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Sachin Kaushal, Lovely Professional University Unit 15: Tensors in Cartesian Coordinates

Unit 15: Tensors in Cartesian Coordinates Notes

CONTENTS
Objectives

Introduction
15.1 Covectors
15.2 Scalar Product of Vector and Covector
15.3 Linear Operators
15.4 Bilinear and Quadratic Forms

15.5 General Definition of Tensors
15.6 Dot Product and Metric Tensor
15.7 Multiplication by Numbers and Addition

15.8 Tensor Product
15.9 Contraction
15.10 Raising and Lowering Indices
15.11 Some Special Tensors and some useful Formulas
15.12 Summary

15.13 Keywords
15.14 Self Assessment
15.15 Review Questions
15.16 Further Readings

Objectives

After studying this unit, you will be able to:
Define convectors

Discuss the scalar products of vector and convectors

Describe bilinear and quadratic forms

Explain the dot product and metric tensor

Introduction

In the last unit, you have studied about notation and summation of convention. A tensor written
in component form is an indexed array. The order of a tensor is the number of indices required.
(The rank of tensor used to mean the order, but now it means something different). The rank of
the tensor is the minimal number of rank-one tensor that you need to sum up to obtain this
higher-rank tensor. Rank-one tensors are given the generalization of outer product to m-vectors,
where m is the order of the tensor.

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