Geometry of Differential Forms

Morita, Shigeyuki

doi:10.1090/mmono/201

Cited by 173 publications

(148 citation statements)

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“…As a consequence, several recent initiatives have been aimed at formulating physical laws in terms of differential forms. For recent work along these lines, the reader is invited to refer to [5,1,27,13,32,6,16] for books offering a theoretical treatment of various physical theories using differential forms.…”

Section: Geometry-based Exterior Calculusmentioning

confidence: 99%

Discrete differential forms for computational modeling

Desbrun

Kanso

Tong

2006

ACM SIGGRAPH 2006 Courses on - SIGGRAPH '06

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Section: Geometry-based Exterior Calculusmentioning

confidence: 99%

Discrete differential forms for computational modeling

Desbrun

Kanso

Tong

2006

ACM SIGGRAPH 2006 Courses on - SIGGRAPH '06

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“…One should stress that spacetime algebra provides an invariant formulation of physics -as opposed to the paravector covariant model of the so-called algebra of physical space advocated by William E. Baylis [27]. Other problems of electromagnetic waves and applications can be addressed within geometric algebra [28], providing a geometric treatment for special relativity that is coordinate-free and generalizes the more elementary ordinary vector analysis stemming from the works of Gibbs and Heaviside, without meandering into more mathematically involved formalisms -such as differential forms [29][30][31] or just classical plain tensors.…”

Section: Resultsmentioning

confidence: 99%

Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

Paiva¹,

Ribeiro²

2006

Journal of Electromagnetic Waves and Applications

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Abstract-As is well-known, Lorentz boosts do not form a group: the composite of two boosts is, in general, equivalent to the composite of a rotation -the so-called Thomas (or Wigner) rotation -and a single boost. Using spacetime algebra, the geometric algebra for Minkowski spacetime, we show in this paper how to relate the Thomas rotation to the Doppler shift experienced between two observers in relative motion. Both observers are moving away from a third observer, set in the laboratory, with relative non-parallel velocities.

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“…We assume that all considerations are local and all objects are smooth. For details in the subject we refer to [28][29][30].…”

Section: Basics Of Riemannian Geometry In An Orthonormal Framementioning

confidence: 99%

Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

Bayrakdar

2018

Advances in Mathematical Physics

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We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers' equations describe surfaces attached to an ODE of the form / = ( , ) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.

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Geometry of Differential Forms

Cited by 173 publications

References 0 publications

Discrete differential forms for computational modeling

Discrete differential forms for computational modeling

Doppler Shift from a Composition of Boosts with Thomas Rotation: a Spacetime Algebra Approach

Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

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