Field Mathematics for Electromagnetics, Photonics, and by Bernard Maxum

By Bernard Maxum

As photonics and fabrics technology span new horizons, it really is paramount that one's mathematical abilities be honed. the first goal of this ebook is to provide a assessment of vector calculus wanted for the actual sciences and engineering. This assessment contains precious tours into tensor research meant because the reader's first publicity to tensors, making features of tensors comprehensible on the undergraduate point. A secondary target of this booklet is to organize the reader for extra complex stories in those areas.

Contents

- record of Examples and Applications
- checklist of Figures
- Preface
- Acknowledgments
- Introduction
- Vector Algebra Review
- straight forward Tensor Analysis
- Vector Calculus Differential Forms
- Vector Calculus necessary Forms
- Appendix A: Vector Arithmetics and Applications
- Appendix B: Vector Calculus in Orthogonal Coordinate Systems
- Appendix C: Intermediate Tensor Calculus in help of Chapters three and 4
- Appendix D: Coordinate Expansions of Vector Differential Operators
- Glossary
- Index

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

Example text

4: Arithmetic Vector Operations 2-7 where the inverse ( ε ) is performed the same as the inverse of a 3 × 3 matrix, −1 [except that the unit dyads in ε are transposed in ( ε ) ]. 4. 3 Vector-vector products We next look at vector-vector products in orthogonal systems. There are three fundamentally distinct ways to perform product operations between two vectors. Each has an entirely different result. The first is the vector dot product. * Since the vector-vector dot product always yields a directionless scalar, it is also called a scalar product.

We first discuss the case of a partial derivative of a scalar function and then do the same for vector fields, ending with a mention of extending this to higher-rank tensors. In the event that the total rate of change is required allowing all of the variables to change, the total derivative is needed. We therefore examine the chain rule for determining the total derivative of a multivariate function. We shall see that the chain rule requires the partial derivatives with respect to each of the variables.

Thus, a convention must be established to remove this two-to-one ambiguity. The one that is chosen depends on the nature of the problem. The usual convention is to use the right-hand rule by pointing the fingers of the right hand in the direction of uˆ1 and then rolling them into the direction uˆ2 . The thumb will then point in the “normal” direction. On a closed surface, it is customary to construct uˆ1 and uˆ2 such that the normal will be directed outward. 2-5) Conveniently, the unit normal given in Eq.

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