A mathematical bridge: an intuitive journey in higher mathematics
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey u.a.
World Scientific
2005
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| Ausgabe: | Repr. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XX, 526 S. Ill., graph. Darst. |
| ISBN: | 981238555x 9812385541 |
Internformat
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| 100 | 1 | |a Hewson, Stephen Fletcher |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a A mathematical bridge |b an intuitive journey in higher mathematics |c Stephen Fletcher Hewson |
| 250 | |a Repr. | ||
| 264 | 1 | |a New Jersey u.a. |b World Scientific |c 2005 | |
| 300 | |a XX, 526 S. |b Ill., graph. Darst. | ||
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| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
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| 655 | 7 | |0 (DE-588)4151278-9 |a Einführung |2 gnd-content | |
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Datensatz im Suchindex
| _version_ | 1804136473287458816 |
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| adam_text | STEPHEN FLETCHER HEWSON : - * , . BRIDGE AN INTUITIVE JOURNEY IN HIGHER
MATHEMATICS WORLD SCIENTIFIC NEW JERSEY * LONDON * SINGAPORE * HONG KONG
CONTENTS PREFACE VII 1. NUMBERS 1 1.1 COUNTING 2 1.1.1 THE NATURAL
NUMBERS 2 1.1.1.1 CONSTRUCTION OF THE NATURAL NUMBERS . . . . 3 1.1.1.2
ARITHMETIC 4 1.1.2 THE INTEGERS 5 1.1.2.1 PROPERTIES OF ZERO AND THE
NEGATIVE INTEGERS 7 1.1.3 THE RATIONAL NUMBERS 7 1.1.4 ORDER 8 1.1.4.1
ORDERING N, Z AND Q 9 1.1.5 1,2,3, INFINITY 10 1.1.5.1 COMPARISON OF
INFINITE SETS 11 1.1.6 THE ARITHMETIC OF INFINITIES 11 1.1.7 BEYOND OO
16 1.2 THE REAL NUMBERS 18 1.2.1 HOW TO CREATE THE IRRATIONAL NUMBERS 20
1.2.1.1 ALGEBRAIC DESCRIPTION OF THE REAL NUMBERS . . 22 1.2.2 HOW MANY
REAL NUMBERS ARE THERE? 24 1.2.3 ALGEBRAIC AND TRANSCENDENTAL NUMBERS 25
1.2.3.1 TRANSCENDENTAL EXAMPLES 27 1.2.4 THE CONTINUUM HYPOTHESIS AND AN
EVEN BIGGER INFINITY 28 1.3 COMPLEX NUMBERS AND THEIR HIGHER DIMENSIONAL
PARTNERS . 30 1.3.1 TH E DISCOVERY OF I 30 1.3.2 THE COMPLEX PLANE 32
XII A MATHEMATICAL BRIDGE 1.3.2.1 USING COMPLEX NUMBERS IN GEOMETRY . .
. . 33 1.3.3 DE MOIVRE S THEOREM 35 1.3.4 POLYNOMIALS AND THE
FUNDAMENTAL THEOREM OF ALGEBRA 36 1.3.4.1 FINDING SOLUTIONS TO
POLYNOMIAL EQUATIONS . 36 1.3.5 ANY MORE NUMBERS? 39 1.3.5.1 THE
QUATERNIONS 40 1.3.5.2 CAYLEY NUMBERS 42 1.4 PRIME NUMBERS 43 1.4.1
COMPUTERS, ALGORITHMS AND MATHEMATICS 45 1.4.2 PROPERTIES OF PRIME
NUMBERS 46 1.4.3 HOW MANY PRIME NUMBERS ARE THERE? 47 1.4.3.1
DISTRIBUTION OF THE PRIME NUMBERS 48 1.4.4 EUCLID S ALGORITHM 48 1.4.4.1
THE SPEED OF THE EUCLID ALGORITHM 50 1.4.4.2 CONTINUED FRACTIONS 51
1.4.5 BEZOUT S LEMMA AND THE FUNDAMENTAL THEOREM OF ARITHMETIC 53 1.5
MODULAR NUMBERS 56 1.5.1 ARITHMETIC MODULO A PRIME NUMBER 57 1.5.1.1 A
FORMULA FOR THE PRIME NUMBERS 58 1.5.1.2 FERMAT S LITTLE THEOREM 59
1.5.2 RSA CRYPTOGRAPHY 60 1.5.2.1 CREATING THE RSA SYSTEM 61 1.5.2.2 AN
RSA CRYPTOSYSTEM 63 2. ANALYSIS 65 2.1 INFINITE LIMITS 66 2.1.1 THREE
EXAMPLES 66 2.1.1.1 ACHILLES AND THE TORTOISE 66 2.1.1.2 CONTINUOUSLY
COMPOUNDED INTEREST RATES . . 68 2.1.1.3 ITERATIVE SOLUTION OF EQUATIONS
71 2.1.2 THE MATHEMATICAL DESCRIPTION OF A LIMIT 73 2.1.2.1 THE GENERAL
PRINCIPLE OF CONVERGENCE . . . . 77 2.1.3 LIMITS APPLIED TO INFINITE
SUMS 77 2.1.3.1 AN EXAMPLE: GEOMETRIC PROGRESSION 78 2.2 CONVERGENCE AND
DIVERGENCE OF INFINITE SUMS 79 2.2.1 THE HARMONIC SERIES 80 2.2.2
TESTING FOR CONVERGENCE 81 2.2.2.1 THE COMPARISON TEST 81 CONTENTS
2.2.2.2 THE ALTERNATING SERIES TEST 82 2.2.2.3 ABSOLUTELY CONVERGENT
SERIES CONVERGE .... 83 2.2.2.4 THE RATIO TEST 84 2.2.3 POWER SERIES AND
THE RADIUS OF CONVERGENCE 85 2.2.3.1 DETERMINING THE RADIUS OF
CONVERGENCE ... 87 2.2.4 REARRANGEMENT OF INFINITE SERIES 87 2.3 REAL
FUNCTIONS 89 2.3.1 LIMITS OF REAL VALUED FUNCTIONS 90 2.3.2 CONTINUOUS
FUNCTIONS 91 2.3.3 DIFFERENTIATION 94 2.3.3.1 EXAMPLES 97 2.3.3.2 THE
MEAN VALUE THEOREM 100 2.3.3.3 L HOPITAL S RULE 102 2.3.4 AREAS AND
INTEGRATION 103 2.3.5 THE FUNDAMENTAL THEOREM OF CALCULUS 105 2.4 THE
LOGARITHM AND EXPONENTIAL FUNCTIONS AND E 107 2.4.1 THE DEFINITION OF
LOG(X) 108 2.4.2 THE DEFINITION OF EXP(A;) ILL 2.4.3 EULER S NUMBER E
113 2.4.3.1 THE IRRATIONALITY OF E 116 2.5 POWER SERIES 118 2.5.1 THE
TAYLOR SERIES 120 2.5.1.1 CAUTIONARY EXAMPLE 123 2.5.1.2 COMPLEX
EXTENSIONS OF REAL FUNCTIONS .... 124 2.6 7 T AND ANALYTICAL VIEWS OF
TRIGONOMETRY 125 2.6.1 ANGLES AND THE AREA OF CIRCLE SECTORS 126 2.6.1.1
A SERIES EXPRESSION FOR T T 128 2.6.2 TANGENT, SINES AND COSINES 130
2.6.2.1 DEFINING SIN(AR) AND COS(A;) THROUGH THEIR POWER SERIES 132
2.6.3 FOURIER SERIES 134 2.7 COMPLEX FUNCTIONS 137 2.7.1 EXPONENTIAL AND
TRIGONOMETRICAL FUNCTIONS 137 2.7.2 SOME BASIC PROPERTIES OF COMPLEX
FUNCTIONS 139 2.7.3 THE LOGARITHM AND MULTIVALUED FUNCTIONS 140 2.7.4
POWERS OF COMPLEX NUMBERS 141 3. ALGEBRA 145 1 3.1 LINEARITY 147 A
MATHEMATICAL BRIDGE 3.1.1 LINEAR EQUATIONS 147 3.1.1.1 SYSTEMS OF
MULTIPLE LINEAR EQUATIONS . . . . 148 3.1.2 VECTOR SPACES 151 3.1.2.1
PLANES, LINES AND OTHER VECTOR SPACES . . . . 153 3.1.2.2 SUBSPACES AND
INTERSECTION OF VECTOR SPACES . 154 3.1.2.3 PHYSICAL EXAMPLES OF VECTORS
155 3.1.2.4 HOW MANY VECTOR SPACES ARE THERE? 156 3.1.2.5 FURTHER
EXAMPLES OF VECTORS 160 3.1.3 PUTTING VECTOR SPACES TO WORK: LINEAR MAPS
AND MATRICES 162 3.1.3.1 SIMULTANEOUS LINEAR EQUATIONS REVISITED . . .
163 3.1.3.2 PROPERTIES OF MATRIX ALGEBRA 163 3.1.4 SOLVING LINEAR
SYSTEMS 164 3.1.4.1 HOMOGENEOUS EQUATIONS 165 3.1.4.2 LINEAR
DIFFERENTIAL OPERATORS 165 3.1.4.3 INHOMOGENEOUS LINEAR EQUATIONS 166
3.1.4.4 INVERTING SQUARE MATRICES 167 3.1.4.5 DETERMINANTS 168 3.1.4.6
PROPERTIES OF DETERMINANTS 169 3.1.4.7 FORMULA FOR THE INVERSE OF A
SQUARE MATRIX . 170 3.2 OPTIMISATION 170 3.2.1 LINEAR CONSTRAINTS 171
3.2.2 THE SIMPLEX ALGORITHM 172 3.2.2.1 AN EXAMPLE 175 3.2.2.2 THE DIET
PROBLEM 178 3.2.2.3 THE TRANSPORTATION PROBLEM 178 3.2.2.4 GAMES 179 3.3
DISTANCE, LENGTH AND ANGLE 180 3.3.1 SCALAR PRODUCTS 180 3.3.1.1
STANDARD GEOMETRY AND THE EUCLIDEAN SCALAR PRODUCT 181 3.3.1.2
POLYNOMIALS AND SCALAR PRODUCTS 183 3.3.2 GENERAL SCALAR PRODUCTS 185
3.3.2.1 THE CAUCHY-SCHWARZ INEQUALITY 186 3.3.2.2 GENERAL PROPERTIES OF
LENGTHS AND DISTANCES . 188 3.3.2.3 LENGTHS NOT ARISING FROM SCALAR
PRODUCTS . . 189 3.4 GEOMETRY AND ALGEBRA 190 3.4.1 QUADRATIC FORMS IN
TWO DIMENSIONS 191 3.4.2 QUADRATIC SURFACES IN THREE DIMENSIONS 193
CONTENTS 4. 3.4.3 EIGENVECTORS AND EIGENVALUES 195 3.4.3.1 FINDING
EIGENVECTORS AND EIGENVALUES . . . . 195 3.4.3.2 THE SPECIAL PROPERTIES
OF REAL SYMMETRIC MA- TRICES 196 3.4.3.3 QUADRATIC FORMS REVISITED 198
3.4.3.4 EXAMPLES REVISITED 199 3.4.4 ISOMETRIES 202 3.4.4.1 TRANSLATIONS
205 3.4.4.2 DETERMINANTS, VOLUMES AND ISOMETRIES . . . 205 3.5 SYMMETRY
207 3.5.1 GROUPS OF SYMMETRIES 210 3.5.1.1 THE GROUP AXIOMS 210 3.5.1.2
QUATERNIONS AGAIN 211 3.5.1.3 MULTIPLICATION OF INTEGERS MODULO P 212
3.5.2 SUBGROUPS-SYMMETRY WITHIN SYMMETRY 213 3.5.2.1 SPECIAL PROPERTIES
OF FINITE GROUPS 214 3.5.3 GROUP ACTIONS 217 3.5.4 TWO- AND
THREE-DIMENSIONAL WALLPAPER 220 3.5.4.1 WALLPAPER ON A LATTICE 221
3.5.4.2 HANGING THE WALLPAPER 223 3.5.4.3 APPLICATION TO CRYSTALLOGRAPHY
226 CALCULUS AND DIFFERENTIAL EQUATIONS 229 4.1 THE WHY AND HOW OF
CALCULUS 229 4.1.1 ACCELERATION, VELOCITY AND POSITION 229 4.1.1.1
INTEGRATION 231 4.1.2 BACK TO NEWTON 233 4.1.2.1 A SIMPLE PENDULUM 233
4.1.2.2 COMPLICATING THE SIMPLE PENDULUM 235 4.1.2.3 DEVELOPMENT OF
CALCULUS FROM NEWTON S LAW 236 4.2 ORDINARY LINEAR DIFFERENTIAL
EQUATIONS 237 4.2.1 COMPLETE SOLUTION OF ORDINARY LINEAR DIFFERENTIAL
EQUA- TIONS 237 4.2.2 INHOMOGENEOUS EQUATIONS 239 4.2.3 SOLVING
HOMOGENEOUS LINEAR EQUATIONS 240 4.2.3.1 EQUATIONS WITH CONSTANT
COEFFICIENTS 240 4.2.4 POWER SERIES METHOD OF SOLUTION 241 4.2.4.1
BESSEL FUNCTIONS 244 4.2.4.2 GENERAL METHOD OF SOLUTION BY SERIES ....
246 A MATHEMATICAL BRIDGE 4.3 PARTIAL DIFFERENTIAL EQUATIONS 246 4.3.1
DEFINITION OF THE PARTIAL DERIVATIVE 247 4.3.2 THE EQUATIONS OF MOTION
FOR A VIBRATING STRING . . . . 248 4.3.2.1 THE WAVE INTERPRETATION 250
4.3.2.2 SEPARABLE SOLUTIONS 250 4.3.2.3 INITIAL AND BOUNDARY CONDITION
252 4.3.2.4 MUSICAL STRINGED INSTRUMENTS 252 4.3.3 THE DIFFUSION
EQUATION 254 4.3.3.1 SOLAR HEATING 257 4.3.4 A REAL LOOK AT COMPLEX
DIFFERENTIATION 258 4.3.4.1 LAPLACE EQUATIONS 260 4.4 CALCULUS MEETS
GEOMETRY 261 4.4.1 TANGENT VECTORS AND NORMALS 262 4.4.2 GRAD, DIV AND
CURL 266 4.4.3 INTEGRATION OVER SURFACES AND VOLUMES 267 4.4.3.1
GAUSSIAN INTEGRALS 268 4.4.3.2 GEOMETRIC UNDERSTANDING OF DIVERGENCE . .
. 271 4.4.3.3 GEOMETRIC UNDERSTANDING OF CURL 273 4.4.3.4 FOURIER
REVISITED 274 4.4.3.5 DIVERGENCE THEOREM IN ACTION 274 4.4.4 LAPLACE AND
POISSON EQUATIONS 276 4.4.4.1 SOLVING LAPLACE S EQUATION 277 4.4.4.2
POISSON EQUATIONS 277 4.4.4.3 BOUNDARY CONDITIONS AND UNIQUENESS OF
SOLU- TIONS 278 4.5 NON-LINEARITY 280 4.5.1 THE NAVIER-STOKES EQUATION
FOR FLUID MOTION 280 4.5.2 PERTURBATION OF DIFFERENTIAL EQUATIONS 282
4.5.2.1 BALLISTICS 283 4.5.2.2 THE SIMPLE PENDULUM IS NOT SO SIMPLE . .
. 287 4.6 QUALITATIVE METHODS: SOLUTION WITHOUT SOLUTION 290 4.6.1 WHAT
DOES IT MEAN TO SOLVE A DIFFERENTIAL EQUATION? . 292 4.6.2 PHASE SPACE
AND ORBITS 293 4.6.3 CONSTRUCTION OF THE PHASE SPACE PORTRAIT 295
4.6.3.1 FIRST ORDER NON-LINEAR DIFFERENTIAL EQUATIONS . 295 4.6.3.2
SECOND ORDER NON-LINEAR DIFFERENTIAL EQUATIONS 295 4.6.3.3 SHM IN WOLF S
CLOTHING 296 4.6.3.4 NON-LINEAR EXAMPLE 298 4.6.4 GENERAL FORMS OF FLOW
NEAR TO A FIXED POINT 301 CONTENTS 4.6.5 PREDATOR PREY EXAMPLE 303 4.6.6
COMPETING HERBIVORES 305 5. PROBABILITY 309 5.1 THE BASIC IDEAS OF
PROBABILITY . 310 5.1.0.1 THE MATCHING BIRTHDAY PROBLEM 311 5.1.1 TWO
CAUTIONARY EXAMPLES 313 5.1.1.1 THE PROBLEM OF THE TERMINATED MATCH . .
. . 313 5.1.1.2 THE PROBLEM OF THE DOORS AND THE GOAT . . . 315 5.2
PRECISE PROBABILITY 316 5.2.1 INCLUSION-EXCLUSION 317 5.2.1.1 THE COATS
PROBLEM 318 5.2.2 CONDITIONAL PROBABILITY 320 5.2.2.1 THE BAYESIAN
STATISTICIAN 321 5.2.3 THE LAW OF TOTAL PROBABILITY AND BAYES FORMULA .
. . 322 5.2.3.1 RELIABILITY OF DRUG TESTING 324 5.3 FUNCTIONS ON SAMPLES
SPACES: RANDOM VARIABLES 325 5.3.1 THE BINOMIAL DISTRIBUTION 326 5.3.2
THE POISSON APPROXIMATION TO THE BINOMIAL 329 5.3.2.1 ERROR DISTRIBUTION
IN NOISY DATA 330 5.3.3 THE POISSON DISTRIBUTION 332 5.3.3.1
INTERPRETATION OF THE POISSON DISTRIBUTION . . 332 5.3.4 CONTINUOUS
RANDOM VARIABLES 334 5.3.4.1 THE NORMAL DISTRIBUTION 335 5.3.4.2 THE
UNIFORM DISTRIBUTION 337 5.3.4.3 THE GAMMA RANDOM VARIABLE 338 5.3.5 AN
APPLICATION OF PROBABILITY TO PRIME NUMBERS . . . 339 5.3.6 AVERAGING
AND EXPECTATION 341 5.3.6.1 WHAT DO WE EXPECT TO OBTAIN IN A POISSON OR
BINOMIAL TRIAL? 343 5.3.6.2 WHAT DO WE EXPECT TO OBTAIN IN A NORMAL
TRIAL? 344 5.3.6.3 THE COLLECTION PROBLEM 345 5.3.6.4 THE CAUCHY
DISTRIBUTION 346 5.3.7 DISPERSION AND VARIANCE 346 5.3.7.1 A DYNAMICAL
INTERPRETATION OF EXPECTATION AND VARIANCE 348 5.4 LIMIT THEOREMS 348
5.4.1 CHEBYSHEV S INEQUALITY 349 XVIII A MATHEMATICAL BRIDGE 5.4.1.1
CHEBYSHEV AS THE BEST POSSIBLE INEQUALITY . 350 5.4.1.2 STANDARDISING
DEVIATIONS FROM THE AVERAGE . 351 5.4.1.3 STANDARDISED VARIABLES 352
5.4.2 THE LAW OF LARGE NUMBERS 352 5.4.2.1 MONTE CARLO INTEGRATION 353
5.4.3 THE CENTRAL LIMIT THEOREM AND THE NORMAL DISTRIBUTION 355 5.4.3.1
THE CENTRAL LIMIT THEOREM 356 6. THEORETICAL PHYSICS 361 6.1 THE
NEWTONIAN WORLD 363 6.1.1 THE MOTION OF THE PLANETS AROUND THE SUN 364
6.1.1.1 TRANSFORMING THE EQUATION OF MOTION . . . . 364 6.1.1.2 SOLUTION
OF THE PROBLEM 366 6.1.1.3 NEWTONIAN ANTI-GRAVITY 369 6.1.2 PROVING
CONSERVATION OF ENERGY 370 6.1.3 PLANETARY CATASTROPHE FOR OTHER TYPES
OF FORCES . . . . 372 6.1.4 EARTH, SUN AND MOON? 374 6.2 LIGHT,
ELECTRICITY AND MAGNETISM 376 6.2.1 STATIC ELECTRICITY 377 6.2.1.1 THE
EQUATION FOR A MAGNET 378 6.2.2 CURRENT ELECTRICITY AND MAGNETISM 380
6.2.3 MAXWELL S EQUATIONS FOR ELECTROMAGNETIC WAVES .... 381 6.2.3.1
ELECTROMAGNETIC WAVE SOLUTIONS IN THE VAC- UUM OF SPACE 382 6.3
RELATIVITY AND THE GEOMETRY OF THE UNIVERSE 384 6.3.1 SPECIAL RELATIVITY
386 6.3.1.1 LENGTH CONTRACTION AND TIME DILATION .... 390 6.3.1.2
LORENTZ TRANSFORMATION AS A ROTATION IN SPACE-TIME 391 ; 6.3.1.3 THE
LORENTZ TRANSFORMATIONS AS THE GROUP OF I. SYMMETRIES OF SPACETIME 393 |
6.3.1.4 RELATIVISTIC MOMENTUM 395 6.3.2 GENERAL RELATIVITY AND
GRAVITATION 397 6.3.2.1 THE SCHWARZSCHILD BLACK HOLE 399 | 6.4 QUANTUM
MECHANICS 401 I; 6.4.1 QUANTISATION 401J 6.4.1.1 THE WAVE-PARTICLE
PARADOX 402 I; 6.4.2 THE FORMULATION OF QUANTUM MECHANICS 4041 6.4.2.1
THE UNDERLYING EQUATION 4051 CONTENTS XIX 6.4.3 THE BASIC QUANTUM
MECHANICAL SETUP 408 6.4.3.1 PARTICLE TRAPPED IN A ONE-DIMENSIONAL BOX .
410 6.4.3.2 MOMENTUM EIGENSTATES 412 6.4.3.3 GENERALISATION TO THREE
DIMENSIONS 413 6.4.4 HEISENBERG S UNCERTAINTY PRINCIPLE 414 6.4.4.1
UNCERTAINTY IN ACTION 416 6.4.5 WHERE NEXT? 417 APPENDIX A EXERCISES FOR
THE READER 419 A.I NUMBERS 420 A.2 ANALYSIS 429 A.3 ALGEBRA 439 A.4
CALCULUS AND DIFFERENTIAL EQUATIONS 453 A.5 PROBABILITY 471 A.6
THEORETICAL PHYSICS 486 APPENDIX B FURTHER READING 499 APPENDIX C BASIC
MATHEMATICAL BACKGROUND 503 C.I SETS 503 C.I.I NOTATION 503 C.I.2
OPERATIONS ON SETS 504 C.2 LOGIC AND PROOF 504 C.2.1 FORMS OF PROOF 505
C.3 FUNCTIONS 506 C.3.1 COMPOSITION OF FUNCTIONS 506 C.3.2 FACTORIALS
506 C.3.3 POWERS, INDICES AND THE BINOMIAL THEOREM 506 C.3.4 THE
EXPONENTIAL, E AND THE NATURAL LOGARITHM .... 507 C.3.5 THE
TRIGONOMETRICAL FUNCTIONS 508 C.3.6 THE HYPERBOLIC FUNCTIONS 508 C.4
VECTORS AND MATRICES 509 C.4.1 COMBINING VECTORS TOGETHER 509 C.4.2
POLAR COORDINATES 510 C.4.3 MATRICES 510 C.5 CALCULUS 512 C.5.1
DIFFERENTIATION 512 C.5.2 INTEGRATION 513 L XX A MATHEMATICAL BRIDGE
C.5.3 POSITION, VELOCITY AND ACCELERATION 513 C.5.4 SIMPLE HARMONIC
MOTION 514 APPENDIX D DICTIONARY OF SYMBOLS 515 D.I THE GREEK LETTERS
515 D.2 MATHEMATICAL SYMBOLS 516 INDEX 519
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| adam_txt |
STEPHEN FLETCHER HEWSON :\ - * , . BRIDGE AN INTUITIVE JOURNEY IN HIGHER
MATHEMATICS WORLD SCIENTIFIC NEW JERSEY * LONDON * SINGAPORE * HONG KONG
CONTENTS PREFACE VII 1. NUMBERS 1 1.1 COUNTING 2 1.1.1 THE NATURAL
NUMBERS 2 1.1.1.1 CONSTRUCTION OF THE NATURAL NUMBERS . . . . 3 1.1.1.2
ARITHMETIC 4 1.1.2 THE INTEGERS 5 1.1.2.1 PROPERTIES OF ZERO AND THE
NEGATIVE INTEGERS 7 1.1.3 THE RATIONAL NUMBERS 7 1.1.4 ORDER 8 1.1.4.1
ORDERING N, Z AND Q 9 1.1.5 1,2,3, INFINITY 10 1.1.5.1 COMPARISON OF
INFINITE SETS 11 1.1.6 THE ARITHMETIC OF INFINITIES 11 1.1.7 BEYOND OO
16 1.2 THE REAL NUMBERS 18 1.2.1 HOW TO CREATE THE IRRATIONAL NUMBERS 20
1.2.1.1 ALGEBRAIC DESCRIPTION OF THE REAL NUMBERS . . 22 1.2.2 HOW MANY
REAL NUMBERS ARE THERE? 24 1.2.3 ALGEBRAIC AND TRANSCENDENTAL NUMBERS 25
1.2.3.1 TRANSCENDENTAL EXAMPLES 27 1.2.4 THE CONTINUUM HYPOTHESIS AND AN
EVEN BIGGER INFINITY 28 1.3 COMPLEX NUMBERS AND THEIR HIGHER DIMENSIONAL
PARTNERS . 30 1.3.1 TH E DISCOVERY OF I 30 1.3.2 THE COMPLEX PLANE 32
XII A MATHEMATICAL BRIDGE 1.3.2.1 USING COMPLEX NUMBERS IN GEOMETRY . .
. . 33 1.3.3 DE MOIVRE'S THEOREM 35 1.3.4 POLYNOMIALS AND THE
FUNDAMENTAL THEOREM OF ALGEBRA 36 1.3.4.1 FINDING SOLUTIONS TO
POLYNOMIAL EQUATIONS . 36 1.3.5 ANY MORE NUMBERS? 39 1.3.5.1 THE
QUATERNIONS 40 1.3.5.2 CAYLEY NUMBERS 42 1.4 PRIME NUMBERS 43 1.4.1
COMPUTERS, ALGORITHMS AND MATHEMATICS 45 1.4.2 PROPERTIES OF PRIME
NUMBERS 46 1.4.3 HOW MANY PRIME NUMBERS ARE THERE? 47 1.4.3.1
DISTRIBUTION OF THE PRIME NUMBERS 48 1.4.4 EUCLID'S ALGORITHM 48 1.4.4.1
THE SPEED OF THE EUCLID ALGORITHM 50 1.4.4.2 CONTINUED FRACTIONS 51
1.4.5 BEZOUT'S LEMMA AND THE FUNDAMENTAL THEOREM OF ARITHMETIC 53 1.5
MODULAR NUMBERS 56 1.5.1 ARITHMETIC MODULO A PRIME NUMBER 57 1.5.1.1 A
FORMULA FOR THE PRIME NUMBERS 58 1.5.1.2 FERMAT'S LITTLE THEOREM 59
1.5.2 RSA CRYPTOGRAPHY 60 1.5.2.1 CREATING THE RSA SYSTEM 61 1.5.2.2 AN
RSA CRYPTOSYSTEM 63 2. ANALYSIS 65 2.1 INFINITE LIMITS 66 2.1.1 THREE
EXAMPLES 66 2.1.1.1 ACHILLES AND THE TORTOISE 66 2.1.1.2 CONTINUOUSLY
COMPOUNDED INTEREST RATES . . 68 2.1.1.3 ITERATIVE SOLUTION OF EQUATIONS
71 2.1.2 THE MATHEMATICAL DESCRIPTION OF A LIMIT 73 2.1.2.1 THE GENERAL
PRINCIPLE OF CONVERGENCE . . . . 77 2.1.3 LIMITS APPLIED TO INFINITE
SUMS 77 2.1.3.1 AN EXAMPLE: GEOMETRIC PROGRESSION 78 2.2 CONVERGENCE AND
DIVERGENCE OF INFINITE SUMS 79 2.2.1 THE HARMONIC SERIES 80 2.2.2
TESTING FOR CONVERGENCE 81 2.2.2.1 THE COMPARISON TEST 81 CONTENTS
2.2.2.2 THE ALTERNATING SERIES TEST 82 2.2.2.3 ABSOLUTELY CONVERGENT
SERIES CONVERGE . 83 2.2.2.4 THE RATIO TEST 84 2.2.3 POWER SERIES AND
THE RADIUS OF CONVERGENCE 85 2.2.3.1 DETERMINING THE RADIUS OF
CONVERGENCE . 87 2.2.4 REARRANGEMENT OF INFINITE SERIES 87 2.3 REAL
FUNCTIONS 89 2.3.1 LIMITS OF REAL VALUED FUNCTIONS 90 2.3.2 CONTINUOUS
FUNCTIONS 91 2.3.3 DIFFERENTIATION 94 2.3.3.1 EXAMPLES 97 2.3.3.2 THE
MEAN VALUE THEOREM 100 2.3.3.3 L'HOPITAL'S RULE 102 2.3.4 AREAS AND
INTEGRATION 103 2.3.5 THE FUNDAMENTAL THEOREM OF CALCULUS 105 2.4 THE
LOGARITHM AND EXPONENTIAL FUNCTIONS AND E 107 2.4.1 THE DEFINITION OF
LOG(X) 108 2.4.2 THE DEFINITION OF EXP(A;) ILL 2.4.3 EULER'S NUMBER E
113 2.4.3.1 THE IRRATIONALITY OF E 116 2.5 POWER SERIES 118 2.5.1 THE
TAYLOR SERIES 120 2.5.1.1 CAUTIONARY EXAMPLE 123 2.5.1.2 COMPLEX
EXTENSIONS OF REAL FUNCTIONS . 124 2.6 7 T AND ANALYTICAL VIEWS OF
TRIGONOMETRY 125 2.6.1 ANGLES AND THE AREA OF CIRCLE SECTORS 126 2.6.1.1
A SERIES EXPRESSION FOR T T 128 2.6.2 TANGENT, SINES AND COSINES 130
2.6.2.1 DEFINING SIN(AR) AND COS(A;) THROUGH THEIR POWER SERIES 132
2.6.3 FOURIER SERIES 134 2.7 COMPLEX FUNCTIONS 137 2.7.1 EXPONENTIAL AND
TRIGONOMETRICAL FUNCTIONS 137 2.7.2 SOME BASIC PROPERTIES OF COMPLEX
FUNCTIONS 139 2.7.3 THE LOGARITHM AND MULTIVALUED FUNCTIONS 140 2.7.4
POWERS OF COMPLEX NUMBERS 141 3. ALGEBRA 145 1 3.1 LINEARITY 147 A
MATHEMATICAL BRIDGE 3.1.1 LINEAR EQUATIONS 147 3.1.1.1 SYSTEMS OF
MULTIPLE LINEAR EQUATIONS . . . . 148 3.1.2 VECTOR SPACES 151 3.1.2.1
PLANES, LINES AND OTHER VECTOR SPACES . . . . 153 3.1.2.2 SUBSPACES AND
INTERSECTION OF VECTOR SPACES . 154 3.1.2.3 PHYSICAL EXAMPLES OF VECTORS
155 3.1.2.4 HOW MANY VECTOR SPACES ARE THERE? 156 3.1.2.5 FURTHER
EXAMPLES OF VECTORS 160 3.1.3 PUTTING VECTOR SPACES TO WORK: LINEAR MAPS
AND MATRICES 162 3.1.3.1 SIMULTANEOUS LINEAR EQUATIONS REVISITED . . .
163 3.1.3.2 PROPERTIES OF MATRIX ALGEBRA 163 3.1.4 SOLVING LINEAR
SYSTEMS 164 3.1.4.1 HOMOGENEOUS EQUATIONS 165 3.1.4.2 LINEAR
DIFFERENTIAL OPERATORS 165 3.1.4.3 INHOMOGENEOUS LINEAR EQUATIONS 166
3.1.4.4 INVERTING SQUARE MATRICES 167 3.1.4.5 DETERMINANTS 168 3.1.4.6
PROPERTIES OF DETERMINANTS 169 3.1.4.7 FORMULA FOR THE INVERSE OF A
SQUARE MATRIX . 170 3.2 OPTIMISATION 170 3.2.1 LINEAR CONSTRAINTS 171
3.2.2 THE SIMPLEX ALGORITHM 172 3.2.2.1 AN EXAMPLE 175 3.2.2.2 THE DIET
PROBLEM 178 3.2.2.3 THE TRANSPORTATION PROBLEM 178 3.2.2.4 GAMES 179 3.3
DISTANCE, LENGTH AND ANGLE 180 3.3.1 SCALAR PRODUCTS 180 3.3.1.1
STANDARD GEOMETRY AND THE EUCLIDEAN SCALAR PRODUCT 181 3.3.1.2
POLYNOMIALS AND SCALAR PRODUCTS 183 3.3.2 GENERAL SCALAR PRODUCTS 185
3.3.2.1 THE CAUCHY-SCHWARZ INEQUALITY 186 3.3.2.2 GENERAL PROPERTIES OF
LENGTHS AND DISTANCES . 188 3.3.2.3 LENGTHS NOT ARISING FROM SCALAR
PRODUCTS . . 189 3.4 GEOMETRY AND ALGEBRA 190 3.4.1 QUADRATIC FORMS IN
TWO DIMENSIONS 191 3.4.2 QUADRATIC SURFACES IN THREE DIMENSIONS 193
CONTENTS 4. 3.4.3 EIGENVECTORS AND EIGENVALUES 195 3.4.3.1 FINDING
EIGENVECTORS AND EIGENVALUES . . . . 195 3.4.3.2 THE SPECIAL PROPERTIES
OF REAL SYMMETRIC MA- TRICES 196 3.4.3.3 QUADRATIC FORMS REVISITED 198
3.4.3.4 EXAMPLES REVISITED 199 3.4.4 ISOMETRIES 202 3.4.4.1 TRANSLATIONS
205 3.4.4.2 DETERMINANTS, VOLUMES AND ISOMETRIES . . . 205 3.5 SYMMETRY
207 3.5.1 GROUPS OF SYMMETRIES 210 3.5.1.1 THE GROUP AXIOMS 210 3.5.1.2
QUATERNIONS AGAIN 211 3.5.1.3 MULTIPLICATION OF INTEGERS MODULO P 212
3.5.2 SUBGROUPS-SYMMETRY WITHIN SYMMETRY 213 3.5.2.1 SPECIAL PROPERTIES
OF FINITE GROUPS 214 3.5.3 GROUP ACTIONS 217 3.5.4 TWO- AND
THREE-DIMENSIONAL WALLPAPER 220 3.5.4.1 WALLPAPER ON A LATTICE 221
3.5.4.2 HANGING THE WALLPAPER 223 3.5.4.3 APPLICATION TO CRYSTALLOGRAPHY
226 CALCULUS AND DIFFERENTIAL EQUATIONS 229 4.1 THE WHY AND HOW OF
CALCULUS 229 4.1.1 ACCELERATION, VELOCITY AND POSITION 229 4.1.1.1
INTEGRATION 231 4.1.2 BACK TO NEWTON 233 4.1.2.1 A SIMPLE PENDULUM 233
4.1.2.2 COMPLICATING THE SIMPLE PENDULUM 235 4.1.2.3 DEVELOPMENT OF
CALCULUS FROM NEWTON'S LAW 236 4.2 ORDINARY LINEAR DIFFERENTIAL
EQUATIONS 237 4.2.1 COMPLETE SOLUTION OF ORDINARY LINEAR DIFFERENTIAL
EQUA- TIONS 237 4.2.2 INHOMOGENEOUS EQUATIONS 239 4.2.3 SOLVING
HOMOGENEOUS LINEAR EQUATIONS 240 4.2.3.1 EQUATIONS WITH CONSTANT
COEFFICIENTS 240 4.2.4 POWER SERIES METHOD OF SOLUTION 241 4.2.4.1
BESSEL FUNCTIONS 244 4.2.4.2 GENERAL METHOD OF SOLUTION BY SERIES .
246 A MATHEMATICAL BRIDGE 4.3 PARTIAL DIFFERENTIAL EQUATIONS 246 4.3.1
DEFINITION OF THE PARTIAL DERIVATIVE 247 4.3.2 THE EQUATIONS OF MOTION
FOR A VIBRATING STRING . . . . 248 4.3.2.1 THE WAVE INTERPRETATION 250
4.3.2.2 SEPARABLE SOLUTIONS 250 4.3.2.3 INITIAL AND BOUNDARY CONDITION
252 4.3.2.4 MUSICAL STRINGED INSTRUMENTS 252 4.3.3 THE DIFFUSION
EQUATION 254 4.3.3.1 SOLAR HEATING 257 4.3.4 A REAL LOOK AT COMPLEX
DIFFERENTIATION 258 4.3.4.1 LAPLACE EQUATIONS 260 4.4 CALCULUS MEETS
GEOMETRY 261 4.4.1 TANGENT VECTORS AND NORMALS 262 4.4.2 GRAD, DIV AND
CURL 266 4.4.3 INTEGRATION OVER SURFACES AND VOLUMES 267 4.4.3.1
GAUSSIAN INTEGRALS 268 4.4.3.2 GEOMETRIC UNDERSTANDING OF DIVERGENCE . .
. 271 4.4.3.3 GEOMETRIC UNDERSTANDING OF CURL 273 4.4.3.4 FOURIER
REVISITED 274 4.4.3.5 DIVERGENCE THEOREM IN ACTION 274 4.4.4 LAPLACE AND
POISSON EQUATIONS 276 4.4.4.1 SOLVING LAPLACE'S EQUATION 277 4.4.4.2
POISSON EQUATIONS 277 4.4.4.3 BOUNDARY CONDITIONS AND UNIQUENESS OF
SOLU- TIONS 278 4.5 NON-LINEARITY 280 4.5.1 THE NAVIER-STOKES EQUATION
FOR FLUID MOTION 280 4.5.2 PERTURBATION OF DIFFERENTIAL EQUATIONS 282
4.5.2.1 BALLISTICS 283 4.5.2.2 THE SIMPLE PENDULUM IS NOT SO SIMPLE . .
. 287 4.6 QUALITATIVE METHODS: SOLUTION WITHOUT SOLUTION 290 4.6.1 WHAT
DOES IT MEAN TO SOLVE A DIFFERENTIAL EQUATION? . 292 4.6.2 PHASE SPACE
AND ORBITS 293 4.6.3 CONSTRUCTION OF THE PHASE SPACE PORTRAIT 295
4.6.3.1 FIRST ORDER NON-LINEAR DIFFERENTIAL EQUATIONS . 295 4.6.3.2
SECOND ORDER NON-LINEAR DIFFERENTIAL EQUATIONS 295 4.6.3.3 SHM IN WOLF'S
CLOTHING 296 4.6.3.4 NON-LINEAR EXAMPLE 298 4.6.4 GENERAL FORMS OF FLOW
NEAR TO A FIXED POINT 301 CONTENTS 4.6.5 PREDATOR PREY EXAMPLE 303 4.6.6
COMPETING HERBIVORES 305 5. PROBABILITY 309 5.1 THE BASIC IDEAS OF
PROBABILITY . 310 5.1.0.1 THE MATCHING BIRTHDAY PROBLEM 311 5.1.1 TWO
CAUTIONARY EXAMPLES 313 5.1.1.1 THE PROBLEM OF THE TERMINATED MATCH . .
. . 313 5.1.1.2 THE PROBLEM OF THE DOORS AND THE GOAT . . . 315 5.2
PRECISE PROBABILITY 316 5.2.1 INCLUSION-EXCLUSION 317 5.2.1.1 THE COATS
PROBLEM 318 5.2.2 CONDITIONAL PROBABILITY 320 5.2.2.1 THE BAYESIAN
STATISTICIAN 321 5.2.3 THE LAW OF TOTAL PROBABILITY AND BAYES FORMULA .
. . 322 5.2.3.1 RELIABILITY OF DRUG TESTING 324 5.3 FUNCTIONS ON SAMPLES
SPACES: RANDOM VARIABLES 325 5.3.1 THE BINOMIAL DISTRIBUTION 326 5.3.2
THE POISSON APPROXIMATION TO THE BINOMIAL 329 5.3.2.1 ERROR DISTRIBUTION
IN NOISY DATA 330 5.3.3 THE POISSON DISTRIBUTION 332 5.3.3.1
INTERPRETATION OF THE POISSON DISTRIBUTION . . 332 5.3.4 CONTINUOUS
RANDOM VARIABLES 334 5.3.4.1 THE NORMAL DISTRIBUTION 335 5.3.4.2 THE
UNIFORM DISTRIBUTION 337 5.3.4.3 THE GAMMA RANDOM VARIABLE 338 5.3.5 AN
APPLICATION OF PROBABILITY TO PRIME NUMBERS . . . 339 5.3.6 AVERAGING
AND EXPECTATION 341 5.3.6.1 WHAT DO WE EXPECT TO OBTAIN IN A POISSON OR
BINOMIAL TRIAL? 343 5.3.6.2 WHAT DO WE EXPECT TO OBTAIN IN A NORMAL
TRIAL? 344 5.3.6.3 THE COLLECTION PROBLEM 345 5.3.6.4 THE CAUCHY
DISTRIBUTION 346 5.3.7 DISPERSION AND VARIANCE 346 5.3.7.1 A DYNAMICAL
INTERPRETATION OF EXPECTATION AND VARIANCE 348 5.4 LIMIT THEOREMS 348
5.4.1 CHEBYSHEV'S INEQUALITY 349 XVIII A MATHEMATICAL BRIDGE 5.4.1.1
CHEBYSHEV AS THE BEST POSSIBLE INEQUALITY . 350 5.4.1.2 STANDARDISING
DEVIATIONS FROM THE AVERAGE . 351 5.4.1.3 STANDARDISED VARIABLES 352
5.4.2 THE LAW OF LARGE NUMBERS 352 5.4.2.1 MONTE CARLO INTEGRATION 353
5.4.3 THE CENTRAL LIMIT THEOREM AND THE NORMAL DISTRIBUTION 355 5.4.3.1
THE CENTRAL LIMIT THEOREM 356 6. THEORETICAL PHYSICS 361 6.1 THE
NEWTONIAN WORLD 363 6.1.1 THE MOTION OF THE PLANETS AROUND THE SUN 364
6.1.1.1 TRANSFORMING THE EQUATION OF MOTION . . . . 364 6.1.1.2 SOLUTION
OF THE PROBLEM 366 6.1.1.3 NEWTONIAN ANTI-GRAVITY 369 6.1.2 PROVING
CONSERVATION OF ENERGY 370 6.1.3 PLANETARY CATASTROPHE FOR OTHER TYPES
OF FORCES . . . . 372 6.1.4 EARTH, SUN AND MOON? 374 6.2 LIGHT,
ELECTRICITY AND MAGNETISM 376 6.2.1 STATIC ELECTRICITY 377 6.2.1.1 THE
EQUATION FOR A MAGNET 378 6.2.2 CURRENT ELECTRICITY AND MAGNETISM 380
6.2.3 MAXWELL'S EQUATIONS FOR ELECTROMAGNETIC WAVES . 381 6.2.3.1
ELECTROMAGNETIC WAVE SOLUTIONS IN THE VAC- UUM OF SPACE 382 6.3
RELATIVITY AND THE GEOMETRY OF THE UNIVERSE 384 6.3.1 SPECIAL RELATIVITY
386 6.3.1.1 LENGTH CONTRACTION AND TIME DILATION . 390 6.3.1.2
LORENTZ TRANSFORMATION AS A ROTATION IN SPACE-TIME 391 ; 6.3.1.3 THE
LORENTZ TRANSFORMATIONS AS THE GROUP OF I. SYMMETRIES OF SPACETIME 393 |
6.3.1.4 RELATIVISTIC MOMENTUM 395 \ 6.3.2 GENERAL RELATIVITY AND
GRAVITATION 397 6.3.2.1 THE SCHWARZSCHILD BLACK HOLE 399 | 6.4 QUANTUM
MECHANICS 401 I; 6.4.1 QUANTISATION 401J 6.4.1.1 THE WAVE-PARTICLE
PARADOX 402 I; 6.4.2 THE FORMULATION OF QUANTUM MECHANICS 4041 6.4.2.1
THE UNDERLYING EQUATION 4051 CONTENTS XIX 6.4.3 THE BASIC QUANTUM
MECHANICAL SETUP 408 6.4.3.1 PARTICLE TRAPPED IN A ONE-DIMENSIONAL BOX .
410 6.4.3.2 MOMENTUM EIGENSTATES 412 6.4.3.3 GENERALISATION TO THREE
DIMENSIONS 413 6.4.4 HEISENBERG'S UNCERTAINTY PRINCIPLE 414 6.4.4.1
UNCERTAINTY IN ACTION 416 6.4.5 WHERE NEXT? 417 APPENDIX A EXERCISES FOR
THE READER 419 A.I NUMBERS 420 A.2 ANALYSIS 429 A.3 ALGEBRA 439 A.4
CALCULUS AND DIFFERENTIAL EQUATIONS 453 A.5 PROBABILITY 471 A.6
THEORETICAL PHYSICS 486 APPENDIX B FURTHER READING 499 APPENDIX C BASIC
MATHEMATICAL BACKGROUND 503 C.I SETS 503 C.I.I NOTATION 503 C.I.2
OPERATIONS ON SETS 504 C.2 LOGIC AND PROOF 504 C.2.1 FORMS OF PROOF 505
C.3 FUNCTIONS 506 C.3.1 COMPOSITION OF FUNCTIONS 506 C.3.2 FACTORIALS
506 C.3.3 POWERS, INDICES AND THE BINOMIAL THEOREM 506 C.3.4 THE
EXPONENTIAL, E AND THE NATURAL LOGARITHM . 507 C.3.5 THE
TRIGONOMETRICAL FUNCTIONS 508 C.3.6 THE HYPERBOLIC FUNCTIONS 508 C.4
VECTORS AND MATRICES 509 C.4.1 COMBINING VECTORS TOGETHER 509 C.4.2
POLAR COORDINATES 510 C.4.3 MATRICES 510 C.5 CALCULUS 512 C.5.1
DIFFERENTIATION 512 C.5.2 INTEGRATION 513 L XX A MATHEMATICAL BRIDGE
C.5.3 POSITION, VELOCITY AND ACCELERATION 513 C.5.4 SIMPLE HARMONIC
MOTION 514 APPENDIX D DICTIONARY OF SYMBOLS 515 D.I THE GREEK LETTERS
515 D.2 MATHEMATICAL SYMBOLS 516 INDEX 519 |
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| index_date | 2024-07-02T17:21:19Z |
| indexdate | 2024-07-09T20:56:57Z |
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| spelling | Hewson, Stephen Fletcher Verfasser aut A mathematical bridge an intuitive journey in higher mathematics Stephen Fletcher Hewson Repr. New Jersey u.a. World Scientific 2005 XX, 526 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Mathematik (DE-588)4037944-9 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Mathematik (DE-588)4037944-9 s DE-604 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015617303&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hewson, Stephen Fletcher A mathematical bridge an intuitive journey in higher mathematics Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4037944-9 (DE-588)4151278-9 |
| title | A mathematical bridge an intuitive journey in higher mathematics |
| title_auth | A mathematical bridge an intuitive journey in higher mathematics |
| title_exact_search | A mathematical bridge an intuitive journey in higher mathematics |
| title_exact_search_txtP | A mathematical bridge an intuitive journey in higher mathematics |
| title_full | A mathematical bridge an intuitive journey in higher mathematics Stephen Fletcher Hewson |
| title_fullStr | A mathematical bridge an intuitive journey in higher mathematics Stephen Fletcher Hewson |
| title_full_unstemmed | A mathematical bridge an intuitive journey in higher mathematics Stephen Fletcher Hewson |
| title_short | A mathematical bridge |
| title_sort | a mathematical bridge an intuitive journey in higher mathematics |
| title_sub | an intuitive journey in higher mathematics |
| topic | Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Mathematik Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015617303&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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