This article summarizes several identities in exterior calculus, a mathematical calculus used in differential geometry.
Notation The following notation is used in this article.
Manifold
M {\displaystyle M} , N {\displaystyle N} are n {\displaystyle n} -dimensional smooth manifolds, where n ∈ N {\displaystyle n\in \mathbb {N} } . That is, differentiable manifolds that can be differentiated enough times for the purposes on this page.
p ∈ M {\displaystyle p\in M} , q ∈ N {\displaystyle q\in N} denote one point on each of the manifolds. The boundary of a manifold M {\displaystyle M} is a manifold ∂ M {\displaystyle \partial M} , which has dimension n − 1 {\displaystyle n-1} . An orientation on M {\displaystyle M} induces an orientation on ∂ M {\displaystyle \partial M} . We usually denote a submanifold by Σ ⊂ M {\displaystyle \Sigma \subset M} .
Tangent and cotangent bundles
T M {\displaystyle TM} , T ∗ M {\displaystyle T^{*}M} denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold M {\displaystyle M} .
T p M {\displaystyle T_{p}M} , T q N {\displaystyle T_{q}N} denote the tangent spaces of M {\displaystyle M} , N {\displaystyle N} at the points p {\displaystyle p} , q {\displaystyle q} , respectively. T p ∗ M {\displaystyle T_{p}^{*}M} denotes the cotangent space of M {\displaystyle M} at the point p {\displaystyle p} . Sections of the tangent bundles, also known as vector fields, are typically denoted as X , Y , Z ∈ Γ ( T M ) {\displaystyle X,Y,Z\in \Gamma (TM)} such that at a point p ∈ M {\displaystyle p\in M} we have X | p , Y | p , Z | p ∈ T p M {\displaystyle X|_{p},Y|_{p},Z|_{p}\in T_{p}M} . Sections of the cotangent bundle, also known as differential 1-forms (or covector fields), are typically denoted as α , β ∈ Γ ( T ∗ M ) {\displaystyle \alpha ,\beta \in \Gamma (T^{*}M)} such that at a point p ∈ M {\displaystyle p\in M} we have α | p , β | p ∈ T p ∗ M {\displaystyle \alpha |_{p},\beta |_{p}\in T_{p}^{*}M} . An alternative notation for Γ ( T ∗ M ) {\displaystyle \Gamma (T^{*}M)} is Ω 1 ( M ) {\displaystyle \Omega ^{1}(M)} .
Differential k-forms Differential k {\displaystyle k} -forms, which we refer to simply as k {\displaystyle k} -forms here, are differential forms defined on T M {\displaystyle TM} . We denote the set of all k {\displaystyle k} -forms as Ω k ( M ) {\displaystyle \Omega ^{k}(M)} . For 0 ≤ k , l , m ≤ n {\displaystyle 0\leq k,\ l,\ m\leq n} we usually write α ∈ Ω k ( M ) {\displaystyle \alpha \in \Omega ^{k}(M)} , β ∈ Ω l ( M ) {\displaystyle \beta \in \Omega ^{l}(M)} , γ ∈ Ω m ( M ) {\displaystyle \gamma \in \Omega ^{m}(M)} .
0 {\displaystyle 0} -forms f ∈ Ω 0 ( M ) {\displaystyle f\in \Omega ^{0}(M)} are just scalar functions C ∞ ( M ) {\displaystyle C^{\infty }(M)} on M {\displaystyle M} . 1 ∈ Ω 0 ( M ) {\displaystyle \mathbf {1} \in \Omega ^{0}(M)} denotes the constant 0 {\displaystyle 0} -form equal to 1 {\displaystyle 1} everywhere.
Omitted elements of a sequence When we are given ( k + 1 ) {\displaystyle (k+1)} inputs X 0 , … , X k {\displaystyle X_{0},\ldots ,X_{k}} and a k {\displaystyle k} -form α ∈ Ω k ( M ) {\displaystyle \alpha \in \Omega ^{k}(M)} we denote omission of the i {\displaystyle i} th entry by writing
α ( X 0 , … , X ^ i , … , X k ) := α ( X 0 , … , X i − 1 , X i + 1 , … , X k ) . {\displaystyle \alpha (X_{0},\ldots ,{\hat {X}}_{i},\ldots ,X_{k}):=\alpha (X_{0},\ldots ,X_{i-1},X_{i+1},\ldots ,X_{k}).}
Exterior product The exterior product is also known as the wedge product. It is denoted by ∧ : Ω k ( M ) × Ω l ( M ) → Ω k + l ( M ) {\displaystyle \wedge :\Omega ^{k}(M)\times \Omega ^{l}(M)\rightarrow \Omega ^{k+l}(M)} . The exterior product of a k {\displaystyle k} -form α ∈ Ω k ( M ) {\displaystyle \alpha \in \Omega ^{k}(M)} and an l {\displaystyle l} -form β ∈ Ω l ( M ) {\displaystyle \beta \in \Omega ^{l}(M)} produce a ( k + l ) {\displaystyle (k+l)} -form α ∧ β ∈ Ω k + l ( M ) {\displaystyle \alpha \wedge \beta \in \Omega ^{k+l}(M)} . It can be written using the set S ( k , k + l ) {\displaystyle S(k,k+l)} of all permutations σ {\displaystyle \sigma } of { 1 , … , n } {\displaystyle \{1,\ldots ,n\}} such that σ ( 1 ) < … < σ ( k ) , σ ( k + 1 ) < … < σ ( k + l ) {\displaystyle \sigma (1)<\ldots <\sigma (k),\ \sigma (k+1)<\ldots <\sigma (k+l)} as
( α ∧ β ) ( X 1 , … , X k + l ) = ∑ σ ∈ S ( k , k + l ) sign ( σ ) α ( X σ ( 1 ) , … , X σ ( k ) ) ⊗ β ( X σ ( k + 1 ) , … , X σ ( k + l ) ) . {\displaystyle (\alpha \wedge \beta )(X_{1},\ldots ,X_{k+l})=\sum _{\sigma \in S(k,k+l)}{\text{sign}}(\sigma )\alpha (X_{\sigma (1)},\ldots ,X_{\sigma (k)})\otimes \beta (X_{\sigma (k+1)},\ldots ,X_{\sigma (k+l)}).}
Directional derivative The directional derivative of a 0-form f ∈ Ω 0 ( M ) {\displaystyle f\in \Omega ^{0}(M)} along a section X ∈ Γ ( T M ) {\displaystyle X\in \Gamma (TM)} is a 0-form denoted ∂ X f . {\displaystyle \partial _{X}f.}
Exterior derivative The exterior derivative d k : Ω k ( M ) → Ω k + 1 ( M ) {\displaystyle d_{k}:\Omega ^{k}(M)\rightarrow \Omega ^{k+1}(M)} is defined for all 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} . We generally omit the subscript when it is clear from the context. For a 0 {\displaystyle 0} -form f ∈ Ω 0 ( M ) {\displaystyle f\in \Omega ^{0}(M)} we have d 0 f ∈ Ω 1 ( M ) {\displaystyle d_{0}f\in \Omega ^{1}(M)} as the 1 {\displaystyle 1} -form that gives the directional derivative, i.e., for the section X ∈ Γ ( T M ) {\displaystyle X\in \Gamma (TM)} we have ( d 0 f ) ( X ) = ∂ X f {\displaystyle (d_{0}f)(X)=\partial _{X}f} , the directional derivative of f {\displaystyle f} along X {\displaystyle X} . For 0 < k ≤ n {\displaystyle 0<k\leq n} ,
( d k ω ) ( X 0 , … , X k ) = ∑ 0 ≤ j ≤ k ( − 1 ) j d 0 ( ω ( X 0 , … , X ^ j , … , X k ) ) ( X j ) + ∑ 0 ≤ i < j ≤ k ( − 1 ) i + j ω ( [ X i , X j ] , X 0 , … , X ^ i , … , X ^ j , … , X k ) . {\displaystyle (d_{k}\omega )(X_{0},\ldots ,X_{k})=\sum _{0\leq j\leq k}(-1)^{j}d_{0}(\omega (X_{0},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}))(X_{j})+\sum _{0\leq i<j\leq k}(-1)^{i+j}\omega ([X_{i},X_{j}],X_{0},\ldots ,{\hat {X}}_{i},\ldots ,{\hat {X}}_{j},\ldots ,X_{k}).}
Lie bracket The Lie bracket of sections X , Y ∈ Γ ( T M ) {\displaystyle X,Y\in \Gamma (TM)} is defined as the unique section [ X , Y ] ∈ Γ ( T M ) {\displaystyle [X,Y]\in \Gamma (TM)} that satisfies
∀ f ∈ Ω 0 ( M ) ⇒ ∂ [ X , Y ] f = ∂ X ∂ Y f − ∂ Y ∂ X f . {\displaystyle \forall f\in \Omega ^{0}(M)\Rightarrow \partial _{[X,Y]}f=\partial _{X}\partial _{Y}f-\partial _{Y}\partial _{X}f.}
Tangent maps If ϕ : M → N {\displaystyle \phi :M\rightarrow N} is a smooth map, then d ϕ | p : T p M → T ϕ ( p ) N {\displaystyle d\phi |_{p}:T_{p}M\rightarrow T_{\phi (p)}N} defines a tangent map from M {\displaystyle M} to N {\displaystyle N} . It is defined through curves γ {\displaystyle \gamma } on M {\displaystyle M} with derivative γ ′ ( 0 ) = X ∈ T p M {\displaystyle \gamma '(0)=X\in T_{p}M} such that
d ϕ ( X ) := ( ϕ ∘ γ ) ′ . {\displaystyle d\phi (X):=(\phi \circ \gamma )'.}
Note that ϕ {\displaystyle \phi } is a 0 {\displaystyle 0} -form with values in N {\displaystyle N} .
Pull-back If ϕ : M → N {\displaystyle \phi :M\rightarrow N} is a smooth map, then the pull-back of a k {\displaystyle k} -form α ∈ Ω k ( N ) {\displaystyle \alpha \in \Omega ^{k}(N)} is defined such that for any k {\displaystyle k} -dimensional submanifold Σ ⊂ M {\displaystyle \Sigma \subset M}
∫ Σ ϕ ∗ α = ∫ ϕ ( Σ ) α . {\displaystyle \int _{\Sigma }\phi ^{*}\alpha =\int _{\phi (\Sigma )}\alpha .}
The pull-back can also be expressed as
( ϕ ∗ α ) ( X 1 , … , X k ) = α ( d ϕ ( X 1 ) , … , d ϕ ( X k ) ) . {\displaystyle (\phi ^{*}\alpha )(X_{1},\ldots ,X_{k})=\alpha (d\phi (X_{1}),\ldots ,d\phi (X_{k})).}
Interior product Also known as the interior derivative, the interior product given a section Y ∈ Γ ( T M ) {\displaystyle Y\in \Gamma (TM)} is a map ι Y : Ω k + 1 ( M ) → Ω k ( M ) {\displaystyle \iota _{Y}:\Omega ^{k+1}(M)\rightarrow \Omega ^{k}(M)} that effectively substitutes the first input of a ( k + 1 ) {\displaystyle (k+1)} -form with Y {\displaystyle Y} . If α ∈ Ω k + 1 ( M ) {\displaystyle \alpha \in \Omega ^{k+1}(M)} and X i ∈ Γ ( T M ) {\displaystyle X_{i}\in \Gamma (TM)} then
( ι Y α ) ( X 1 , … , X k ) = α ( Y , X 1 , … , X k ) . {\displaystyle (\iota _{Y}\alpha )(X_{1},\ldots ,X_{k})=\alpha (Y,X_{1},\ldots ,X_{k}).}
Metric tensor Given a nondegenerate bilinear form g p ( ⋅ , ⋅ ) {\displaystyle g_{p}(\cdot ,\cdot )} on each T p M {\displaystyle T_{p}M} that is continuous on M {\displaystyle M} , the manifold becomes a pseudo-Riemannian manifold. We denote the metric tensor g {\displaystyle g} , defined pointwise by g ( X , Y ) | p = g p ( X | p , Y | p ) {\displaystyle g(X,Y)|_{p}=g_{p}(X|_{p},Y|_{p})} . We call s = sign ( g ) {\displaystyle s=\operatorname {sign} (g)} the signature of the metric. A Riemannian manifold has s = 1 {\displaystyle s=1} , whereas Minkowski space has s = − 1 {\displaystyle s=-1} .
Musical isomorphisms The metric tensor g ( ⋅ , ⋅ ) {\displaystyle g(\cdot ,\cdot )} induces duality mappings between vector fields and one-forms: these are the musical isomorphisms flat ♭ {\displaystyle \flat } and sharp ♯ {\displaystyle \sharp } . A section A ∈ Γ ( T M ) {\displaystyle A\in \Gamma (TM)} corresponds to the unique one-form A ♭ ∈ Ω 1 ( M ) {\displaystyle A^{\flat }\in \Omega ^{1}(M)} such that for all sections X ∈ Γ ( T M ) {\displaystyle X\in \Gamma (TM)} , we have:
A ♭ ( X ) = g ( A , X ) . {\displaystyle A^{\flat }(X)=g(A,X).}
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)} .
Importa