--- title: "Exterior calculus identities" chunk: 3/7 source: "https://en.wikipedia.org/wiki/Exterior_calculus_identities" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T08:14:13.652508+00:00" instance: "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