scholarly journals A primer on exterior differential calculus

2003 ◽  
pp. 85-162 ◽  
Author(s):  
D.A. Burton
Keyword(s):  
Continuum Mechanics ◽  
Differential Calculus ◽  
Differential Forms ◽  
Lie Derivative ◽  
Exterior Differential ◽  
Exterior Calculus ◽  
Vector Calculus ◽  
Exterior Forms ◽  
Pedagogical Application

A pedagogical application-oriented introduction to the cal?culus of exterior differential forms on differential manifolds is presented. Stokes' theorem, the Lie derivative, linear con?nections and their curvature, torsion and non-metricity are discussed. Numerous examples using differential calculus are given and some detailed comparisons are made with their tradi?tional vector counterparts. In particular, vector calculus on R3 is cast in terms of exterior calculus and the traditional Stokes' and divergence theorems replaced by the more powerful exterior expression of Stokes' theorem. Examples from classical continuum mechanics and spacetime physics are discussed and worked through using the language of exterior forms. The numerous advantages of this calculus, over more traditional ma?chinery, are stressed throughout the article. .

scholarly journals A PROPOSAL FOR A DIFFERENTIAL CALCULUS IN QUANTUM MECHANICS

1994 ◽  
Vol 09 (13) ◽  
pp. 2191-2227 ◽  
Author(s):  
E. GOZZI ◽  
M. REUTER
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Hamiltonian System ◽  
Differential Calculus ◽  
Cotangent Bundle ◽  
Differential Forms ◽  
Lie Derivative ◽  
Exterior Derivative ◽  
Tensor Calculus ◽  
Exterior Calculus

In this paper, using, the Weyl-Wigner-Moyal formalism for quantum mechanics, we develop a quantum-deformed exterior calculus on the phase space of an arbitrary Hamiltonian system. Introducing additional bosonic and fermionic coordinates, we construct a supermanifold which is closely related to the tangent and cotangent bundle over phase space. Scalar functions on the supermanifold become equivalent to differential forms on the standard phase space. The algebra of these functions is equipped with a Moyal superstar product which deforms the pointwise product of the classical tensor calculus. We use the Moyal bracket algebra to derive a set of quantum-deformed rules for the exterior derivative, Lie derivative, contraction, and similar operations of the Cartan calculus.

On the Vector-Valued Cauchy Stress 2-Form and Its Stress Rate

2007 ◽  
Author(s):  
H. Murakami
Keyword(s):  
Shear Stress ◽  
Differential Forms ◽  
Shear Loading ◽  
Stress Rate ◽  
Curvilinear Coordinates ◽  
Lie Derivative ◽  
Cauchy Stress ◽  
Exterior Differential ◽  
Correct Definition ◽  
Vector Valued

Using exterior differential forms, basic equations of continuum mechanics are presented in direct notation. To this end, Elie Cartan’s vector-valued Cauchy stress 2-form is introduced. Its Lie derivative along the world line becomes the Truesdell stress rate. In the presentation, the notation adopted by Theodore Frankel (The Geometry of Physics, Cambridge, New York, 1997) is utilized. With the use of exterior differential forms, complicated computations in tensor analyses in curvilinear coordinates are dramatically simplified. As specific examples, the following subjects are presented: (i) Piola transformations of the Cauchy stress 2-form and (ii) simple shear deformation using the Lie derivative of the Cauchy stress 2-form, i.e., the Truesdell stress rate. It is known that under monotonic shear loading, if inappropriate stress-rates are used, shear stress oscillates. With the use of geometrically correct stress-rate, the shear stress monotonically increases. Thereby, the search for an appropriate stress rate reduces to the correct definition of the stress 2-form and the computation of its Lie derivative with respect to velocity.

scholarly journals Differential form representation of stochastic electromagnetic fields

2017 ◽  
Vol 15 ◽  
pp. 21-28 ◽  
Author(s):  
Michael Haider ◽  
Johannes A. Russer
Keyword(s):  
Electromagnetic Fields ◽  
Method Of Moments ◽  
Differential Forms ◽  
Electromagnetic Theory ◽  
Discretization Scheme ◽  
Algebraic Equations ◽  
Exterior Calculus ◽  
Vector Calculus ◽  
Form Representation ◽  
Continuous Quantities

Abstract. In this work, we revisit the theory of stochastic electromagnetic fields using exterior differential forms. We present a short overview as well as a brief introduction to the application of differential forms in electromagnetic theory. Within the framework of exterior calculus we derive equations for the second order moments, describing stochastic electromagnetic fields. Since the resulting objects are continuous quantities in space, a discretization scheme based on the Method of Moments (MoM) is introduced for numerical treatment. The MoM is applied in such a way, that the notation of exterior calculus is maintained while we still arrive at the same set of algebraic equations as obtained for the case of formulating the theory using the traditional notation of vector calculus. We conclude with an analytic calculation of the radiated electric field of two Hertzian dipole, excited by uncorrelated random currents.

Basic Forms

2020 ◽  
pp. 97-102
Author(s):  
Loring W. Tu
Keyword(s):  
Lie Group ◽  
Principal Bundle ◽  
Differential Forms ◽  
Total Space ◽  
Lie Derivative ◽  
Connected Lie Group

This chapter describes basic forms. On a principal bundle π‎: P → M, the differential forms on P that are pullbacks of forms ω‎ on the base M are called basic forms. The chapter characterizes basic forms in terms of the Lie derivative and interior multiplication. It shows that basic forms on a principal bundle are invariant and horizontal. To understand basic forms better, the chapter considers a simple example. The plane ℝ2 may be viewed as the total space of a principal ℝ-bundle. A connected Lie group is generated by any neighborhood of the identity. This example shows the necessity of the connectedness hypothesis.

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