Applied mathematics: body and soul 1 Derivatives and geometry in IR3
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2004
|
| Schriftenreihe: | Applied mathematics
1 |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019281270&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XLIII, 425 S. Ill., graph. Darst. |
| ISBN: | 354000890X |
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| 245 | 1 | 0 | |a Applied mathematics |b body and soul |n 1 |p Derivatives and geometry in IR3 |c K. Eriksson ; D. Estep ; C. Johnson |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2004 | |
| 300 | |a XLIII, 425 S. |b Ill., graph. Darst. | ||
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| 490 | 0 | |a Applied mathematics : body and soul |v ... | |
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Datensatz im Suchindex
| _version_ | 1819376307274776576 |
|---|---|
| adam_text | K. ERIKSSON * D. ESTEP * C.JOHNSON APPLIED MATHEMATICS: BODY AND SOUL
[VOLUME 1] DERIVATIVES AND GEOMETRY IN IR 3 SPRINGER CONTENTS VOLUME 1
DERIVATIVES AND GEOMETRY IN R 3 1 1 WHAT IS MATHEMATICS? 3 1.1
INTRODUCTION 3 1.2 THE MODERN WORLD 3 1.3 THE ROLE OF MATHEMATICS 6 1.4
DESIGN AND PRODUCTION OF CARS 11 1.5 NAVIGATION: FROM STARS TO GPS 11
1.6 MEDICAL TOMOGRAPHY 11 1.7 MOLECULAR DYNAMICS AND MEDICAL DRUG DESIGN
. . . . 12 1.8 WEATHER PREDICTION AND GLOBAL WARMING 13 1.9 ECONOMY:
STOCKS AND OPTIONS 13 1.10 LANGUAGES 14 1.11 MATHEMATICS AS THE LANGUAGE
OF SCIENCE 15 1.12 THE BASIC AREAS OF MATHEMATICS 16 1.13 WHAT IS
SCIENCE? 17 1.14 WHAT IS CONSCIENCE? 17 1.15 HOW TO VIEW THIS BOOK AS A
FRIEND 18 2 THE MATHEMATICS LABORATORY 21 2.1 INTRODUCTION 21 2.2 MATH
EXPERIENCE 22 XVIII CONTENTS VOLUME 1 3 INTRODUCTION TO MODELING 25 3.1
INTRODUCTION 25 3.2 THE DINNER SOUP MODEL 25 3.3 THE MUDDY YARD MODEL 28
3.4 A SYSTEM OF EQUATIONS 29 3.5 FORMULATING AND SOLVING EQUATIONS 30 4
A VERY SHORT CALCULUS COURSE 33 4.1 INTRODUCTION 33 4.2 ALGEBRAIC
EQUATIONS 34 4.3 DIFFERENTIAL EQUATIONS 34 4.4 GENERALIZATION 39 4.5
LEIBNIZ TEEN-AGE DREAM 41 4.6 SUMMARY 43 4.7 LEIBNIZ 44 5 NATURAL
NUMBERS AND INTEGERS 47 5.1 INTRODUCTION 47 5.2 THE NATURAL NUMBERS 48
5.3 IS THERE A LARGEST NATURAL NUMBER? 51 5.4 THE SET N OF ALL NATURAL
NUMBERS 52 5.5 INTEGERS 53 5.6 ABSOLUTE VALUE AND THE DISTANCE BETWEEN
NUMBERS . . 56 5.7 DIVISION WITH REMAINDER 57 5.8 FACTORIZATION INTO
PRIME FACTORS 58 5.9 COMPUTER REPRESENTATION OF INTEGERS 59 6
MATHEMATICAL INDUCTION 63 6.1 INDUCTION 63 6.2 CHANGES IN A POPULATION
OF INSECTS 68 7 RATIONAL NUMBERS 71 7.1 INTRODUCTION 71 7.2 HOW TO
CONSTRUCT THE RATIONAL NUMBERS 72 7.3 ON THE NEED FOR RATIONAL NUMBERS
75 7.4 DECIMAL EXPANSIONS OF RATIONAL NUMBERS 75 7.5 PERIODIC DECIMAL
EXPANSIONS OF RATIONAL NUMBERS . . 76 7.6 SET NOTATION 80 7.7 THE SET Q
OF ALL RATIONAL NUMBERS 81 7.8 THE RATIONAL NUMBER LINE AND INTERVALS 82
7.9 GROWTH OF BACTERIA 83 7.10 CHEMICAL EQUILIBRIUM 85 CONTENTS VOLUME 1
XIX 8 PYTHAGORAS AND EUCLID 87 8.1 INTRODUCTION 87 8.2 PYTHAGORAS
THEOREM 87 8.3 THE SUM OF THE ANGLES OF A TRIANGLE IS 180 89 8.4
SIMILAR TRIANGLES 91 8.5 WHEN ARE TWO STRAIGHT LINES ORTHOGONAL? 91 8.6
THE GPS NAVIGATOR 94 8.7 GEOMETRIC DEFINITION OF SIN(I ) AND COS(I ) 96
8.8 GEOMETRIC PROOF OF ADDITION FORMULAS FOR COS(V) ... 97 8.9
REMEMBERING SOME AREA FORMULAS 98 8.10 GREEK MATHEMATICS 98 8.11 THE
EUCLIDEAN PLANE Q 2 99 8.12 FROM PYTHAGORAS TO EUCLID TO DESCARTES 100
8.13 NON-EUCLIDEAN GEOMETRY 101 9 WHAT IS A FUNCTION? 103 9.1
INTRODUCTION 103 9.2 FUNCTIONS IN DAILY LIFE 106 9.3 GRAPHING FUNCTIONS
OF INTEGERS 109 9.4 GRAPHING FUNCTIONS OF RATIONAL NUMBERS 112 9.5 A
FUNCTION OF TWO VARIABLES 114 9.6 FUNCTIONS OF SEVERAL VARIABLES 116 10
POLYNOMIAL FUNCTIONS 119 10.1 INTRODUCTION 119 10.2 LINEAR POLYNOMIALS
120 10.3 PARALLEL LINES 124 10.4 ORTHOGONAL LINES 124 10.5 QUADRATIC
POLYNOMIALS 125 10.6 ARITHMETIC WITH POLYNOMIALS 129 10.7 GRAPHS OF
GENERAL POLYNOMIALS 135 10.8 PIECEWISE POLYNOMIAL FUNCTIONS 137 11
COMBINATIONS OF FUNCTIONS 141 11.1 INTRODUCTION 141 11.2 SUM OF TWO
FUNCTIONS AND PRODUCT OF A FUNCTION WITH A NUMBER 142 11.3 LINEAR
COMBINATIONS OF FUNCTIONS 142 11.4 MULTIPLICATION AND DIVISION OF
FUNCTIONS 143 11.5 RATIONAL FUNCTIONS 143 11.6 THE COMPOSITION OF
FUNCTIONS 145 12 LIPSCHITZ CONTINUITY 149 12.1 INTRODUCTION 149 12.2 THE
LIPSCHITZ CONTINUITY OF A LINEAR FUNCTION 150 XX CONTENTS VOLUME 1 12.3
THE DEFINITION OF LIPSCHITZ CONTINUITY 151 12.4 MONOMIALS 154 12.5
LINEAR COMBINATIONS OF FUNCTIONS 157 12.6 BOUNDED FUNCTIONS 158 12.7 THE
PRODUCT OF FUNCTIONS 159 12.8 THE QUOTIENT OF FUNCTIONS 160 12.9 THE
COMPOSITION OF FUNCTIONS 161 12.10 FUNCTIONS OF TWO RATIONAL VARIABLES
162 12.11 FUNCTIONS OF SEVERAL RATIONAL VARIABLES 163 13 SEQUENCES AND
LIMITS 165 13.1 A FIRST ENCOUNTER WITH SEQUENCES AND LIMITS 165 13.2
SOCKET WRENCH SETS 167 13.3 J.P. JOHANSSON S ADJUSTABLE WRENCHES 169
13.4 THE POWER OF LANGUAGE: FROM INFINITELY MANY TO ONE 169 13.5 THE E -
N DEFINITION OF A LIMIT 170 13.6 A CONVERGING SEQUENCE HAS A UNIQUE
LIMIT 174 13.7 LIPSCHITZ CONTINUOUS FUNCTIONS AND SEQUENCES . . . . 175
13.8 GENERALIZATION TO FUNCTIONS OF TWO VARIABLES 176 13.9 COMPUTING
LIMITS 177 13.10 COMPUTER REPRESENTATION OF RATIONAL NUMBERS . . . . 180
13.11 SONYA KOVALEVSKAYA 181 14 THE SQUARE ROOT OF TWO 185 14.1
INTRODUCTION 185 14.2 Y/2 IS NOT A RATIONAL NUMBER! 187 14.3 COMPUTING
Y/2 BY THE BISECTION ALGORITHM 188 14.4 THE BISECTION ALGORITHM
CONVERGES! 189 14.5 FIRST ENCOUNTERS WITH CAUCHY SEQUENCES 192 14.6
COMPUTING Y/2 BY THE DECA-SECTION ALGORITHM 192 15 REAL NUMBERS 195 15.1
INTRODUCTION 195 15.2 ADDING AND SUBTRACTING REAL NUMBERS 197 15.3
GENERALIZATION TO F(X,X) WITH / LIPSCHITZ 199 15.4 MULTIPLYING AND
DIVIDING REAL NUMBERS 200 15.5 THE ABSOLUTE VALUE 200 15.6 COMPARING TWO
REAL NUMBERS 200 15.7 SUMMARY OF ARITHMETIC WITH REAL NUMBERS 201 15.8
WHY ^2^/2 EQUALS 2 201 15.9 A REFLECTION ON THE NATURE OF [2 202 15.10
CAUCHY SEQUENCES OF REAL NUMBERS 203 15.11 EXTENSION FROM / : Q - Q TO
/ : R - R 204 15.12 LIPSCHITZ CONTINUITY OF EXTENDED FUNCTIONS 205
CONTENTS VOLUME 1 XXI 15.13 GRAPHING FUNCTIONS / : R - R 206 15.14
EXTENDING A LIPSCHITZ CONTINUOUS FUNCTION 206 15.15 INTERVALS OF REAL
NUMBERS 207 15.16 WHAT IS F(X) IF X IS IRRATIONAL? 208 15.17 CONTINUITY
VERSUS LIPSCHITZ CONTINUITY 211 16 THE BISECTION ALGORITHM FOR F(X) = 0
215 16.1 BISECTION 215 16.2 AN EXAMPLE 217 16.3 COMPUTATIONAL COST 219
17 DO MATHEMATICIANS QUARREL?* 221 17.1 INTRODUCTION 221 17.2 THE
FORMALISTS 224 17.3 THE LOGICISTS AND SET THEORY 224 17.4 THE
CONSTRUCTIVISTS 227 17.5 THE PEANO AXIOM SYSTEM FOR NATURAL NUMBERS . .
. . 229 17.6 REAL NUMBERS 229 17.7 CANTOR VERSUS KRONECKER 230 17.8
DECIDING WHETHER A NUMBER IS RATIONAL OR IRRATIONAL . 232 17.9 THE SET
OF ALL POSSIBLE BOOKS 233 17.10 RECIPES AND GOOD FOOD 234 17.11 THE NEW
MATH IN ELEMENTARY EDUCATION 234 17.12 THE SEARCH FOR RIGOR IN
MATHEMATICS 235 17.13 A NON-CONSTRUCTIVE PROOF 236 17.14 SUMMARY 237 18
THE FUNCTION Y = X R 241 18.1 THE FUNCTION Y/X 241 18.2 COMPUTING WITH
THE FUNCTION Y/X 242 18.3 IS Y/X LIPSCHITZ CONTINUOUS ONL + ? 242 18.4
THE FUNCTION X R FOR RATIONAL R = | 243 18.5 COMPUTING WITH THE FUNCTION
X R 243 18.6 GENERALIZING THE CONCEPT OF LIPSCHITZ CONTINUITY . . . 243
18.7 TURBULENT FLOW IS HOLDER (LIPSCHITZ) CONTINUOUS WITH EX- PONENT |
244 19 FIXED POINTS AND CONTRACTION MAPPINGS 245 19.1 INTRODUCTION 245
19.2 CONTRACTION MAPPINGS 246 19.3 REWRITING F(X) = 0 AS X = G{X) 247
19.4 CARD SALES MODEL 248 19.5 PRIVATE ECONOMY MODEL 249 19.6 FIXED
POINT ITERATION IN THE CARD SALES MODEL 250 19.7 A CONTRACTION MAPPING
HAS A UNIQUE FIXED POINT . . 254 XXII CONTENTS VOLUME 1 19.8
GENERALIZATION TO G : [A, B] * » [A, B] 256 19.9 LINEAR CONVERGENCE IN
FIXED POINT ITERATION 257 19.10 QUICKER CONVERGENCE 258 19.11 QUADRATIC
CONVERGENCE 259 20 ANALYTIC GEOMETRY IN R 2 265 20.1 INTRODUCTION 265
20.2 DESCARTES, INVENTOR OF ANALYTIC GEOMETRY 266 20.3 DESCARTES:
DUALISM OF BODY AND SOUL 266 20.4 THE EUCLIDEAN PLANE R 2 267 20.5
SURVEYORS AND NAVIGATORS 269 20.6 A FIRST GLIMPSE OF VECTORS 270 20.7
ORDERED PAIRS AS POINTS OR VECTORS/ARROWS 271 20.8 VECTOR ADDITION 272
20.9 VECTOR ADDITION AND THE PARALLELOGRAM LAW 273 20.10 MULTIPLICATION
OF A VECTOR BY A REAL NUMBER 274 20.11 THE NORM OF A VECTOR 275 20.12
POLAR REPRESENTATION OF A VECTOR 275 20.13 STANDARD BASIS VECTORS 277
20.14 SCALAR PRODUCT 278 20.15 PROPERTIES OF THE SCALAR PRODUCT 278
20.16 GEOMETRIC INTERPRETATION OF THE SCALAR PRODUCT . . . . 279 20.17
ORTHOGONALITY AND SCALAR PRODUCT 280 20.18 PROJECTION OF A VECTOR ONTO A
VECTOR 281 20.19 ROTATION BY 90 283 20.20 ROTATION BY AN ARBITRARY
ANGLE 9 285 20.21 ROTATION BY 0 AGAIN! 286 20.22 ROTATING A COORDINATE
SYSTEM 286 20.23 VECTOR PRODUCT 287 20.24 THE AREA OF A TRIANGLE WITH A
CORNER AT THE ORIGIN . . 290 20.25 THE AREA OF A GENERAL TRIANGLE 290
20.26 THE AREA OF A PARALLELOGRAM SPANNED BY TWO VECTORS 291 20.27
STRAIGHT LINES 292 20.28 PROJECTION OF A POINT ONTO A LINE 294 20.29
WHEN ARE TWO LINES PARALLEL? 294 20.30 A SYSTEM OF TWO LINEAR EQUATIONS
IN TWO UNKNOWNS 295 20.31 LINEAR INDEPENDENCE AND BASIS 297 20.32 THE
CONNECTION TO CALCULUS IN ONE VARIABLE 298 20.33 LINEAR MAPPINGS / : R 2
-»* R 299 20.34 LINEAR MAPPINGS / : R 2 - R 2 299 20.35 LINEAR MAPPINGS
AND LINEAR SYSTEMS OF EQUATIONS . . 300 20.36 A FIRST ENCOUNTER WITH
MATRICES 300 20.37 FIRST APPLICATIONS OF MATRIX NOTATION 302 CONTENTS
VOLUME 1 XXIII 20.38 ADDITION OF MATRICES 303 20.39 MULTIPLICATION OF A
MATRIX BY A REAL NUMBER 303 20.40 MULTIPLICATION OF TWO MATRICES 303
20.41 THE TRANSPOSE OF A MATRIX 305 20.42 THE TRANSPOSE OF A 2-COLUMN
VECTOR 305 20.43 THE IDENTITY MATRIX 305 20.44 THE INVERSE OF A MATRIX
306 20.45 ROTATION IN MATRIX FORM AGAIN! 306 20.46 A MIRROR IN MATRIX
FORM 307 20.47 CHANGE OF BASIS AGAIN! 308 20.48 QUEEN CHRISTINA 309 21
ANALYTIC GEOMETRY IN R 3 313 21.1 INTRODUCTION 313 21.2 VECTOR ADDITION
AND MULTIPLICATION BY A SCALAR . . . . 315 21.3 SCALAR PRODUCT AND NORM
315 21.4 PROJECTION OF A VECTOR ONTO A VECTOR 316 21.5 THE ANGLE BETWEEN
TWO VECTORS 316 21.6 VECTOR PRODUCT 317 21.7 GEOMETRIC INTERPRETATION OF
THE VECTOR PRODUCT . . . . 319 21.8 CONNECTION BETWEEN VECTOR PRODUCTS
IN R 2 AND R 3 . . 320 21.9 VOLUME OF A PARALLELEPIPED SPANNED BY THREE
VECTORS 320 21.10 THE TRIPLE PRODUCT A-BXC 321 21.11 A FORMULA FOR THE
VOLUME SPANNED BY THREE VECTORS 322 21.12 LINES 323 21.13 PROJECTION OF
A POINT ONTO A LINE 324 21.14 PLANES 324 21.15 THE INTERSECTION OF A
LINE AND A PLANE 326 21.16 TWO INTERSECTING PLANES DETERMINE A LINE 327
21.17 PROJECTION OF A POINT ONTO A PLANE 328 21.18 DISTANCE FROM A POINT
TO A PLANE 328 21.19 ROTATION AROUND A GIVEN VECTOR 329 21.20 LINES AND
PLANES THROUGH THE ORIGIN ARE SUBSPACES . 330 21.21 SYSTEMS OF 3 LINEAR
EQUATIONS IN 3 UNKNOWNS 330 21.22 SOLVING A 3 X 3-SYSTEM BY GAUSSIAN
ELIMINATION . . . 332 21.23 3X3 MATRICES: SUM, PRODUCT AND TRANSPOSE 333
21.24 WAYS OF VIEWING A SYSTEM OF LINEAR EQUATIONS . . . . 335 21.25
NON-SINGULAR MATRICES 336 21.26 THE INVERSE OF A MATRIX 336 21.27
DIFFERENT BASES 337 21.28 LINEARLY INDEPENDENT SET OF VECTORS 337 21.29
ORTHOGONAL MATRICES 338 21.30 LINEAR TRANSFORMATIONS VERSUS MATRICES 338
XXIV CONTENTS VOLUME 1 21.31 THE SCALAR PRODUCT IS INVARIANT UNDER
ORTHOGONAL TRANSFORMATIONS 339 21.32 LOOKING AHEAD TO FUNCTIONS / : R 3
- R 3 340 22 COMPLEX NUMBERS 345 22.1 INTRODUCTION 345 22.2 ADDITION
AND MULTIPLICATION 346 22.3 THE TRIANGLE INEQUALITY 347 22.4 OPEN
DOMAINS 348 22.5 POLAR REPRESENTATION OF COMPLEX NUMBERS 348 22.6
GEOMETRICAL INTERPRETATION OF MULTIPLICATION 348 22.7 COMPLEX
CONJUGATION 349 22.8 DIVISION 350 22.9 THE FUNDAMENTAL THEOREM OF
ALGEBRA 350 22.10 ROOTS 351 22.11 SOLVING A QUADRATIC EQUATION W 2 + 2BW
+ C = 0 . . . . 351 22.12 GOSTA MITTAG-LEMER 352 23 THE DERIVATIVE 355
23.1 RATES OF CHANGE 355 23.2 PAYING TAXES 356 23.3 HIKING 359 23.4
DEFINITION OF THE DERIVATIVE 359 23.5 THE DERIVATIVE OF A LINEAR
FUNCTION IS CONSTANT . . . 362 23.6 THE DERIVATIVE OF X 2 IS 2X 362 23.7
THE DERIVATIVE OF X N IS NX N ~ L 364 23.8 THE DERIVATIVE OF IS *
FOR X 0 365 23.9 THE DERIVATIVE AS A FUNCTION 365 23.10 DENOTING THE
DERIVATIVE OF F(X) BY DF(X) 365 23.11 DENOTING THE DERIVATIVE OF F(X) BY
367 23.12 THE DERIVATIVE AS A LIMIT OF DIFFERENCE QUOTIENTS . . . 367
23.13 HOW TO COMPUTE A DERIVATIVE? 369 23.14 UNIFORM DIFFERENTIABILITY
ON AN INTERVAL 371 23.15 A BOUNDED DERIVATIVE IMPLIES LIPSCHITZ
CONTINUITY . . 372 23.16 A SLIGHTLY DIFFERENT VIEWPOINT 374 23.17
SWEDENBORG 374 24 DIFFERENTIATION RULES 377 24.1 INTRODUCTION 377 24.2
THE LINEAR COMBINATION RULE 378 24.3 THE PRODUCT RULE 379 24.4 THE CHAIN
RULE 380 24.5 THE QUOTIENT RULE 381 24.6 DERIVATIVES OF DERIVATIVES: F^
= D N F = | 382 24.7 ONE-SIDED DERIVATIVES * 383 CONTENTS VOLUME 1 XXV
24.8 QUADRATIC APPROXIMATION 384 24.9 THE DERIVATIVE OF AN INVERSE
FUNCTION 387 24.10 IMPLICIT DIFFERENTIATION 388 24.11 PARTIAL
DERIVATIVES 389 24.12 A SUM UP SO FAR 390 25 NEWTON S METHOD 393 25.1
INTRODUCTION 393 25.2 CONVERGENCE OF FIXED POINT ITERATION 393 25.3
NEWTON S METHOD 394 25.4 NEWTON S METHOD CONVERGES QUADRATICALLY 395
25.5 A GEOMETRIC INTERPRETATION OF NEWTON S METHOD . . . 396 25.6 WHAT
IS THE ERROR OF AN APPROXIMATE ROOT? 397 25.7 STOPPING CRITERION 400
25.8 GLOBALLY CONVERGENT NEWTON METHODS 400 26 GALILEO, NEWTON, HOOKE,
MALTHUS AND FOURIER 403 26.1 INTRODUCTION 403 26.2 NEWTON S LAW OF
MOTION 404 26.3 GALILEO S LAW OF MOTION 404 26.4 HOOKE S LAW 407 26.5
NEWTON S LAW PLUS HOOKE S LAW 408 26.6 FOURIER S LAW FOR HEAT FLOW 409
26.7 NEWTON AND ROCKET PROPULSION 410 26.8 MALTHUS AND POPULATION GROWTH
412 26.9 EINSTEIN S LAW OF MOTION 413 26.10 SUMMARY 414 REFERENCES 417
INDEX 419
|
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| id | DE-604.BV025676938 |
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| indexdate | 2024-12-20T14:12:01Z |
| institution | BVB |
| isbn | 354000890X |
| language | English |
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| publisher | Springer |
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| series | Applied mathematics |
| series2 | Applied mathematics : body and soul |
| spellingShingle | Eriksson, Kenneth Estep, Donald Johnson, Claes Applied mathematics body and soul Applied mathematics |
| title | Applied mathematics body and soul |
| title_auth | Applied mathematics body and soul |
| title_exact_search | Applied mathematics body and soul |
| title_full | Applied mathematics body and soul 1 Derivatives and geometry in IR3 K. Eriksson ; D. Estep ; C. Johnson |
| title_fullStr | Applied mathematics body and soul 1 Derivatives and geometry in IR3 K. Eriksson ; D. Estep ; C. Johnson |
| title_full_unstemmed | Applied mathematics body and soul 1 Derivatives and geometry in IR3 K. Eriksson ; D. Estep ; C. Johnson |
| title_short | Applied mathematics |
| title_sort | applied mathematics body and soul derivatives and geometry in ir3 |
| title_sub | body and soul |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019281270&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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