Tensor Analysis
Suitable for students and practicing engineers, this concise text provides a straightforward treatment of those formulas needed to describe objects in three-dimensional space.
TABLE OF CONTENTS Tensor Analysis
Appendix A: Formulary
Overview
For convenience we list here the main tensor formulas obtained in each chapter. Note: the symbol ? denotes the universal quantifier (read as "for all" or "for every").
Chapter 2
Reciprocal (dual) basis
Kronecker delta symbol
Definition of reciprocal basis
Components of a vector
Relations between dual bases
where
and
Metric coefficients
In Cartesian frames,
Dot products in mixed and unmixed bases
Raising and lowering of indices
Frame transformation
Equations of transformation
where
Vector components and transformation laws
and
Miscellaneous
Permutation (Levi-Civita) symbol
Useful identities
Determinant of Gram matrix
Cross product
Chapter 3
Dyad product
Properties
Dot products of dyad with vector
Tensors from operator viewpoint
Equality of tensors
Components
Definition of sum A+B
Definition of scalar multiple c A
Definition of dot product A B
Definition of pre-multiplication y A
Definition of unit tensor E
Unit tensor components
Inverse tensor
Nonsingular tensor A
Determinant of a tensor
Dyadic components under transformation
Transformation to reciprocal basis
More general transformation
where
More dyadic operations
Dot product
Double dot product
Second rank tensor topics
Transpose
Tensors raised to powers
Symmetric and antisymmetric tensors
Eigenpair
Vi te formulas for invariants
Orthogonal tensor
Polar decompositions
Chapter 4
Vector fields
Some rules for differentiating vector functions
and
where
Tangent vectors to coordinate lines
Jacobian
Pointwise definition of reciprocal basis
Definition of metric coefficients
Transformation laws
Differentials and the nabla operator
Metric forms
Nabla operator
Gradient of a vector function
Divergence of vector
Rotation and curl of vector
Divergence and rotation of second-rank tensor
Differentiation of a...
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