High-Order Method with Moving Frames to Compute the Covariant Derivatives of Vectors on General 2D Curved Surfaces

The covariant derivative is a generalization of differentiating vectors.The Euclidean derivative is a special case of the covariant derivative in Euclidean space.The covariant derivative gathers broad attention,particularly when computing vector derivatives on curved surfaces and volumes in various applications.Covariant derivatives have been computed using the metric tensor from the analytically known curved axes.However,deriving the global axis for the domain has been mathematically and computationally challenging for an arbitrary two-dimensional(2D)surface.Consequently,computing the covariant derivative has been difficult or even impossible.A novel high-order numerical scheme is proposed for computing the covariant derivative on any 2D curved surface.A set of orthonormal vectors,known as moving frames,expand vectors to compute accurately covariant derivatives on 2D curved surfaces.The proposed scheme does not require the construction of curved axes for the metric tensor or the Christoffel symbols.The connectivity given by the Christoffel symbols is equivalently provided by the attitude matrix of orthonormal moving frames.Consequently,the proposed scheme can be extended to the general 2D curved surface.As an application,the Helmholtz‐Hodge decomposition is considered for a realistic atrium and a bunny.

Extension of covariant derivative(Ⅰ): From component form to objective form

This paper extends the covariant derivative un der curved coordinate systems in 3D Euclid space. Based on the axiom of the covariant form invariability, the classical covariant derivative that can only act on components is ex tended to the generalized covariant derivative that can act on any geometric quantity including base vectors, vectors and tensors. Under the axiom, the algebra structure of the gen eralized covariant derivative is proved to be covariant dif ferential ring. Based on the powerful operation capabilities and simple analytical properties of the generalized covariant derivative, the tensor analysis in curved coordinate systems is simplified to a large extent.

Extension of covariant derivative(Ⅱ): From flat space to curved space

This paper extends the classical covariant deriva tive to the generalized covariant derivative on curved sur faces. The basement for the extension is similar to the pre vious paper, i.e., the axiom of the covariant form invariabil ity. Based on the generalized covariant derivative, a covari ant differential transformation group with orthogonal duality is set up. Through such orthogonal duality, tensor analy sis on curved surfaces is simplified intensively. Under the covariant differential transformation group, the differential invariabilities and integral invariabilities are constructed on curved surfaces.

Extension of covariant derivative(Ⅲ): From classical gradient to shape gradient

This paper further extends the generalized covari ant derivative from the first covariant derivative to the sec ond one on curved surfaces. Through the linear transforma tion between the first generalized covariant derivative and the second one, the second covariant differential transformation group is set up. Under this transformation group, the sec ond class of differential invariants and integral invariants on curved surfaces is made clear. Besides, the symmetric struc ture of the tensor analysis on curved surfaces are revealed.

On the uniqueness of Einstein-Cartan theory:Lagrangian,covariant derivative and equation of motion

In the standard Einstein-Cartan theory,matter fields couple to gravity through the Minimal Coupling Procedure(MCP),and yet leave the theory an ambiguity.Applying MCP to the action or to the equation of motion would lead to different gravitational couplings.We propose a new covariant derivative to remove the ambiguity and discuss the relation between our proposal and previous treatments on this subject.

Generalized Covariant Derivative with Respect to Time in Flat Space(Ⅰ):Eulerian Description

This paper reports a new derivative in the Eulerian description in flat space-the generalized covariant derivative with respect to time. The following contents are included:(a) the restricted covariant derivative with respect to time for Eulerian component is defined;(b) the postulate of the covariant form invariability in time field is set up;(c) the generalized covariant derivative with respect to time for generalized Eulerian component is defined;(d) the algebraic structure of the generalized covariant derivative with respect to time is made clear;(e) the covariant differential transformation group in time filed is derived. These progresses reveal the covariant form invariability of Eulerian space and time.

Generalized Covariant Derivative with Respect to Time in Flat Space(Ⅱ):Lagrangian Description

The previous paper reported a new derivative in the Eulerian description in flat space—the generalized covariant derivative of generalized Eulerian component with respect to time. This paper extends the thought from the Eulerian description to the Lagrangian description:on the basis of the postulate of covariant form invariability in time field, we define a new derivative in the Lagrangian description in flat space—the generalized covariant derivative of generalized Lagrangian component with respect to time. Besides, the covariant differential transformation group is set up. The covariant form invariability of Lagrangian space-time is ascertained.

Property of Tensor Satisfying Binary Law 2

I have already reported “Property of Tensor Satisfying Binary Law”. This article is the article that I revise the contents of “Property of Tensor Satisfying Binary Law”, and increase the report about new characteristics. We may arrive at the deeper understanding in this about “Property of Tensor Satisfying Binary Law”.