Page 193 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Sachin Kaushal, Lovely Professional University

Notes Unit 17: Tensor Fields in Curvilinear Coordinates

CONTENTS
Objectives

Introduction
17.1 General idea of Curvilinear Coordinates
17.2 Auxiliary Cartesian Coordinate System
17.3 Coordinate Lines and the Coordinate Grid
17.4 Moving Frame of Curvilinear Coordinates

17.5 Dynamics of Moving Frame
17.6 Formula for Christoffel Symbols
17.7 Tensor Fields in Curvilinear Coordinates

17.8 Differentiation of Tensor Fields in Curvilinear Coordinates
17.9 Concordance of Metric and Connection
17.10 Summary
17.11 Keywords
17.12 Self Assessment

17.13 Review Questions
17.14 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the general idea of curvilinear coordinates

Describe the auxiliary Cartesian coordinate system

Explain the coordinate lines and coordinate grid.

Discuss the moving frame of curvilinear coordinates

Explain the formula for Christoffel symbols

Introduction

In the last unit, you have studied about tensor fields differentiation of tensors and tensor fields
in Cartesian coordinates. Curvilinear coordinates are a coordinate system for Euclidean space in
which the coordinate lines may be curved. These coordinates may be derived from a set of
Cartesian coordinates by using a transformation that is locally invertible (a one-to-one map) at
each point. This means that one can convert a point given in a Cartesian coordinate system to its
curvilinear coordinates and back. The name curvilinear coordinates, coined by the French
mathematician Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems
are curved.

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