Page 157 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 14: Notation and Summation Convention

CONTENTS
Objectives

Introduction
14.1 Rectangular Cartesian Coordinate Systems
14.2 Suffix and Symbolic Notation
14.3 Orthogonal Transformations
14.4 Summary

14.5 Keywords
14.6 Self Assessment
14.7 Review Questions

14.8 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the concept rectangular Cartesian coordinate systems

Describe the suffix and symbolic notation

Discuss the orthogonal transformation

Introduction

In this unit, we will discuss an elementary introduction to Cartesian tensor analysis in a three-
dimensional Euclidean point space or a two-dimensional subspace. A Euclidean point space is
the space of position vectors of points. The term vector is used in the sense of classical vector
analysis, and scalars and polar vectors are zeroth- and first-order tensors, respectively. The
distinction between polar and axial vectors is discussed later in this chapter. A scalar is a single
quantity that possesses magnitude and does not depend on any particular coordinate system,
and a vector is a quantity that possesses both magnitude and direction and has components, with
respect to a particular coordinate system, which transform in a definite manner under change of
coordinate system. Also vectors obey the parallelogram law of addition. There are quantities
that possess both magnitude and direction but are not vectors, for example, the angle of finite
rotation of a rigid body about a fixed axis.
A second-order tensor can be defined as a linear operator that operates on a vector to give
another vector. That is, when a second-order tensor operates on a vector, another vector, in the
same Euclidean space, is generated, and this operation can be illustrated by matrix multiplication.
The components of a vector and a second-order tensor, referred to the same rectangular Cartesian
coordinate system, in a three-dimensional Euclidean space, can be expressed as a (3 × 1) matrix
and a (3 × 3) matrix, respectively. When a second-order tensor operates on a vector, the components
of the resulting vector are given by the matrix product of the (3 × 3) matrix of components of the

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