Quaternionic and Clifford calculus for physicists and engineers.

*(English)*Zbl 0897.30023
Mathematical Methods in Practice. Chichester: Wiley. xi, 371 p. (1997).

Quaternionic and Clifford analysis started in the thirtieth but began to flourish in the seventieth. During the last ten years many books appeared on various aspects of the theory. The authors of this book try to give an introduction convenient for physicist and engineers.

Let us start with a glimpse on the content: Chapter 1: ‘Quaternions and multivectors’ (34 p.) gives the necessary introduction into quaternions and Clifford algebras together with the relations to classical vector calculus. Chapter 2: ‘Clifford valued functions and forms’ (40 p.) introduces the generalization of holomorphic functions to regular functions in Clifford analysis with an excursion to exterior forms. Chapter 3: ‘Clifford operator calculus’ (53 p.) presents the various operators (Cauchy, Teodorescu, Fourier etc.) with their most important properties. This is applied in Chapter 4: ‘Boundary value problems’ (106 p.) to a broad range of examples which show the importance and usefulness of the Clifford methods. This and the next Chapter 5: ‘Numerical Clifford analysis’ (66 p.) are the special research areas of both the authors, see also their book ‘Quaternionic analysis and elliptic boundary value problems (Berlin: Akademie Verlag 1989; Zbl 0699.35007) (also published by Birkhäuser/Basel 1990). Chapter 5 gives an overview of the different numerical methods within the Clifford frame, which enables one to come to numerical solutions of the different boundary value problems. Chapter 6: ‘Further results and research problems’ (24 p.) is an interesting account also of open problems. Three appendices, a bibliography of 340 items, and an index complete the work.

The authors deserve well of ‘popularizing’ Clifford analysis and one may hope that this book will show the advantages of these hypercomplex methods to many possible interested users.

Let us start with a glimpse on the content: Chapter 1: ‘Quaternions and multivectors’ (34 p.) gives the necessary introduction into quaternions and Clifford algebras together with the relations to classical vector calculus. Chapter 2: ‘Clifford valued functions and forms’ (40 p.) introduces the generalization of holomorphic functions to regular functions in Clifford analysis with an excursion to exterior forms. Chapter 3: ‘Clifford operator calculus’ (53 p.) presents the various operators (Cauchy, Teodorescu, Fourier etc.) with their most important properties. This is applied in Chapter 4: ‘Boundary value problems’ (106 p.) to a broad range of examples which show the importance and usefulness of the Clifford methods. This and the next Chapter 5: ‘Numerical Clifford analysis’ (66 p.) are the special research areas of both the authors, see also their book ‘Quaternionic analysis and elliptic boundary value problems (Berlin: Akademie Verlag 1989; Zbl 0699.35007) (also published by Birkhäuser/Basel 1990). Chapter 5 gives an overview of the different numerical methods within the Clifford frame, which enables one to come to numerical solutions of the different boundary value problems. Chapter 6: ‘Further results and research problems’ (24 p.) is an interesting account also of open problems. Three appendices, a bibliography of 340 items, and an index complete the work.

The authors deserve well of ‘popularizing’ Clifford analysis and one may hope that this book will show the advantages of these hypercomplex methods to many possible interested users.

Reviewer: Klaus Habetha (Aachen)

##### MSC:

30G35 | Functions of hypercomplex variables and generalized variables |

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |

35J25 | Boundary value problems for second-order elliptic equations |

35J56 | Boundary value problems for first-order elliptic systems |

35J57 | Boundary value problems for second-order elliptic systems |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |

31B20 | Boundary value and inverse problems for harmonic functions in higher dimensions |

30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |

35F15 | Boundary value problems for linear first-order PDEs |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

47F05 | General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) |