In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed...

similar duality in higher dimensions; this duality is now known as the Hodge star operator. He further conjectured that each cohomology class should have a...

provides the Hodge star operator ⋆ {\displaystyle \star } and thus makes it possible to define J = ⋆ d ⋆ F {\displaystyle J={\star }d{\star }F} or the equivalent...

the Hodge star and the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian...

same as the tensor algebra on the vector space with X as basis. Hodge star operator Exterior power The wedge product is the anti-symmetric form of the...

}&=\mathbf {0} \\d{\star \mathbf {F} }&=\mathbf {J} ,\end{aligned}}} where ⋆ {\displaystyle \star } denotes the Hodge star operator. Similar considerations...

Hodge star operator maps two-forms to two-forms, ⋆ : Ω 2 ( X ) → Ω 2 ( X ) {\displaystyle \star :\Omega ^{2}(X)\to \Omega ^{2}(X)} . The Hodge star operator...

{ ∗ } {\displaystyle \{\ast \}} . as a unary operator, denoted in prefix notation The Hodge star operator on vector spaces ∗ : A k → A n − k {\displaystyle...

(Gauss's law and the Ampère-Maxwell equation), the Hodge dual of this 2-form is needed. The Hodge star operator takes a p-form to a (n − p)-form, where n is...

{\displaystyle d^{*}=-(-1)^{n(r+1)}\star d\,\star } , where ⋆ {\displaystyle \star } is the Hodge star operator. (Equivalently, d ∗ {\displaystyle d^{*}}...

In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional...

further. In this case, φ can be understood as the composite of the Hodge star operator and dualization. Specifically, if ω is the volume form, then it,...

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of...

{\displaystyle A} on the adjoint bundle, and ⋆ {\displaystyle \star } is the Hodge star operator on M {\displaystyle M} . These equations are named after E...

{\left(F^{\flat }\right)}}\right)^{\sharp },\\\end{array}}} where ⋆ is the Hodge star operator, ♭ and ♯ are the musical isomorphisms,  f  is a scalar field and...

{\displaystyle \star d\star } with d {\displaystyle d} the differential and ⋆ {\displaystyle \star } the Hodge star. The Hodge star, by its construction...

the current 1-form, F is the field strength 2-form and the star denotes the Hodge star operator. This is exactly the same Lagrangian as in the section above...

linear endomorphism. Sometimes the term linear operator refers to this case, but the term "linear operator" can have different meanings for different conventions:...

star ⋆ {\displaystyle \star } denotes the Hodge star operator. The dependence of Maxwell's equation on the metric of spacetime lies in the Hodge star...

rightmost formulation follows from the transformation properties of the Hodge star operator. Note that on restriction to the even Clifford algebra, the paired...

operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative Exterior derivative Exterior product Hodge star...

material (i.e. in rest frame of material), ⋆ {\displaystyle \star } and denotes the Hodge star operator. The Lagrangian density for classical electrodynamics...

product of their lengths). The name "dot product" is derived from the dot operator " ⋅ " that is often used to designate this operation; the alternative name...

notation Flat (music) and Sharp (music) about the signs ♭ and ♯ Hodge star operator Metric tensor Vector bundle Lee 2003, Chapter 11. Lee 1997, Chapter...

phrased equivalently in terms of the exterior derivative and the Hodge star operator ∗. u and v will be isothermal coordinates if ∗du = dv, where ∗ is...

{\displaystyle \operatorname {ad} (P)} and ⋆ {\displaystyle \star } is the Hodge star operator. Such solutions are called Yang–Mills connections and are...

identities Harmonic function Helmholtz decomposition Hessian matrix Hodge star operator Inverse function theorem Irrotational vector field Isoperimetry Jacobian...

if one defines the conjugate-linear Hodge star operator by ⋆ ¯ ω = ⋆ ω ¯ {\displaystyle {\bar {\star }}\omega =\star {\bar {\omega }}} then we have: ⋆ ¯...

coordinates Differential form Differential geometry Exterior algebra Hodge star operator Holonomic basis Matrix calculus Metric tensor Multilinear algebra...

Alfred Clement (1912). An Introduction to Algebraical Geometry. Clarendon. Hodge, W.V.D.; D. Pedoe (1994) [1947]. Methods of Algebraic Geometry, Volume I...