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Geomathematics: Modelling and Solving Mathematical Problems in Geodesy and Geophysics [Kõva köide]

(Universität Siegen, Germany)
  • Formaat: Hardback, 466 pages, kõrgus x laius x paksus: 251x174x26 mm, kaal: 1000 g, Worked examples or Exercises
  • Ilmumisaeg: 28-Apr-2022
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1108419445
  • ISBN-13: 9781108419444
  • Formaat: Hardback, 466 pages, kõrgus x laius x paksus: 251x174x26 mm, kaal: 1000 g, Worked examples or Exercises
  • Ilmumisaeg: 28-Apr-2022
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1108419445
  • ISBN-13: 9781108419444
"Eratosthenes of Cyrene, who lived basically in the third century B.C., was one of those first mathematicians whose knowledge and abilities at these early stages of human civilization was remarkable. Besides his method for seeking prime numbers, he particularly also contributed to the measurement of the Earth by, for example, determining its circumference. In this respect, he might have been the geomathematician, or at least one of the. Many more followed him, where definitely Carl Friedrich Gauss must be mentioned here, who can be seen as the greatest genius in mathematical history. His works and their in uence are widespread in mathematics and they are also of essential importance in various applications, in particular and (in the author's possibly biased point of view) first of all in Earth sciences, especially geomagnetics and potential theory. The awareness, which reaches back to the classical antiquity, that mathematics is the foremostly required skill and toolbox for understanding the objects and processes that surround us has been preserved up to the presence. It has nicely and more generally been put in a nutshell by the quotation above, which is from Kant (1786), for the English translation, see Kant (1883). Over the centuries, Earth sciences and mathematics have both advanced. While the achievements at the time of Eratosthenes and his fellows are nowadays parts of the curricula at schools, many modern challenges in geosciences are equally challenges to 21st century mathematics.Eratosthenes of Cyrene, who lived basically in the third century B.C., was one of those first mathematicians whose knowledge and abilities at these early stages of human civilization was remarkable. Besides his method for seeking prime numbers, he particularly also contributed to the measurement of the Earth by, for example, determining its circumference. In this respect, he might have been the first geomathematician, or at least one of the first. Many more followed him, where definitely Carl Friedrich Gauss must be mentioned here, who can be seen as the greatest genius in mathematical history. His works and their in uence are widespread in mathematics and they are also of essential importance in various applications, in particular and (in the author's possibly biased point of view) first of all in Earth sciences, especially geomagnetics and potential theory. The awareness, which reaches back to the classical antiquity, that mathematics is the foremostly required skill and toolbox for understanding the objects and processes that surround us has been preserved up to the presence. It has nicely and more generally been put in a nutshell by the quotation above, which is from Kant (1786), for the English translation, see Kant (1883). Over the centuries, Earth sciences and mathematics have both advanced. While the achievements at the time of Eratosthenes and his fellows are nowadays parts of the curricula at schools, many modern challenges in geosciences are equally challenges to 21st century mathematics"--

Arvustused

'Volker does a good job of presenting thorough associated proofs. These chapters serve much as a mathematical methods survey course would in a physics curriculum, presenting the needed mathematics without unduly burdening the reader with the breadth that would normally be offered in a mathematics course Recommended.' E. Kincanon, Choice

Muu info

A comprehensive summary of the fundamental mathematical principles behind key topics in geophysics and geodesy.
1 Introduction
1(3)
2 Required Mathematical Basics
4(46)
2.1 Some Important Definitions
4(3)
2.2 A Short Course on Tensors
7(4)
2.3 Derivatives: Notations and More
11(2)
2.4 Some Theorems on Integration
13(4)
2.5 Selected Topics of Functional Analysis
17(28)
2.6 Curves and Surfaces
45(5)
3 On Gravitation, Harmonic Functions, and Related Topics
50(92)
3.1 The Gravitational Potential
50(20)
3.2 Some Fundamental Properties of Harmonic Functions
70(9)
3.3 Boundary-Value Problems, Green's Function, and Layer Potentials
79(36)
3.4 The Sphere as a Particular Boundary
115(22)
3.5 A Brief Excursion to Other Dimensions
137(5)
Exercises
139(3)
4 Basis Functions
142(122)
4.1 Spherical Analysis
142(9)
4.2 Spherical Harmonics and Legendre Polynomials
151(17)
4.3 Vector Spherical Harmonics
168(15)
4.4 Tensor Spherical Harmonics
183(8)
4.5 On the Multitude of Trial Functions
191(3)
4.6 Slepian Functions on the Sphere
194(17)
4.7 Radial Basis Functions and Sobolev Spaces
211(21)
4.8 Basis Functions on the 3D Ball - Briefly
232(5)
4.9 A Best Basis Algorithm
237(27)
Exercises
263(1)
5 Inverse Problems
264(61)
5.1 Example 1: Downward Continuation
264(2)
5.2 Example 2: Inverse Gravimetry
266(5)
5.3 Basic Theory of Inverse Problems and Their Regularization
271(34)
5.4 Best Bases for Ill-Posed Inverse Problems
305(20)
Exercises
322(3)
6 The Magnetic Field
325(25)
6.1 The Governing Equations
325(2)
6.2 The GauB Representation and Inner and Outer Sources
327(2)
6.3 The Spherical Helmholtz Decomposition and the Mie Representation
329(15)
6.4 Internal and External Fields under the Mie Representation
344(6)
Exercises
349(1)
7 Mathematical Models in Seismology
350(97)
7.1 Continuum Mechanics with Focus on Elasticity
350(24)
7.2 Specifics and Simplifications for the Elastic Body Earth
374(12)
7.3 Propagation of Body Waves and Normal Modes
386(20)
7.4 An Inverse Problem in Seismology
406(41)
Exercises
423(2)
Appendix A Hints for the Exercises
425(3)
Appendix B Questions for Understanding
428(5)
References
433(14)
Index 447
Volker Michel is a mathematics professor at the University of Siegen where he founded the geomathematics group. He is also an editor-in-chief of the 'International Journal on Geomathematics' and a member of the editorial boards of the Journal of Geodesy and Mathematics of Computation and Data Science. Previous works include Lectures on Constructive Approximation (Springer, 2013) and 'Multiscale Potential Theory' (Birkhauser, 2004) with Willi Freeden. Michel has organised several conferences on inverse problems and geomathematics.