Field Mathematics for Electromagnetics, Photonics, and Materials Science: A Guide for the Scientist and Engineer

Author(s): Bernard Maxum

Published: 2004

PDF ISBN: 9780819478689 | Print ISBN: 9780819455239

DESCRIPTION

As electromagnetics, photonics, and materials science evolve, it is increasingly important for students and practitioners in the physical sciences and engineering to understand vector calculus and tensor analysis. This book provides a review of vector calculus. This review includes necessary excursions into tensor analysis intended as the reader's first exposure to tensors, making aspects of tensors understandable to advanced undergraduate students. This book will also prepare the reader for more advanced studies in vector calculus and tensor analysis.

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Front Matter

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CHAPTER 1.

Introduction

Chapter Outline +

1.1 Notation
1.1.1 Scalars
1.1.2 Vectors
1.1.3 Unit vectors
1.1.4 r -space notation: the vector-like r used in the argument of a field function
1.1.5 Phasors
1.1.6 Dyadics
1.1.7 Tensors
1.2 Spatial Differentials
1.2.1 Differential length vectors
1.2.2 Differential area
1.2.3 Differential volume
1.3 Partial and Total Derivatives
1.3.1 Partial derivative of a scalar function
1.3.2 Total derivative of a scalar function: chain rules
1.3.3 A dimensionally consistent formulation of partial derivatives
1.3.4 Partial derivative of a vector function
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CHAPTER 2.

Vector Algebra Review

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2.1 Variant and Invariant Scalars
2.2 Scalar Fields
2.3 Vector Fields
2.4 Arithmetic Vector Operations
2.4.1 Commutative and associative laws in vector addition and subtraction
2.4.2 Multiplication or division of a vector by a scalar
2.4.3 Vector-vector products
2.5 Scalars, Vectors, Dyadics, and Tensors as Phasors
2.6 Vector Field Direction Lines
2.6.1 Cartesian (rectangular) coordinates
2.6.2 Cylindrical coordinates
2.6.3 Spherical coordinates
2.6.4 Example of field direction lines
2.7 Scalar Field Equivalue Surfaces
References

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CHAPTER 3.

Elementary Tensor Analysis

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The tensor/dyadic issue
3.1 Directional Compoundedness, Rank, and Order of Tensors
The rank/order issue
3.2 Tensor Components
3.3 Dyadics and the Unit Dyad
3.4 Dyadic Dot Products
3.4.1 Vector-dyadic dot products
3.4.2 Dyadic-dyadic dot and double-dot products
3.5 The Four-Rank Elastic Modulus Tensor
3.6 The Use of Tensors in Nonlinear Optics
3.7 Term-by-Term Rank Consistency and the Rules for Determining the Rank after Performing Inner-Product Operations with Tensors
3.8 Summary of Tensors
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CHAPTER 4.

Vector Calculus Differential Forms

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4.1 Introduction to Differential Operators
AND SOME ADDITIONAL TENSOR RULES
4.2 Scalar Differential Operators, Differential Equations, and Eigenvalues
4.3 The Gradient Differential Operator
4.3.1 The gradient of a scalar field—a physical description
4.3.2 The gradient of a vector field
4.4 The Divergence Differential Operator
4.4.1 The divergence of a vector field—a physical description
4.4.2 The divergence in GOCCs
4.5 The Curl Differential Operator
4.5.1 The curl of a vector field—a physical description
4.5.2 The curl as a vorticity vector
4.5.3 The expansion of the curl in GOCCs
4.5.4 The expansion of the curl in cylindrical coordinates
4.6 Tensorial Resultants of First-Order Vector Differential Operators
4.7 Second-Order Vector Differential Operators—Differential Operators of Differential Operators
4.7.1 Resultant forms from second-order vector differential operators—a tabular summary of tensorial resultants
4.7.2 Two important second-order vector differential operators that vanish
4.7.3 The divergence of the gradient of a scalar field—the scalar Laplacian
4.7.4 The divergence of the gradient of a vector field—the vector Laplacian
4.7.5 The curl of the curl of a vector field and the Lagrange identity
4.7.6 The gradient of the divergence of a vector field
4.7.7 The gradient of the divergence minus the curl of the curl—the vector Laplacian
References

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CHAPTER 5.

Vector Calculus Integral Forms

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5.1 Line Integrals of Vector (and Other Tensor) Fields
5.1.1 Line integrals of scalar, vector, and tensor fields with dot-, cross-, and direct-product integrands
5.1.2 Examples of form (5.1-1): Line integral of the tangential component of F along path L
5.1.3 Other line-integral examples
5.2 Surface Integrals of Vector (and Other Tensor) Fields
5.2.1 Surface integrals of scalar, vector and other tensor fields with dot-, cross-, and tensor-product integrands
5.2.2 Surface integral applications
5.3 Gauss’ (Divergence) Theorem
5.3.1 Gauss’ law
5.3.2 Derivation of Gauss’ divergence theorem
5.3.3 Implications of divergence theorem on the source distribution
5.3.4 Application: The energy in electromagnetic fields—Pointing’s theorem
5.4 Stokes’ (Curl) Theorem
5.4.1 Ampere’s circuital law
5.4.2 Derivation of Stokes’ theorem
5.4.3 Implications of Stokes’ theorem
5.5 Green’s Mathematics
5.5.1 Green’s identities
5.5.2 Green’s function
5.5.3 Applications of Green’s mathematics
References

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Appendix A Vector Arithmetics and Applications

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Appendix B Vector Calculus in Orthogonal Coordinate Systems

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B.1 Cartesian Coordinate Geometry for the Divergence
B.2 Cartesian Coordinate Geometry for the Curl
B.3 Cylindrical Coordinate Geometry for the Divergence
B.4 Summary of the Geometries for Divergence, Curl, and Gradient
B.5 Orthogonal Coordinate System Parameters and Surface Graphics
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Appendix C Intermediate Tensor Calculus in Support of Chapters 3 and 4

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C.1 Explicit Standard Notation for General Rank Tensors
C.2 Properties of First- and Second-Order Vector Differential Operators on Tensors
C.2.1 First-order vector differential operators with vector and generalized tensor operands
C.2.2 Proof that the divergence of the curl of any tensor is zero
C.2.3 Proof that the curl of the gradient of any tensor is zero
C.2.4 Demonstration that the curl of the divergence of any tensor is in general nonzero
C.2.5 Demonstration that the gradient of the curl of any tensor is in general nonzero
C.2.6 Demonstration of the Lagrange identity applied to tensors
C.3 Generalization of the Divergence Operator of Eq. (4.7-7)
C.4 The Dual Nature of the Nabla Operator
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Appendix D Coordinate Expansions of Vector Differential Operators

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D.1 Cartesian Coordinate Expansions
D.1.1 Cartesian coordinate expansions of vector differential operators
D.1.2 Cartesian coordinate expansions of second-order vector differential operators
D.2 Cylindrical Coordinate Expansions
D.2.1 Cylindrical coordinate expansions of first-order vector differential operators
D.2.2 Cylindrical coordinate expansions of second-order vector differential operators

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Glossary

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Back Matter

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Back Matter — Errata

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