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c partial b partial x frac partial a partial y right in the third case f a b c displaystyle mathbf f a b c again corresponds to φ a d x b d y c d z displaystyle varphi a dx b dy c dz applying hodge star exterior derivative and hodge star again φ a d y d z b d x d z c d x d y d φ a x b y c z d x d y d z d φ a x b y c z div f displaystyle begin aligned star varphi a dy wedge dz b dx wedge dz c dx wedge dy d star varphi left frac partial a partial x frac partial b partial y frac partial c partial z right dx wedge dy wedge dz star d star varphi frac partial a partial x frac partial b partial y frac partial c partial z operatorname div mathbf f end aligned one advantage of this expression is that the identity d 2 0 which is true in all cases has as special cases two other identities 1 curl grad f 0 and 2 div curl f 0 in particular maxwell s equations take on a particularly simple and elegant form when expressed in terms of the exterior derivative and the hodge star the expression d displaystyle star d star multiplied by an appropriate power of 1 is called the codifferential it is defined in full generality for any dimension further in the article below one can also obtain the laplacian δ f div grad f in terms of the above operations δ f d d f 2 f x 2 2 f y 2 2 f z 2 displaystyle delta f star d star df frac partial 2 f partial x 2 frac partial 2 f partial y 2 frac partial 2 f partial z 2 the laplacian can also be seen as a special case of the more general laplace derham operator δ d δ δ d displaystyle delta d delta delta d where in three dimensions δ 1 k d displaystyle delta 1 k star d star is the codifferential for k displaystyle k forms any function f displaystyle f is a 0 form and δ f 0 displaystyle delta f 0 and so this reduces to the ordinary laplacian for the 1 form φ displaystyle varphi above the codifferential is δ d displaystyle delta star d star and after some straightforward calculations one obtains the laplacian acting on φ displaystyle varphi duality edit when the bilinear form is nondegenerate applying the hodge star twice leaves a k vector unchanged up to a sign for η k v displaystyle eta in textstyle bigwedge k v in an n dimensional space v one has η 1 k n k s η displaystyle star star eta 1 k n k s eta where s is the parity of the signature of the scalar product on v that is the sign of the determinant of the matrix of the scalar product with respect to any basis for example if n 4 and the signature of the scalar product is either or then s 1 for riemannian manifolds including euclidean spaces we always have s 1 the above identity implies that the inverse of displaystyle star can be given as 1 k v n k v η 1 k n k s η displaystyle begin aligned star 1 textstyle bigwedge k v to textstyle bigwedge n k v eta mapsto 1 k n k s star eta end aligned if n is odd then k n k is even for any k whereas if n is even then k n k has the parity of k therefore 1 s n is odd 1 k s n is even displaystyle star 1 begin cases s star n text is odd 1 k s star n text is even end cases where k is the degree of the element operated on on manifolds edit for an n dimensional oriented pseudo riemannian manifold m we apply the construction above to each cotangent space t p m displaystyle text t _ p m and its exterior powers k t p m textstyle bigwedge k text t _ p m and hence to the differential k forms ζ ω k m γ k t m textstyle zeta in omega k m gamma left bigwedge k text t m right the global sections of the bundle k t m m textstyle bigwedge k mathrm t m to m the riemannian metric induces a scalar product on k t p m textstyle bigwedge k text t _ p m at each point p m displaystyle p in m we define the hodge dual of a k form ζ displaystyle zeta defining ζ displaystyle star zeta as the unique n k form satisfying η ζ η ζ ω displaystyle eta wedge star zeta langle eta zeta rangle omega for every k form η displaystyle eta where η ζ displaystyle langle eta zeta rangle is a real valued function on m displaystyle m and the volume form ω displaystyle omega is induced by the pseudo riemannian metric integrating this equation over m displaystyle m the right side becomes the l 2 displaystyle l 2 square integrable scalar product on k forms and we obtain m η ζ m η ζ ω displaystyle int _ m eta wedge star zeta int _ m langle eta zeta rangle omega more generally if m displaystyle m is non orientable one can define the hodge star of a k form as a n k pseudo differential form that is a differential form with values in the canonical line bundle computation in index notation edit we compute in terms of tensor index notation with respect to a not necessarily orthonormal basis x 1 x n textstyle left frac partial partial x_ 1 ldots frac partial partial x_ n right in a tangent space v t p m displaystyle v t_ p m and its dual basis d x 1 d x n displaystyle dx_ 1 ldots dx_ n in v t p m displaystyle v t_ p m having the metric matrix g i j x i x j textstyle g_ ij left left langle frac partial partial x_ i frac partial partial x_ j right rangle right and its inverse matrix g i j d x i d x j displaystyle g ij langle dx i dx j rangle the hodge dual of a decomposable k form is d x i 1 d x i k det g i j n k g i 1 j 1 g i k j k ε j 1 j n d x j k 1 d x j n displaystyle star left dx i_ 1 wedge dots wedge dx i_ k right frac sqrt left det g_ ij right n k g i_ 1 j_ 1 cdots g i_ k j_ k varepsilon _ j_ 1 dots j_ n dx j_ k 1 wedge dots wedge dx j_ n here ε j 1 j n displaystyle varepsilon _ j_ 1 dots j_ n is the levi civita symbol with ε 1 n 1 displaystyle varepsilon _ 1 dots n 1 and we implicitly take the sum over all values of the repeated indices j 1 j n displaystyle j_ 1 ldots j_ n the factorial n k displaystyle n k accounts for double counting and is not present if the summation indices are restricted so that j k 1 j n displaystyle j_ k 1 dots j_ n the absolute value of the determinant is necessary since it may be negative as for tangent spaces to lorentzian manifolds an arbitrary differential form can be written as follows α 1 k α i 1 i k d x i 1 d x i k i 1 i k α i 1 i k d x i 1 d x i k displaystyle alpha frac 1 k alpha _ i_ 1 dots i_ k dx i_ 1 wedge dots wedge dx i_ k sum _ i_ 1 dots i_ k alpha _ i_ 1 dots i_ k dx i_ 1 wedge dots wedge dx i_ k the factorial k displaystyle k is again included to account for double counting when we allow non increasing indices we would like to define the dual of the component α i 1 i k displaystyle alpha _ i_ 1 dots i_ k so that the hodge dual of the form is given by α 1 n k α i k 1 i n d x i k 1 d x i n displaystyle star alpha frac 1 n k star alpha _ i_ k 1 dots i_ n dx i_ k 1 wedge dots wedge dx i_ n using the above expression for the hodge dual of d x i 1 d x i k displaystyle dx i_ 1 wedge dots wedge dx i_ k we find 4 α j k 1 j n det g a b k α i 1 i k g i 1 j 1 g i k j k ε j 1 j n displaystyle star alpha _ j_ k 1 dots j_ n frac sqrt left det g_ ab right k alpha _ i_ 1 dots i_ k g i_ 1 j_ 1 cdots g i_ k j_ k varepsilon _ j_ 1 dots j_ n although one can apply this expression to any tensor α displaystyle alpha the result is antisymmetric since contraction with the completely anti symmetric levi civita symbol cancels all but the totally antisymmetric part of the tensor it is thus equivalent to antisymmetrization followed by applying the hodge star the unit volume form ω 1 n v textstyle omega star 1 in bigwedge n v is given by ω det g i j d x 1 d x n displaystyle omega sqrt left det g_ ij right dx 1 wedge cdots wedge dx n codifferential edit the most important application of the hodge star on manifolds is to define the codifferential δ displaystyle delta on k displaystyle k forms let δ 1 n k 1 1 s d 1 k 1 d displaystyle delta 1 n k 1 1 s star d star 1 k star 1 d star where d displaystyle d is the exterior derivative or differential and s 1 displaystyle s 1 for riemannian manifolds then d ω k m ω k 1 m displaystyle d omega k m to omega k 1 m while δ ω k m ω k 1 m displaystyle delta omega k m to omega k 1 m the codifferential is not an antiderivation on the exterior algebra in contrast to the exterior derivative the codifferential is the adjoint of the exterior derivative with respect to the square integrable scalar product η δ ζ d η ζ displaystyle langle langle eta delta zeta rangle rangle langle langle d eta zeta rangle rangle where ζ displaystyle zeta is a k displaystyle k form and η displaystyle eta a k 1 displaystyle k 1 form this property is useful as it can be used to define the codifferential even when the manifold is non orientable and the hodge star operator not defined the identity can be proved from stokes theorem for smooth forms 0 m d η ζ m d η ζ 1 k 1 η 1 d ζ d η ζ η δ ζ displaystyle 0 int _ m d eta wedge star zeta int _ m left d eta wedge star zeta 1 k 1 eta wedge star star 1 d star zeta right langle langle d eta zeta rangle rangle langle langle eta delta zeta rangle rangle provided m displaystyle m has empty boundary or η displaystyle eta or ζ displaystyle star zeta has zero boundary values the proper definition of the above requires specifying a topological vector space that is closed and complete on the space of smooth forms the sobolev space is conventionally used it allows the convergent sequence of forms ζ i ζ displaystyle zeta _ i to zeta as i displaystyle i to infty to be interchanged with the combined differential and integral operations so that η δ ζ i η δ ζ displaystyle langle langle eta delta zeta _ i rangle rangle to langle langle eta delta zeta rangle rangle and likewise for sequences converging to η displaystyle eta since the differential satisfies d 2 0 displaystyle d 2 0 the codifferential has the corresponding property δ 2 1 n s 2 d d 1 n k k 1 s 3 d 2 0 displaystyle delta 2 1 n s 2 star d star star d star 1 nk k 1 s 3 star d 2 star 0 the laplace derham operator is given by δ δ d 2 δ d d δ displaystyle delta delta d 2 delta d d delta and lies at the heart of hodge theory it is symmetric δ ζ η ζ δ η displaystyle langle langle delta zeta eta rangle rangle langle langle zeta delta eta rangle rangle and non negative δ η η 0 displaystyle langle langle delta eta eta rangle rangle geq 0 the hodge star sends harmonic forms to harmonic forms as a consequence of hodge theory the de rham cohomology is naturally isomorphic to the space of harmonic k forms and so the hodge star induces an isomorphism of cohomology groups h δ k m h δ n k m displaystyle star h_ delta k m to h_ delta n k m which in turn gives canonical identifications via poincaré duality of h k m with its dual space in coordinates with notation as above the codifferential of the form α displaystyle alpha may be written as δ α 1 k g m l x l α m i 1 i k 1 γ m l j α j i 1 i k 1 d x i 1 d x i k 1 displaystyle delta alpha frac 1 k g ml left frac partial partial x_ l alpha _ m i_ 1 dots i_ k 1 gamma _ ml j alpha _ j i_ 1 dots i_ k 1 right dx i_ 1 wedge dots wedge dx i_ k 1 where here γ m l j displaystyle gamma _ ml j denotes the christoffel symbols of x 1 x n textstyle left frac partial partial x_ 1 ldots frac partial partial x_ n right poincaré lemma for codifferential edit in analogy to the poincaré lemma for exterior derivative one can define its version for codifferential which reads 5 if δ ω 0 displaystyle delta omega 0 for ω λ k u displaystyle omega in lambda k u where u displaystyle u is a star domain on a manifold then there is α λ k 1 u displaystyle alpha in lambda k 1 u such that ω δ α displaystyle omega delta alpha a practical way of finding α displaystyle alpha is to use cohomotopy operator h displaystyle h that is a local inverse of δ displaystyle delta one has to define a homotopy operator 5 h β 0 1 k β f t x t k d t displaystyle h beta int _ 0 1 mathcal k lrcorner beta _ f t x t k dt where f t x x 0 t x x 0 displaystyle f t x x_ 0 t x x_ 0 is the linear homotopy between its center x 0 u displaystyle x_ 0 in u and a point x u displaystyle x in u and the euler vector k i 1 n x x 0 i x i displaystyle mathcal k sum _ i 1 n x x_ 0 i partial _ x i for n dim u displaystyle n dim u is inserted into the form β λ u displaystyle beta in lambda u we can then define cohomotopy operator as 5 h λ u λ u h η 1 h displaystyle h lambda u rightarrow lambda u quad h eta star 1 h star where η β 1 k β displaystyle eta beta 1 k beta for β λ k u displaystyle beta in lambda k u the cohomotopy operator fulfills co homotopy invariance formula 5 δ h h δ i s x 0 displaystyle delta h h delta i s_ x_ 0 where s x 0 1 s x 0 displaystyle s_ x_ 0 star 1 s_ x_ 0 star and s x 0 displaystyle s_ x_ 0 is the pullback along the constant map s x 0 x x 0 displaystyle s_ x_ 0 x rightarrow x_ 0 therefore if we want to solve the equation δ ω 0 displaystyle delta omega 0 applying cohomotopy invariance formula we get ω δ h ω s x 0 ω displaystyle omega delta h omega s_ x_ 0 omega where h ω λ k 1 u displaystyle h omega in lambda k 1 u is a differential form we are looking for and constant of integration s x 0 ω displaystyle s_ x_ 0 omega vanishes unless ω displaystyle omega is a top form cohomotopy operator fulfills the following properties 5 h 2 0 δ h δ δ h δ h h displaystyle h 2 0 quad delta h delta delta quad h delta h h they make it possible to use it to define 5 anticoexact forms on u displaystyle u by y u ω λ u ω h δ ω displaystyle mathcal y u omega in lambda u omega h delta omega which together with exact forms c u ω λ u ω δ h ω displaystyle mathcal c u omega in lambda u omega delta h omega make a direct sum decomposition 5 λ u c u y u displaystyle lambda u mathcal c u oplus mathcal y u this direct sum is another way of saying that the cohomotopy invariance formula is a decomposition of unity and the projector operators on the summands fulfills idempotence formulas 5 h δ 2 h δ δ h 2 δ h displaystyle h delta 2 h delta quad delta h 2 delta h these results are extension of similar results for exterior derivative 6 citations edit 1 2 harley flanders 1963 differential forms with applications to the physical sciences academic press eric lengyel 2024 projective geometric algebra illuminated terathon software p 81 isbn 979 8 9853582 5 4 1 2 pertti lounesto 2001 3 6 the hodge dual clifford algebras and spinors volume 286 of london mathematical society lecture note series 2nd ed cambridge university press p 39 isbn 0 521 00551 5 frankel t 2012 the geometry of physics 3rd ed cambridge university press isbn 978 1 107 60260 1 1 2 3 4 5 6 7 8 kycia radosław antoni 2022 07 29 the poincare lemma for codifferential anticoexact forms and applications to physics results in mathematics 77 5 182 arxiv 2009 08542 doi 10 1007 s00025 022 01646 z issn 1420 9012 s2cid 221802588 edelen dominic g b 2005 applied exterior calculus revised ed mineola n y isbn 978 0 486 43871 9 oclc 56347718 cite book cs1 maint location missing publisher link references edit david bleecker 1981 gauge theory and variational principles addison wesley publishing isbn 0 201 10096 7 chpt 0 contains a condensed review of non riemannian differential geometry jost jürgen 2002 riemannian geometry and geometric analysis springer verlag isbn 3 540 426...