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Vector analysis
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Differential forms and the exterior derivative operator d simplify many of the ideas in vector analysis. There is a second operator *, called the Hodge star operator. It takes k-forms to (n − k)-forms and ** = ±1. On a k-form in ℝn, we have ** = (−1)k(n − k). If one thinks of a k-form ω as describing a k-dimensional plane, then *ω describes the orthogonal (n − k)-dimensional plane. The volume form (commonly written dV, but not really d of something!) can be regarded as *(1). We also have 1 = *(dV). The operator * is linear. Thus *(ω + η) = * ω + *η and *(fω) = f(*ω) when f is a function and ω is a differential form.
Transformation Optics in a Nutshell
Published in Didier Felbacq, Guy Bouchitté, Metamaterials Modeling and Design, 2017
In the exterior calculus formalism, the only task associated to changing a coordinate system is to determine an explicit expression for the Hodge star operator (Nicolet et al., 1994, 2010; Zolla et al., 2005). A very useful point of view is to consider weak formulations where integrals of volume forms (3-forms) are built with scalar products of forms, i.e., exterior products together with the Hodge operator acting on one of the factors.
A geometric look at MHD and the Braginsky dynamo
Published in Geophysical & Astrophysical Fluid Dynamics, 2021
Andrew D. Gilbert, Jacques Vanneste
Using these and similar rules, we can replace quantities in the above integral by their equivalents. To keep notation light we denote this by ≃. We define the key quantities, the 1-forms and where ⋆ is the Hodge star operator. We then have for the kinetic energy terms, using (15) for (37). For the internal energy terms, we have where we have used thermodynamic relations for pressure p, temperature T and enthalpy h, Finally, for the magnetic terms we have using that . These expressions are inserted into (32) which must hold for arbitrary w, and from this we obtain the momentum equation in the form for ideal MHD flow in a general setting.
Electromagnetic curves of the linearly polarized light wave along an optical fiber in a 3D semi-Riemannian manifold
Published in Journal of Modern Optics, 2019
In the case of the semi-Riemannian manifold vector fields and 2-forms can be described via the volume form and the Hodge star operator ★ of the manifold. Hence, divergence-free vector fields and magnetic fields are in correspondence. Therefore, for any vector field on the semi-Riemannian manifold, Lorentz equation can be given by the following formula where (19).