- Title Pages
- Preface
- Contents
- 1 Some essential mathematics
- 2 Static electric fields in vacuum
- 3 The electrostatics of conductors
- 4 Static magnetic fields in vacuum
- 5 Quasi-static electric and magnetic fields in vacuum
- 6 Ohm’s law and electric circuits
- 7 Electromagnetic fields and waves in vacuum
- 8 The electromagnetic potentials
- 9 Static electric and magnetic fields in matter
- 10 Some applications of Maxwell’s equations in matter
- 11 Electromagnetic radiation
- 12 Electromagnetism and special relativity
- Appendix A Vectors and Cartesian tensors
- Appendix A Vectors and Cartesian tensors
- Appendix B Cartesian coordinates
- Appendix C Spherical polar coordinates
- Appendix D Cylindrical polar coordinates
- Appendix E The Dirac delta function
- Appendix E The Dirac delta function
- Appendix F Legendre polynomials
- Appendix F Legendre polynomials
- Appendix G Bessel functions
- Appendix G Bessel functions
- Appendix H Parametric representation of a surface
- Appendix H Parametric representation of a surface
- Appendix I The Cauchy–Riemann equations
- Appendix I The Cauchy–Riemann equations
- Appendix J Questions involving computational work
- Glossary of symbols
- Index

# Some essential mathematics

# Some essential mathematics

- Chapter:
- (p.1) 1 Some essential mathematics
- Source:
- Solved Problems in Classical Electromagnetism
- Author(s):
### J. Pierrus

- Publisher:
- Oxford University Press

This chapter introduces most of the important mathematics that will be used repeatedly throughout this book. Whilst scalars like the electric potential *Ф* and the vector fields **E** and **B** are familiar quantities in electricity and magnetism, it is not always known that they are examples of a mathematical form termed a tensor. More sophisticated tensors are required in some descriptions and theories in electrodynamics. As tensor notation is compact and usually allows for a simple derivation of a result, the chapter begins with a series of questions involving Cartesian tensors. The results obtained here will facilitate the solution of many questions in subsequent chapters. Readers who are not familiar with tensors are advised to consult Appendix A before proceeding. Towards the end of this Appendix a ‘checklist for detecting errors when using tensor notation’ is given, which will hopefully provide some help for the uninitiated.

*Keywords:*
vector algebra vector calculus, vector identities Cartesian tensors, Gauss’s theorem Stokes’s theorem, separation of variables Dirac delta function, solid angle time averaging

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- Title Pages
- Preface
- Contents
- 1 Some essential mathematics
- 2 Static electric fields in vacuum
- 3 The electrostatics of conductors
- 4 Static magnetic fields in vacuum
- 5 Quasi-static electric and magnetic fields in vacuum
- 6 Ohm’s law and electric circuits
- 7 Electromagnetic fields and waves in vacuum
- 8 The electromagnetic potentials
- 9 Static electric and magnetic fields in matter
- 10 Some applications of Maxwell’s equations in matter
- 11 Electromagnetic radiation
- 12 Electromagnetism and special relativity
- Appendix A Vectors and Cartesian tensors
- Appendix A Vectors and Cartesian tensors
- Appendix B Cartesian coordinates
- Appendix C Spherical polar coordinates
- Appendix D Cylindrical polar coordinates
- Appendix E The Dirac delta function
- Appendix E The Dirac delta function
- Appendix F Legendre polynomials
- Appendix F Legendre polynomials
- Appendix G Bessel functions
- Appendix G Bessel functions
- Appendix H Parametric representation of a surface
- Appendix H Parametric representation of a surface
- Appendix I The Cauchy–Riemann equations
- Appendix I The Cauchy–Riemann equations
- Appendix J Questions involving computational work
- Glossary of symbols
- Index