--- title: "Exterior calculus identities" chunk: 4/7 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" --- === 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 ===