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### Spie Press Book

Field Mathematics for Electromagnetics, Photonics, and Materials Science: A Guide for the Scientist and Engineer
Author(s): Bernard Maxum
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Book 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.

Book Details

Date Published: 4 November 2004
Pages: 272
ISBN: 9780819455239
Volume: TT64
Errata

The author has prepared a cross reference between this book and SciTech Publishing's Fundamentals of Electromagnetics with MATLAB by Lonngren, Savov, and Jost. The link below provides users of that text with a convenient way to find supplementary information in Field Mathematics for Electromagnetics…

List of Examples and Applications

List of Figures

Preface

Acknowledgements

Chapter 1 Introduction / 1-1
1.1 Notation / 1-1
1.1.1 Scalars / 1-2
1.1.2 Vectors / 1-2
1.1.3 Unit vectors / 1-3
1.1.4 r-space notation: the vector-like r used in the argument of a field function / 1-4
1.1.5 Phasors / 1-5
1.1.6 Dyadics / 1-6
1.1.7 Tensors / 1-10
(a) Explicit standard notation for tensors / 1-11
(b) Multiple-subscript notation for tensors / 1-11
(c) Pre-subscript, pre-superscript notation for tensors / 1-12
(d) Arrow notation for tensors / 1-13
(e) Post-subscript, post-superscript notation for tensors / 1-14
1.2 Spatial Differentials / 1-14
1.2.1 Differential length vectors / 1-15
1.2.2 Differential area / 1-15
1.2.3 Differential volume / 1-17
1.3 Partial and Total Derivatives / 1-18
1.3.1 Partial derivative of a scalar function / 1-19
1.3.2 Total derivative of a scalar function: chain rules / 1-20
(a) Chain rule for functions of three independent variables / 1-20
(b) Chain rule for surface functions / 1-21
1.3.3 A dimensionally consistent formulation of partial derivatives / 1-21
1.3.4 Partial derivative of a vector function / 1-22
References / 1-24
Chapter 2 Vector Algebra Review / 2-1
2.1 Variant and Invariant Scalars / 2-1
2.2 Scalar Fields / 2-1
2.3 Vector Fields / 2-2
2.4 Arithmetic Vector Operations / 2-4
2.4.1 Commutative and associative laws in vector addition and subtraction / 2-4
2.4.2 Multiplication or division of a vector by a scalar / 2-5
2.4.3 Vector-vector products / 2-7
(a) Restricted use of the terms "scalar product" and "vector product" / 2-7
(b) Dot product and the Kroneker delta / 2-8
(c) Cross product and the Levi-Civita symbol / 2-13
(i) Commutative and distributive laws for cross products / 2-13
(ii) Vector cross products and the Levi-Civita symbol / 2-14
(iii) Area formulas using cross products / 2-15
(iv) Cross product coordinate expansion / 2-16
2.4.4 Triple vector products / 2-17
2.5 Scalars, Vectors, Dyadics, and Tensors as Phasors / 2-17
2.6 Vector Field Direction Lines / 2-18
2.6.1 Cartesian (rectangular) coordinates / 2-20
2.6.2 Cylindrical coordinates / 2-21
2.6.3 Spherical coordinates / 2-22
2.6.4 Example of field direction lines/ 2-22
2.7 Scalar Field Equivalue Surfaces / 2-25
References / 2-28
Chapter 3 Elementary Tensor Analysis / 3-1
3.1 Directional Compoundedness, Rank, and Order of Tensors / 3-3
The "rank-order" issue / 3-4
3.2 Tensor Components / 3-4
3.3 Dyadics and the Unit Dyad / 3-5
3.4 Dyadic Dot Products / 3-8
3.4.1 Vector-dyadic dot products / 3-8

(a) Application of the dyadic-vector dot product for anisotropic dielectrics / 3-9
(b) Comparison of the dyadic-vector dot product with the vector- dyadic dot product / 3-10
3.4.2 Dyadic-dyadic dot and double-dot products / 3-12
3.5 The Four-Rank Elastic Modulus Tensor / 3-13
3.6 The Use of Tensors in Nonlinear Optics / 3-15
3.7 Term-by-Term Rank Consistency and the Rules for Determining the Rank after
Performing Inner-Product Operations with Tensors / 3-19
3.8 Summary of Tensors / 3-20
References / 3-22
Chapter 4 Vector Calculus Differential Forms / 4-1 WITH EXCURSIONS INTO TENSOR CALCULUS
4.1 Introduction to Differential Operators / 4-2
AND SOME ADDITIONAL TENSOR RULES
4.2 Scalar Differential Operators, Differential Equations, and
Eigenvalues / 4-5
4.3 The Gradient Differential Operator / 4-8
4.3.1 The gradient of a scalar field - a physical description / 4-8
(a) Why the unit normal is the direction of maximal increase / 4-10
(b) Expansion of the gradient of a scalar field in GOCCs / 4-12
(c) The directional derivative nature of the gradient of a scalar field / 4-13
4.3.2 The gradient of a vector field / 4-14
(a) The gradient of a vector field in GOCCs / 4-14
(b) The gradient of a vector field in cylindrical coordinates / 4-15
4.4 The Divergence Differential Operator / 4-16
4.4.1 The divergence of a vector field - a physical description / 4-17
(a) Vector-field flux tubes and sources / 4-18
(b) Examples of zero and nonzero divergence / 4-19
(c) Significance of a nonzero divergence / 4-23
4.4.2 The divergence in GOCCs / 4-24
4.5 The Curl Differential Operator / 4-27
4.5.1 The curl of a vector field - a physical description / 4-28
4.5.2 The curl as a vorticity vector / 4-30
4.5.3 The expansion of the curl in GOCCs / 4-32
4.5.4 The expansion of the curl in cylindrical coordinates / 4-35
4.6 Tensorial Resultants of First-Order Vector Differential Operators / 4-35
4.7 Second-Order Differential Operators / 4-36
4.7.1 Resultant forms from second-order vector differential operators - a tabular summary of tensorial resultants / 4-37
4.7.2 Two important second-order vector differential operators that vanish / 4-40
4.7.3 The divergence of the gradient of a scalar field - the scalar Laplacian / 4-42
(a) The scalar Laplacian in GOCCs / 4-42
(b) The scalar Laplacian in cylindrical coordinates / 4-43
4.7.4 The divergence of the gradient of a vector field-the vector Laplacian / 4-43
(a) The divergence of a dyadic in GOCCs / 4-43
(b) The vector Laplacian in GOCCs / 4-45
(c) The vector Laplacian in cylindrical coordinates / 4-46
4.7.5 The curl of the curl of a vector field and the Lagrange identity / 4-48
(a) A physical description of the curl of the curl / 4-49
(b) The curl of the curl in GOCCs / 4-51
4.7.6 The gradient of the divergence of a vector field / 4-52
(a) A physical description of the gradient of the divergence / 4-53
(b) The gradient of the divergence in GOCCs / 4-53
4.7.7 The gradient of the divergence minus the curl of the curl - the vector Laplacian / 4-53
References / 4-54
Chapter 5 Vector Calculus Integral Forms / 5-1
5.1 Line Integrals of Vector (and other Tensor) Fields / 5-2
5.1.1 Line integrals of scalar, vector, and tensor fields with dot-, cross-, and direct-product integrands / 5-2
5.1.2 Examples of form: Line integral of the tangential component of along path L / 5-5
(a) Examples in mechanics - force and work / 5-6
(b) Electrostatics - electric field intensity and electric potential / 5-9
(c) Path dependence of tangential line integrals / 5-10
5.1.3 Other line-integral examples / 5-11
5.2 Surface Integrals of Vector (and Other Tensor) Fields / 5-12
5.2.1 Surface integrals of scalar, vector and other tensor fields with dot-, cross-, and tensor-product integrands / 5-12
5.2.2 Surface integral applications / 5-14
5.3 Gauss' (Divergence) Theorem / 5-15
5.3.1 Gauss' law / 5-16
5.3.2 Derivation of Gauss' divergence theorem / 5-17
5.3.3 Implications of divergence theorem on the source distribution / 5-18
5.3.4 Application: The energy in electromagnetic fields - Pointing's theorem / 5-19
5.4 Stokes' (Curl) Theorem / 5-21
5.4.1 Ampere's circuital law / 5-21
5.4.2 Derivation of Stokes' theorem / 5-22
5.4.3 Implications of Stokes' theorem / 5-23
5.5 Green's Mathematics / 5-24
5.5.1 Green's identities / 5-24
5.5.2 Green's function / 5-25
5.5.3 Applications of Green's mathematics / 5-26
(a) Retarded electric scalar potential / 5-27
(b) Retarded magnetic vector potential / 5-30
References / 5-31
Appendix A Vector Arithmetics and Applications / A-1
Appendix B Vector Calculus in Orthogonal Coordinate Systems / B-1
B.1 Cartesian Coordinate Geometry for the Divergence / B-2
B.2 Cartesian Coordinate Geometry for the Curl / B-5
B.3 Cylindrical Coordinate Geometry for the Divergence / B-9
B.4 Summary of the Geometries for Divergence, Curl, and Gradient / B-12
B.5 Orthogonal Coordinate System Parameters and Surface Graphics / B-13
References / B-26
Appendix C Intermediate Tensor Calculus in Support of Chapters 3 and 4 / C-1
C.1 Explicit Standard Notation for General Rank Tensors / C-2
C.2 Properties of First- and Second-Order Vector Differential Operators on Tensors/ C-5
C.2.1 First-order vector differential operators with vector and generalized tensor operands / C-5
C.2.2 Proof that the divergence of the curl of any tensor is zero / C-7
C.2.3 Proof that the curl of the gradient of any tensor is zero / C-9
C.2.4 Demonstration that the curl of the divergence of any tensor is in general nonzero / C-10
C.2.5 Demonstration that the gradient of the curl of any tensor is in general nonzero / C-12
C.2.6 Demonstration of the Lagrange identity applied to tensors / C-13
C.3 Generalization of the Divergence Operator of Eq. (4.7-7) / C-16
C.4 The Dual Nature of the Nabla Operator / C-21
Reference / C-24
Appendix D Coordinate Expansions of Vector Differential Operators / D-1
D.1 Cartesian Coordinate Expansions / D-1
D.1.1 Cartesian coordinate expansions of vector differential operators / D-1
(a) The divergence of vector and dyadic fields / D-1
(b) The curl of vector and dyadic fields / D-2
(c) The gradient of scalar, vector, and dyadic fields / D-3
D.1.2 Cartesian coordinate expansions of second-order vector differential operators / D-4
(a) The scalar and vector Laplacian / D-4
(b) The curl of the curl of a vector field / D-4
(c) The gradient of the divergence / D-5
D.2 Cylindrical Coordinate Expansions / D-5
D.2.1 Cylindrical coordinate expansions of first-order vector differential operators / D-6
(a) The divergence of vector and dyadic fields / D-6
(b) The curl of vector and dyadic fields / D-6
(c) The gradient of scalar, vector, and dyadic fields / D-7
D.2.2 Cylindrical coordinate expansions of second-order vector differential operators / D-9
(a) The scalar and vector Laplacian / D-9
(b) The curl of the curl of a vector field / D-9
(c) The gradient of the divergence / D-9
Glossary
Index