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Spherical Functions of Mathematical Geosciences: A Scalar, Vectorial, and Tensorial Setup Second Edition 2022 [Kietas viršelis]

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  • Formatas: Hardback, 729 pages, aukštis x plotis: 235x155 mm, weight: 1280 g, 71 Illustrations, color; 3 Illustrations, black and white; XV, 729 p. 74 illus., 71 illus. in color., 1 Hardback
  • Serija: Geosystems Mathematics
  • Išleidimo metai: 15-Oct-2022
  • Leidėjas: Springer-Verlag
  • ISBN-10: 3662656914
  • ISBN-13: 9783662656914
Kitos knygos pagal šią temą:
Spherical Functions of Mathematical Geosciences: A Scalar, Vectorial, and Tensorial Setup Second Edition 2022Didesnis paveikslėlis
  • Formatas: Hardback, 729 pages, aukštis x plotis: 235x155 mm, weight: 1280 g, 71 Illustrations, color; 3 Illustrations, black and white; XV, 729 p. 74 illus., 71 illus. in color., 1 Hardback
  • Serija: Geosystems Mathematics
  • Išleidimo metai: 15-Oct-2022
  • Leidėjas: Springer-Verlag
  • ISBN-10: 3662656914
  • ISBN-13: 9783662656914
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783662656914.html
Raktažodžiai:

This book is an enlarged second edition of a monograph published in the Springer AGEM2-Series, 2009. It presents, in a consistent and unified overview, a setup of the theory of spherical functions of mathematical (geo-)sciences. The content shows a twofold transition: First, the natural transition from scalar to vectorial and tensorial theory of spherical harmonics is given in a coordinate-free context, based on variants of the addition theorem, Funk-Hecke formulas, and Helmholtz as well as Hardy-Hodge decompositions. Second, the canonical transition from spherical harmonics via zonal (kernel) functions to the Dirac kernel is given in close orientation to an uncertainty principle classifying the space/frequency (momentum) behavior of the functions for purposes of data analysis and (geo-)application. The whole palette of spherical functions is collected in a well-structured form for modeling and simulating the phenomena and processes occurring in the Earth's system. The result is a work which, while reflecting the present state of knowledge in a time-related manner, claims to be of largely timeless significance in (geo-)mathematical research and teaching.


Part I Introduction
1 Introductional Remarks
3(28)
1.1 From Spherical Harmonics to Zonal Kernels
5(11)
1.2 From Scalar Spherical Harmonics to its Vectorial and Tensorial Extensions
16(7)
1.3 Organization of the Work
23(8)
Part II Background and Nomenclature
2 Basic Settings
31(40)
2.1 Euclidean Nomenclature
31(6)
2.2 Euclidean Differential Calculus
37(3)
2.3 Euclidean Integral Calculus
40(3)
2.4 Spherical Nomenclature
43(4)
2.5 Spherical Differential Calculus
47(5)
2.6 Spherical Integral Calculus
52(10)
2.7 Orthogonal Invariance
62(9)
3 Circular Harmonics
71(12)
3.1 Homogeneous Harmonic Polynomials and Circular Harmonics
72(1)
3.2 Eigenfunctions of the Beltrami Operator
72(1)
3.3 Addition Theorem and Chebyshev Polynomial
73(2)
3.4 Stereographic Projection
75(2)
3.5 Closure and Completeness of Circular Harmonics
77(1)
3.6 Inner/Outer Harmonics
78(5)
Part III Spherical Harmonics
4 Scalar Spherical Harmonics
83(110)
4.1 Homogeneous Harmonic Polynomials
84(8)
4.2 Addition Theorem
92(6)
4.3 Exact Computation of Homogeneous Harmonic Polynomials
98(10)
4.4 Definition of Scalar Spherical Harmonics
108(4)
4.5 Legendre Polynomials
112(10)
4.6 Orthogonal (Fourier) Expansions
122(14)
4.7 Legendre (Spherical) Harmonics
136(4)
4.8 Funk-Hecke Formula
140(3)
4.9 Eigenfunctions of the Beltrami Operator
143(1)
4.10 Irreducibility of Scalar Harmonics
144(4)
4.11 Degree and Order Variances
148(6)
4.12 Associated Legendre Polynomials
154(9)
4.13 Associated Legendre (Spherical) Harmonics
163(8)
4.14 Spherical Harmonics in Cartesian Coordinates
171(13)
4.15 Spherical Harmonics in Terms of Wigner D-Matrices
184(7)
4.16 Bibliographical Notes
191(2)
5 Green's Functions and Integral Formulas
193(58)
5.1 Green's Function with Respect to the Beltrami Operator
193(3)
5.2 Space Mollifier Green's Function with Respect to the Beltrami Operator
196(8)
5.3 Frequency Mollifier Green's Function with Respect to the Beltrami Operator
204(4)
5.4 Modified Green Functions
208(4)
5.5 Integral Formulas
212(4)
5.6 Differential Equations
216(2)
5.7 Classical Understanding of Spline Functions
218(6)
5.8 Integral Formulas with Respect to Iterated Beltrami Operators
224(9)
5.9 Differential Equations Respect to Iterated Beltrami Operators
233(3)
5.10 Pseudodifferential Operators, Green's Functions, and Splines
236(13)
5.11 Bibliographical Notes
249(2)
6 Vector Spherical Harmonics
251(74)
6.1 Normal and Tangential Fields
253(1)
6.2 Helmholtz System of Vector Spherical Harmonics
254(9)
6.3 Orthogonal (Fourier) Expansions
263(8)
6.4 Homogeneous Harmonic Vector Polynomials
271(3)
6.5 Exact Computation of Orthonormal Systems
274(6)
6.6 Irreducibility and Orthogonal Invariance of Vector Spherical Harmonics
280(8)
6.7 Vectorial Beltrami Operator
288(2)
6.8 Vectorial Addition Theorem
290(7)
6.9 Vectorial Funk-Hecke Formulas
297(4)
6.10 Counterparts to the Legendre Polynomial
301(4)
6.11 Degree and Order Variances
305(4)
6.12 Hardy-Hodge System of Vector Spherical Harmonics
309(9)
6.13 Orthogonal Expansion Using Vector Legendre Kernels
318(5)
6.14 Bibliographical Notes
323(2)
7 Tensor Spherical Harmonics
325(70)
7.1 Some Nomenclature
326(2)
7.2 Normal and Tangential Fields
328(3)
7.3 Integral Theorems
331(4)
7.4 Helmholtz System of Tensor Spherical Harmonics
335(7)
7.5 Helmholtz Decomposition Theorem
342(4)
7.6 Closure and Completeness of Tensor Spherical Harmonics
346(8)
7.7 Homogeneous Harmonic Tensor Polynomials
354(6)
7.8 Tensorial Beltrami Operator
360(3)
7.9 Tensorial Addition Theorem
363(9)
7.10 Tensorial Funk-Hecke Formulas
372(5)
7.11 Counterparts to the Legendre Polynomials
377(2)
7.12 Tensor Spherical Harmonics Related to Tensor Homogeneous Harmonic Polynomials
379(4)
7.13 Hardy-Hodge System of Tensor Spherical Harmonics
383(6)
7.14 Orthogonal Expansion Using Tensor Legendre Kernels
389(3)
7.15 Bibliographical Notes
392(3)
8 Spin-Weighted Spherical Harmonics
395(66)
8.1 Spin-Weighted Differential Calculus
396(3)
8.2 Spin-Weighted Integral Calculus
399(10)
8.3 Spin-Weighted Spherical Harmonics in Coordinate Representation
409(14)
8.4 Spin-Weighted Spherical Harmonics and the Wigner D-Function
423(30)
8.5 Relations to Vector and Tensor Spherical Harmonics
453(3)
8.6 Bibliographical Notes
456(5)
Part IV Zonal Functions
9 Scalar Zonal Kernel Functions
461(62)
9.1 Bandlimited/Spacelimited Functions
462(1)
9.2 Zonal Kernel Functions
462(4)
9.3 Zonal Kernel Functions in Scalar Context
466(1)
9.4 Convolutions Involving Scalar Zonal Kernel Functions
467(2)
9.5 Classification of Zonal Kernel Functions
469(9)
9.6 Localization of Representative Zonal Kernels
478(13)
9.7 Dirac Families of Zonal Scalar Kernel Functions
491(9)
9.8 Examples of Dirac Families
500(20)
9.9 Bibliographical Notes
520(3)
10 Vector Zonal Kernel Functions
523(16)
10.1 Preparatory Material
524(1)
10.2 Tensor Zonal Kernel Functions of Rank Two in Vectorial Context
525(6)
10.3 Vector Zonal Kernel Functions in Vectorial Context
531(2)
10.4 Convolutions Involving Vector Zonal Kernel Functions
533(2)
10.5 Dirac Families of Zonal Vector Kernel Functions
535(2)
10.6 Bibliographical Notes
537(2)
11 Tensorial Zonal Kernel Functions
539(14)
11.1 Preparatory Material
540(1)
11.2 Tensor Zonal Kernel Functions of Rank Four in Tensorial Context
540(2)
11.3 Convolutions Involving Zonal Tensor Kernel Functions
542(2)
11.4 Tensor Zonal Kernel Functions of Rank Two in Tensorial Context
544(4)
11.5 Dirac Families of Zonal Tensor Kernel Functions
548(1)
11.6 Bibliographical Notes
549(4)
Part V Geopotentials
12 Potentials in Euclidean Space
553(98)
12.1 Newton's Law of Gravitation
553(1)
12.2 Volume Potential
554(4)
12.3 Interior/Exterior Green's Formula
558(7)
12.4 Volume Potential and Gravitational Field
565(5)
12.5 Scalar Inner/Outer Harmonics
570(5)
12.6 Surface Potentials
575(3)
12.7 Boundary-Value Problems for the Laplace Operator
578(12)
12.8 Existence and Regularity Method
590(5)
12.9 Locally and Globally Uniform Approximation
595(17)
12.10 Vector Outer Harmonics and the Gravitational Gradient
612(4)
12.1 ISatellite-to-Satellite Tracking
616(4)
12.12 Tensor Outer Harmonics and the Gravitational Tensor
620(3)
12.13 Satellite Gravitation Gradiometry
623(5)
12.14 Function Systems for Internal Elastic Fields
628(20)
12.15 Bibliographical Notes
648(3)
13 Potentials on the Sphere
651(18)
13.1 Green's Formulas
651(2)
13.2 Surface Harmonicity
653(1)
13.3 Surface Potentials
654(2)
13.4 Curve Potentials
656(1)
13.5 Limit Formulas and Jump Relations
656(1)
13.6 Boundary-Value Problems for the Beltrami Operator
657(6)
13.7 Complete Function Systems
663(4)
13.8 Bibliographical Notes
667(2)
14 Multiscale Mollifier Potential Systems in Geoapplications
669(20)
14.1 Surface Mollifier Gradient Fields and Gravitational Deflections of the Vertical
669(11)
14.2 Surface Mollifier Curl Gradient Fields and Geostrophic Ocean Flow
680(6)
14.3 Bibliographical Notes
686(3)
Part VI Conclusion
15 Concluding Remarks
689(8)
List of Symbols 697(6)
References 703(20)
Index 723
Willi Freeden born in 1948 in Kaldenkirchen/Germany, Studies in Mathematics, Geography, and Philosophy at the RWTH Aachen, 1971 Diplom in Mathematics, 1972 Staatsexamen in Mathematics and Geography, 1975 PhD in Mathematics, 1979 Habilitation in Mathematics, 1981/1982 Visiting Research Professor at the Ohio State University, Columbus (Department of Geodetic Sciences and Surveying), 1984 Professor of Mathematics at the RWTH Aachen (Institute of Pure and Applied Mathematics), 1989 Professor of Technomathematics, 1994 Head of the Geomathematics Group, 2002-2006 Vice-president for Research and Technology at the University of Kaiserslautern.

Michael Schreiner born in 1966 in Mertesheim/Germany, Studies in Industrial Mathematics, Mechanical Engineering, and Computer Science at the University of Kaiserslautern, 1991 Diplom in Industrial Mathematics, 1994 PhD in Mathematics, 2004 Habilitation in Mathematics. 19972001 researcher and project leader at the Hilti Corp. Schaan,Liechtenstein, 2002 Professor for Industrial Mathematics at the University of Buchs NTB, Buchs, Switzerland. 2004 Head of the Department of Mathematics of the University of Buchs, 2004 also Lecturer at the University of Kaiserslautern.