Tangent Vectors on Tangent Euclidean Spaces

In this paper, we give how to define the basic concepts of differential geometry on Dual space. For this, dual tangent vectors that have p as dual point of application are defined. Then, the dual analytic functions defined by Dimentberg are examined in detail, and by using the derivative of the these functions, dual directional derivatives and dual tangent maps are introduced.

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Differential geometry of curves in centro-Euclidean spaces

Czechoslovak Mathematical Journal ◽

10.21136/cmj.1962.100501 ◽

1962 ◽

Vol 12 (1) ◽

pp. 119-143

Author(s):

Čestmír Vitner

Keyword(s):

Differential Geometry ◽

Euclidean Spaces

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Manifolds and tensors

General Relativity ◽

10.1093/oso/9780198822158.003.0004 ◽

2019 ◽

pp. 25-36

Author(s):

Steven Carlip

Keyword(s):

Differential Geometry ◽

General Relativity ◽

Differential Forms ◽

General Covariance ◽

Mathematical Basis ◽

Metric Tensor ◽

Diffeomorphism Invariance ◽

Starting Point ◽

Tangent Vectors

The mathematical basis of general relativity is differential geometry. This chapter establishes the starting point of differential geometry: manifolds, tangent vectors, cotangent vectors, tensors, and differential forms. The metric tensor is introduced, and its symmetries (isometries) are described. The importance of diffeomorphism invariance (or “general covariance”) is stressed.

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Some Characterizations of the Cobb-Douglas and CES Production Functions in Microeconomics

Abstract and Applied Analysis ◽

10.1155/2013/761832 ◽

2013 ◽

Vol 2013 ◽

pp. 1-6 ◽

Cited By ~ 10

Author(s):

Xiaoshu Wang ◽

Yu Fu

Keyword(s):

Differential Geometry ◽

Economic Analysis ◽

Production Function ◽

Production Functions ◽

Euclidean Spaces ◽

Production Possibility ◽

Constant Return ◽

Production Possibility Frontier ◽

Theory Of Production ◽

Douglas Production Function

It is well known that the study of the shape and the properties of the production possibility frontier is a subject of great interest in economic analysis. Vîlcu (Vîlcu, 2011) proved that the generalized Cobb-Douglas production function has constant return to scale if and only if the corresponding hypersurface is developable. Later on, the authors A. D. Vîlcu and G. E. Vîlcu, 2011 extended this result to the case of CES production function. Both results establish an interesting link between some fundamental notions in the theory of production functions and the differential geometry of hypersurfaces in Euclidean spaces. In this paper, we give some characterizations of minimal generalized Cobb-Douglas and CES production hypersurfaces in Euclidean spaces.

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On in-plane drill rotations for Cosserat surfaces

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2021.0158 ◽

2021 ◽

Vol 477 (2252) ◽

pp. 20210158

Author(s):

Maryam Mohammadi Saem ◽

Peter Lewintan ◽

Patrizio Neff

Keyword(s):

Differential Geometry ◽

Rotation Axis ◽

Initial Surface ◽

Identity Mapping ◽

Shell Surface ◽

The Given ◽

Tangent Vectors

We show under some natural smoothness assumptions that pure in-plane drill rotations as deformation mappings of a C 2 -smooth regular shell surface to another one parametrized over the same domain are impossible provided that the rotations are fixed at a portion of the boundary. Put otherwise, if the tangent vectors of the new surface are obtained locally by only rotating the given tangent vectors, and if these rotations have a rotation axis which coincides everywhere with the normal of the initial surface, then the two surfaces are equal provided they coincide at a portion of the boundary. In the language of differential geometry of surfaces, we show that any isometry which leaves normals invariant and which coincides with the given surface at a portion of the boundary is the identity mapping.

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