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Sachin Kaushal, Lovely Professional University Unit 16: Tensor Fields Differentiation of Tensors

Unit 16: Tensor Fields Differentiation of Tensors Notes

CONTENTS
Objectives

Introduction
16.1 Tensor Fields in Cartesian Coordinates
16.2 Change of Cartesian Coordinate System
16.3 Differentiation of Tensor Fields
16.4 Gradient, Divergency, and Rotor

16.5 Summary
16.6 Keywords
16.7 Self Assessment

16.8 Review Questions
16.9 Further Readings

Objectives

After studying this unit, you will be able to:

Discuss the tensor fields in Cartesian coordinates

Describe the change of Cartesian coordinates system

Explain the differentiation of tensors fields

Discuss the gradient, divergency and rotor

Introduction

Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid
mechanics and elasticity. In classical continuum mechanics, the space of interest is usually
3-dimensional Euclidean space, as is the tangent space at each point. If we restrict the local
coordinates to be Cartesian coordinates with the same scale centered at the point of interest, the
metric tensor is the Kronecker delta. This means that there is no need to distinguish covariant
and contra variant components, and furthermore there is no need to distinguish tensors and
tensor densities. All Cartesian-tensor indices are written as subscripts. Cartesian tensors achieve
considerable computational simplification at the cost of generality and of some theoretical
insight.

16.1 Tensor Fields in Cartesian Coordinates

The tensors that we defined in the previous unit are free tensors. Indeed, their components are
arrays related to bases, while any basis is a triple of free vectors (not bound to any point). Hence,
the tensors previously considered are also not bound to any point.

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