Types of Derivatives: Concepts and Applications (II)

Salma A. Khalil; Mohammed A. Basheer; Tarig A. Abdelhaleem

doi:10.5539/jmr.v9n1p50

Types of Derivatives: Concepts and Applications (II)

Journal of Mathematics Research ◽

10.5539/jmr.v9n1p50 ◽

2017 ◽

Vol 9 (1) ◽

pp. 50

Author(s):

Salma A. Khalil ◽

Mohammed A. Basheer ◽

Tarig A. Abdelhaleem

Keyword(s):

Differential Geometry ◽

Fluid Mechanics ◽

Exterior Calculus ◽

Geometrical Formulation

The notion of differential geometry is known to have played a fundamental role in unifying aspects of the physics of particles and fields, and have completely transformed the study of classical mechanics.In this paper we applied the definitions and concepts which we defined and derived in part (I) of our paper: Types of Derivatives: Concepts and Applications to problems arising in Geometry and Fluid Mechanics using exterior calculus. We analyzed this problem, using the geometrical formulation which is global and free of coordinates.

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Fractional vector calculus and fluid mechanics

Journal of the Mechanical Behavior of Materials ◽

10.1515/jmbm-2017-0012 ◽

2017 ◽

Vol 26 (1-2) ◽

pp. 43-54 ◽

Cited By ~ 7

Author(s):

Konstantinos A. Lazopoulos ◽

Anastasios K. Lazopoulos

Keyword(s):

Differential Geometry ◽

Porous Media ◽

Fluid Mechanics ◽

Research Field ◽

Flow In Porous Media ◽

Navier Stokes ◽

Basic Fluid ◽

Vector Calculus ◽

Fractional Differential ◽

Fractional Continuum

AbstractBasic fluid mechanics equations are studied and revised under the prism of fractional continuum mechanics (FCM), a very promising research field that satisfies both experimental and theoretical demands. The geometry of the fractional differential has been clarified corrected and the geometry of the fractional tangent spaces of a manifold has been studied in Lazopoulos and Lazopoulos (Lazopoulos KA, Lazopoulos AK. Progr. Fract. Differ. Appl. 2016, 2, 85–104), providing the bases of the missing fractional differential geometry. Therefore, a lot can be contributed to fractional hydrodynamics: the basic fractional fluid equations (Navier Stokes, Euler and Bernoulli) are derived and fractional Darcy’s flow in porous media is studied.

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Electromagnetism and differential geometry

10.1093/oso/9780198786399.003.0040 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

Differential Geometry ◽

Maxwell’S Equations ◽

Maxwell's Equations ◽

Exterior Derivative ◽

Exterior Product ◽

Exterior Calculus ◽

Definition Of

This chapter begins by examining p-forms and the exterior product, as well as the dual of a p-form. Meanwhile, the exterior derivative is an operator, denoted d, which acts on a p-form to give a (p + 1)-form. It possesses the following defining properties: if f is a 0-form, df(t) = t f (where t is a vector of Eₙ), which coincides with the definition of differential 1-forms. Moreover, d(α‎ + β‎) = dα‎ + dβ‎, where α‎ and β‎ are forms of the same degree. Moreover, the exterior calculus can be used to obtain a compact and elegant formulation of Maxwell’s equations.

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