathcal {L}}_{X}=d\circ \iota _{X}+\iota _{X}\circ d.}
It describes the change of a
k
{\displaystyle k}
-form along a flow
ϕ
t
{\displaystyle \phi _{t}}
associated to the section
X
{\displaystyle X}
.
=== Laplace–Beltrami operator === The Laplacian
Δ
:
Ω
k
(
M
)
→
Ω
k
(
M
)
{\displaystyle \Delta :\Omega ^{k}(M)\rightarrow \Omega ^{k}(M)}
is defined as
Δ
=
−
(
d
δ
+
δ
d
)
{\displaystyle \Delta =-(d\delta +\delta d)}
.
== Important definitions ==
=== Definitions on Ωk(M) ===
α
∈
Ω
k
(
M
)
{\displaystyle \alpha \in \Omega ^{k}(M)}
is called...
closed if
d
α
=
0
{\displaystyle d\alpha =0}
exact if
α
=
d
β
{\displaystyle \alpha =d\beta }
for some
β
∈
Ω
k
−
1
{\displaystyle \beta \in \Omega ^{k-1}}
coclosed if
δ
α
=
0
{\displaystyle \delta \alpha =0}
coexact if
α
=
δ
β
{\displaystyle \alpha =\delta \beta }
for some
β
∈
Ω
k
+
1
{\displaystyle \beta \in \Omega ^{k+1}}
harmonic if closed and coclosed