Spherical functions of mathematical geosciences: a scalar, vectorial, and tensorial setup
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2009
|
| Series: | Advances in geophysical and environmental mechanics and mathematics
2 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 602 S. Ill., graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783540851110 |
Staff View
MARC
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| 245 | 1 | 0 | |a Spherical functions of mathematical geosciences |b a scalar, vectorial, and tensorial setup |c Willi Freeden ; Michael Schreiner |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2009 | |
| 300 | |a XV, 602 S. |b Ill., graph. Darst. |c 235 mm x 155 mm | ||
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Record in the Search Index
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| adam_text | Contents
Preface
XIII
1 Introduction............................. 1
1.1 Motivation............................ 3
1.2 Layout.............................. 14
2 Basic Settings and Spherical Nomenclature........... 19
2.1 Scalars, Vectors, and Tensors.................. 19
2.2 Differential Operators...................... 24
2.3 Spherical Notation ....................... 30
2.4 Function Spaces......................... 32
2.5 Differential Calculus ...................... 35
2.6 Integral Calculus........................ 39
2.7 Orthogonal Invariance ..................... 48
3 Scalar Spherical Harmonics.................... 57
3.1 Homogeneous Harmonic Polynomials............. 58
3.2 Addition Theorem........................ 65
3.3 Exact Computation of Basis Systems............. 71
3.4 Definition of Scalar Spherical Harmonics........... 81
3.5 Legendre Polynomials...................... 87
3.6 Orthogonal (Fourier) Expansions ............... 97
3.7 Legendre (Spherical) Harmonics................ 110
3.8 Funk-Hecke Formula...................... 115
3.9 Eigenfunctions of the Beltrami Operator........... 117
3.10 Irreducibility of Scalar Harmonics............... 119
3.11 Degree and Order Variances.................. 122
3.12 Associated Legendre Polynomials............... 129
3.13 Associated Legendre (Spherical) Harmonics ......... 138
3.14 Exact Computation of Legendre Basis Systems........ 153
3.15 Bibliographical Notes...................... 158
4 Green s Functions and Integral Formulas............. 159
4.1 Green s Function with Respect to the Beltrami Operator . . 159
4.2 Space Regularized Green Function with Respect to the Bel-
trami Operator......................... 162
IX
Contents
4.3 Frequency Regularized Green Function with Respect to the
Beltrami Operator........................170
4.4 Modified Green Functions ...................173
4.5 Integral Formulas........................176
4.6 Differential Equations......................181
4.7 Approximate Integration and Spline Interpolation......183
4.8 Integral Formulas with Respect to Iterated Beltrami Operatorsl89
4.9 Differential Equations Respect to Iterated Beltrami Operators 198
4.10 Bibliographical Notes......................200
Vector Spherical Harmonics....................201
5.1 Normal and Tangential Fields.................202
5.2 Definition of Vector Spherical Harmonics...........203
•5.3 Helmholtz Decomposition Theorem for Spherical Vector
Fields...............................208
5.4 Orthogonal (Fourier) Expansions ...............212
5.5 Homogeneous Harmonic Vector Polynomials.........220
5.6 Exact Computation of Orthonormal Systems.........223
5.7 Orthogonal Invariance .....................228
5.8 Vectorial Beltrami Operator..................236
5.9 Vectorial Addition Theorem..................238
5.10 Vectorial Funk-Hecke Formulas................244
5.11 Counterparts of the Legendre Polynomial...........248
5.12 Degree and Order Variances..................252
5.13 Vector Homogeneous Harmonic Polynomials.........257
5.14 Alternative Systems of Vector Spherical Harmonics.....260
5.15 Vector Legendre Kernels....................266
5.16 Bibliographical Notes......................271
Tensor Spherical Harmonics....................273
6.1 Some Nomenclature.......................274
6.2 Normal and Tangential Fields.................275
6.3 Integral Theorems........................278
6.4 Definition of Tensor Spherical Harmonics...........283
6.5 Helmholtz Decomposition Theorem..............289
6.6 Orthogonal (Fourier) Expansions ...............293
6.7 Homogeneous Harmonic Tensor Polynomials.........301
6.8 Tensorial Beltrami Operator..................306
6.9 Tensorial Addition Theorem..................309
6.10 Tensorial Funk-Hecke Formulas................318
6.11 Counterparts to the Legendre Polynomials..........323
6.12 Tensor Homogeneous Harmonic Polynomials.........325
6.13 Alternative Systems of Tensor Spherical Harmonics.....328
Contents xi
6.14 Tensor Legendre Kernels....................334
6.15 Bibliographical Notes......................337
7 Scalar Zonal Kernel Functions...................339
7.1 Zonal Kernel Functions in Scalar Context...........339
7.2 Convolutions Involving Scalar Zonal Kernel Functions . . . . . 541
7.3 Classification of Zonal Kernel Functions ...........343
7.4 Dirac Families of Zonal Scalar Kernel Functions........ 557
7.5 Examples of Dirac Families...................366
7.6 Bibliographical Notes......................386
8 Vector Zonal Kernel Functions..................389
8.1 Preparatory Material......................390
8.2 Tensor Zonal Kernel Functions of Rank Two in Vectorial
Context .............................391
8.3 Vector Zonal Kernel Functions in Vectorial Context.....396
8.4 Convolutions Involving Vector Zonal Kernel Functions . . . 399
8.5 Dirac Families of Zonal Vector Kernel Functions.......401
8.6 Bibliographical Notes......................403
9 Tensorial Zonal Kernel Functions.................405
9.1 Preparatory Material......................406
9.2 Tensor Zonal Kernel Functions of Rank Four in Tensorial
Context .............................406
9.3 Convolutions Involving Zonal Tensor Kernel Functions . . . 408
9.4 Tensor Zonal Kernel Functions of Rank Two in Tensorial
Context .............................410
9.5 Dirac Families of Zonal Tensor Kernel Functions.......414
9.6 Bibliographical Notes......................415
10 Zonal Function Modeling of Earth s Mass Distribution.....417
10.1 Key Observables.........................418
10.2 Gravity Potential........................428
10.3 Inner/Outer Harmonics.....................435
10.4 Limit Formulas and Jump Relations..............454
10.5 Gravity Anomalies and Deflections of the Vertical......458
10.6 Geostrophic Ocean Flow and Dynamic Ocean Topography . 482
10.7 Elastic Field...........................496
10.8 Density Distribution ......................515
10.9 Vector Outer Harmonics and the Gravitational Gradient . . 542
10.10 Tensor Outer Harmonics and the Gravitational Tensor . . . 551
10.11 Gravity Quantities in Spherical Nomenclature........560
10.12 Pseudodifferential Operators and Geomathematics .....564
10.13 Bibliographical Notes......................568
xii Contents
Concluding Remarks.......................... 571
List of Symbols............................. 573
Bibliography .............................. 579
Index .................................. 597
|
| adam_txt |
Contents
Preface
XIII
1 Introduction. 1
1.1 Motivation. 3
1.2 Layout. 14
2 Basic Settings and Spherical Nomenclature. 19
2.1 Scalars, Vectors, and Tensors. 19
2.2 Differential Operators. 24
2.3 Spherical Notation . 30
2.4 Function Spaces. 32
2.5 Differential Calculus . 35
2.6 Integral Calculus. 39
2.7 Orthogonal Invariance . 48
3 Scalar Spherical Harmonics. 57
3.1 Homogeneous Harmonic Polynomials. 58
3.2 Addition Theorem. 65
3.3 Exact Computation of Basis Systems. 71
3.4 Definition of Scalar Spherical Harmonics. 81
3.5 Legendre Polynomials. 87
3.6 Orthogonal (Fourier) Expansions . 97
3.7 Legendre (Spherical) Harmonics. 110
3.8 Funk-Hecke Formula. 115
3.9 Eigenfunctions of the Beltrami Operator. 117
3.10 Irreducibility of Scalar Harmonics. 119
3.11 Degree and Order Variances. 122
3.12 Associated Legendre Polynomials. 129
3.13 Associated Legendre (Spherical) Harmonics . 138
3.14 Exact Computation of Legendre Basis Systems. 153
3.15 Bibliographical Notes. 158
4 Green's Functions and Integral Formulas. 159
4.1 Green's Function with Respect to the Beltrami Operator . . 159
4.2 Space Regularized Green Function with Respect to the Bel-
trami Operator. 162
IX
Contents
4.3 Frequency Regularized Green Function with Respect to the
Beltrami Operator.170
4.4 Modified Green Functions .173
4.5 Integral Formulas.176
4.6 Differential Equations.181
4.7 Approximate Integration and Spline Interpolation.183
4.8 Integral Formulas with Respect to Iterated Beltrami Operatorsl89
4.9 Differential Equations Respect to Iterated Beltrami Operators 198
4.10 Bibliographical Notes.200
Vector Spherical Harmonics.201
5.1 Normal and Tangential Fields.202
5.2 Definition of Vector Spherical Harmonics.203
•5.3 Helmholtz Decomposition Theorem for Spherical Vector
Fields.208
5.4 Orthogonal (Fourier) Expansions .212
5.5 Homogeneous Harmonic Vector Polynomials.220
5.6 Exact Computation of Orthonormal Systems.223
5.7 Orthogonal Invariance .228
5.8 Vectorial Beltrami Operator.236
5.9 Vectorial Addition Theorem.238
5.10 Vectorial Funk-Hecke Formulas.244
5.11 Counterparts of the Legendre Polynomial.248
5.12 Degree and Order Variances.252
5.13 Vector Homogeneous Harmonic Polynomials.257
5.14 Alternative Systems of Vector Spherical Harmonics.260
5.15 Vector Legendre Kernels.266
5.16 Bibliographical Notes.271
Tensor Spherical Harmonics.273
6.1 Some Nomenclature.274
6.2 Normal and Tangential Fields.275
6.3 Integral Theorems.278
6.4 Definition of Tensor Spherical Harmonics.283
6.5 Helmholtz Decomposition Theorem.289
6.6 Orthogonal (Fourier) Expansions .293
6.7 Homogeneous Harmonic Tensor Polynomials.301
6.8 Tensorial Beltrami Operator.306
6.9 Tensorial Addition Theorem.309
6.10 Tensorial Funk-Hecke Formulas.318
6.11 Counterparts to the Legendre Polynomials.323
6.12 Tensor Homogeneous Harmonic Polynomials.325
6.13 Alternative Systems of Tensor Spherical Harmonics.328
Contents xi
6.14 Tensor Legendre Kernels.334
6.15 Bibliographical Notes.337
7 Scalar Zonal Kernel Functions.339
7.1 Zonal Kernel Functions in Scalar Context.339
7.2 Convolutions Involving Scalar Zonal Kernel Functions . . . . .'541
7.3 Classification of Zonal Kernel Functions .343
7.4 Dirac Families of Zonal Scalar Kernel Functions."557
7.5 Examples of Dirac Families.366
7.6 Bibliographical Notes.386
8 Vector Zonal Kernel Functions.389
8.1 Preparatory Material.390
8.2 Tensor Zonal Kernel Functions of Rank Two in Vectorial
Context .391
8.3 Vector Zonal Kernel Functions in Vectorial Context.396
8.4 Convolutions Involving Vector Zonal Kernel Functions . . . 399
8.5 Dirac Families of Zonal Vector Kernel Functions.401
8.6 Bibliographical Notes.403
9 Tensorial Zonal Kernel Functions.405
9.1 Preparatory Material.406
9.2 Tensor Zonal Kernel Functions of Rank Four in Tensorial
Context .406
9.3 Convolutions Involving Zonal Tensor Kernel Functions . . . 408
9.4 Tensor Zonal Kernel Functions of Rank Two in Tensorial
Context .410
9.5 Dirac Families of Zonal Tensor Kernel Functions.414
9.6 Bibliographical Notes.415
10 Zonal Function Modeling of Earth's Mass Distribution.417
10.1 Key Observables.418
10.2 Gravity Potential.428
10.3 Inner/Outer Harmonics.435
10.4 Limit Formulas and Jump Relations.454
10.5 Gravity Anomalies and Deflections of the Vertical.458
10.6 Geostrophic Ocean Flow and Dynamic Ocean Topography . 482
10.7 Elastic Field.496
10.8 Density Distribution .515
10.9 Vector Outer Harmonics and the Gravitational Gradient . . 542
10.10 Tensor Outer Harmonics and the Gravitational Tensor . . . 551
10.11 Gravity Quantities in Spherical Nomenclature.560
10.12 Pseudodifferential Operators and Geomathematics .564
10.13 Bibliographical Notes.568
xii Contents
Concluding Remarks. 571
List of Symbols. 573
Bibliography . 579
Index . 597 |
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| id | DE-604.BV035062839 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:01:07Z |
| indexdate | 2024-07-09T21:21:21Z |
| institution | BVB |
| isbn | 9783540851110 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016731337 |
| oclc_num | 233934589 |
| open_access_boolean | |
| owner | DE-29 |
| owner_facet | DE-29 |
| physical | XV, 602 S. Ill., graph. Darst. 235 mm x 155 mm |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | Advances in geophysical and environmental mechanics and mathematics |
| series2 | Advances in geophysical and environmental mechanics and mathematics |
| spelling | Freeden, Willi 1948- Verfasser (DE-588)121167259 aut Spherical functions of mathematical geosciences a scalar, vectorial, and tensorial setup Willi Freeden ; Michael Schreiner Berlin [u.a.] Springer 2009 XV, 602 S. Ill., graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Advances in geophysical and environmental mechanics and mathematics 2 Geologie Mathematik Geology Mathematics Spherical functions Geowissenschaften (DE-588)4020288-4 gnd rswk-swf Kugelfunktion (DE-588)4033494-6 gnd rswk-swf Geowissenschaften (DE-588)4020288-4 s Kugelfunktion (DE-588)4033494-6 s DE-604 Schreiner, Michael Verfasser aut Advances in geophysical and environmental mechanics and mathematics 2 (DE-604)BV035223452 2 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016731337&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Freeden, Willi 1948- Schreiner, Michael Spherical functions of mathematical geosciences a scalar, vectorial, and tensorial setup Advances in geophysical and environmental mechanics and mathematics Geologie Mathematik Geology Mathematics Spherical functions Geowissenschaften (DE-588)4020288-4 gnd Kugelfunktion (DE-588)4033494-6 gnd |
| subject_GND | (DE-588)4020288-4 (DE-588)4033494-6 |
| title | Spherical functions of mathematical geosciences a scalar, vectorial, and tensorial setup |
| title_auth | Spherical functions of mathematical geosciences a scalar, vectorial, and tensorial setup |
| title_exact_search | Spherical functions of mathematical geosciences a scalar, vectorial, and tensorial setup |
| title_exact_search_txtP | Spherical functions of mathematical geosciences a scalar, vectorial, and tensorial setup |
| title_full | Spherical functions of mathematical geosciences a scalar, vectorial, and tensorial setup Willi Freeden ; Michael Schreiner |
| title_fullStr | Spherical functions of mathematical geosciences a scalar, vectorial, and tensorial setup Willi Freeden ; Michael Schreiner |
| title_full_unstemmed | Spherical functions of mathematical geosciences a scalar, vectorial, and tensorial setup Willi Freeden ; Michael Schreiner |
| title_short | Spherical functions of mathematical geosciences |
| title_sort | spherical functions of mathematical geosciences a scalar vectorial and tensorial setup |
| title_sub | a scalar, vectorial, and tensorial setup |
| topic | Geologie Mathematik Geology Mathematics Spherical functions Geowissenschaften (DE-588)4020288-4 gnd Kugelfunktion (DE-588)4033494-6 gnd |
| topic_facet | Geologie Mathematik Geology Mathematics Spherical functions Geowissenschaften Kugelfunktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016731337&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035223452 |
| work_keys_str_mv | AT freedenwilli sphericalfunctionsofmathematicalgeosciencesascalarvectorialandtensorialsetup AT schreinermichael sphericalfunctionsofmathematicalgeosciencesascalarvectorialandtensorialsetup |