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Relativity in Modern Physics$
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Nathalie Deruelle and Jean-Philippe Uzan

Print publication date: 2018

Print ISBN-13: 9780198786399

Published to Oxford Scholarship Online: October 2018

DOI: 10.1093/oso/9780198786399.001.0001

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Differential geometry

Differential geometry

(p.41) 4 Differential geometry
Relativity in Modern Physics

Nathalie Deruelle

Jean-Philippe Uzan

Oxford University Press

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.

Keywords:   differential geometry, curvilinear coordinates, covariant derivative, moving frames, cotangent spaces, differential forms, vector calculus

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