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Categorical Topology: Proceedings of the International Conference, Berlin, August 27th to September 2nd, 1978 1979 book

Categorical Topology: Proceedings of the International Conference, Berlin, August 27th to September 2nd, 1978

Details Of The Book

Categorical Topology: Proceedings of the International Conference, Berlin, August 27th to September 2nd, 1978

Category: Geometry and topology
edition: 1 
Authors: , ,   
serie: Lecture Notes in Mathematics 719 
ISBN : 0387095039 
publisher: Springer-Verlag Berlin Heidelberg 
publish year: 1979 
pages: 432 
language: English 
ebook format : DJVU (It will be converted to PDF, EPUB OR AZW3 if requested by the user) 
file size: 3 MB 

price : $15 20 With 25% OFF



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You can Download Categorical Topology: Proceedings of the International Conference, Berlin, August 27th to September 2nd, 1978 Book After Make Payment, According to the customer's request, this book can be converted into PDF, EPUB, AZW3 and DJVU formats.


Abstract Of The Book



Table Of Contents

Cover......Page 1
Preface to Second Edition......Page 6
Preface......Page 8
Note to the Reader......Page 12
Contents......Page 13
Part I Coordinates and Calculus......Page 22
1.1 Vectors in a Plane and in Space......Page 23
1.1.1 Dot Product......Page 25
1.1.2 Vector or Cross Product......Page 27
1.2 Coordinate Systems......Page 31
1.3 Vectors in Different Coordinate Systems......Page 36
1.3.1 Fields and Potentials......Page 41
1.3.2 Cross Product......Page 48
1.4 Relations Among Unit Vectors......Page 51
1.5 Problems......Page 57
2 Differentiation......Page 63
2.1 The Derivative......Page 64
2.2.1 Definition, Notation, and Basic Properties......Page 67
2.2.2 Differentials......Page 73
2.2.3 Chain Rule......Page 75
2.2.4 Homogeneous Functions......Page 77
2.3 Elements of Length, Area, and Volume......Page 79
2.3.1 Elements in a Cartesian Coordinate System......Page 80
2.3.2 Elements in a Spherical Coordinate System......Page 82
2.3.3 Elements in a Cylindrical Coordinate System......Page 85
2.4 Problems......Page 88
3.1 “∫” Means “∫um”......Page 96
3.2 Properties of Integral......Page 100
3.2.4 Partition of Range of Integration......Page 101
3.2.6 Small Region of Integration......Page 102
3.2.8 Symmetric Range of Integration......Page 103
3.2.9 Differentiating an Integral......Page 104
3.2.10 Fundamental Theorem of Calculus......Page 106
3.3 Guidelines for Calculating Integrals......Page 110
3.3.1 Reduction to Single Integrals......Page 111
3.3.2 Components of Integrals of Vector Functions......Page 114
3.4 Problems......Page 117
4.1.1 An Example from Mechanics......Page 120
4.1.2 Examples from Electrostatics and Gravity......Page 123
4.1.3 Examples from Magnetostatics......Page 128
4.2.1 Cartesian Coordinates......Page 134
4.2.2 Cylindrical Coordinates......Page 137
4.2.3 Spherical Coordinates......Page 139
4.3 Applications: Triple Integrals......Page 141
4.4 Problems......Page 147
5.1 One-Variable Case......Page 157
5.1.1 Linear Densities of Points......Page 161
5.1.2 Properties of the Delta Function......Page 163
5.1.3 The Step Function......Page 170
5.2 Two-Variable Case......Page 172
5.3 Three-Variable Case......Page 177
5.4 Problems......Page 184
Part II Algebra of Vectors......Page 189
6 Planar and Spatial Vectors......Page 190
6.1 Vectors in a Plane Revisited......Page 191
6.1.1 Transformation of Components......Page 193
6.1.2 Inner Product......Page 199
6.1.3 Orthogonal Transformation......Page 207
6.2 Vectors in Space......Page 209
6.2.1 Transformation of Vectors......Page 211
6.2.2 Inner Product......Page 215
6.3 Determinant......Page 219
6.4 The Jacobian......Page 224
6.5 Problems......Page 228
7 Finite-Dimensional Vector Spaces......Page 232
7.1 Linear Transformations......Page 233
7.2 Inner Product......Page 235
7.3 The Determinant......Page 239
7.4 Eigenvectors and Eigenvalues......Page 241
7.5 Orthogonal Polynomials......Page 244
7.6 Systems of Linear Equations......Page 247
7.7 Problems......Page 251
8 Vectors in Relativity......Page 254
8.1 Proper and Coordinate Time......Page 256
8.2 Spacetime Distance......Page 257
8.3 Lorentz Transformation......Page 260
8.4 Four-Velocity and Four-Momentum......Page 264
8.4.1 Relativistic Collisions......Page 267
8.4.2 Second Law of Motion......Page 270
8.5 Problems......Page 271
Part III Infinite Series......Page 273
9.1 Infinite Sequences......Page 274
9.2 Summations......Page 277
9.2.1 Mathematical Induction......Page 280
9.3 Infinite Series......Page 281
9.3.1 Tests for Convergence......Page 282
9.3.2 Operations on Series......Page 288
9.4 Sequences and Series of Functions......Page 289
9.4.1 Properties of Uniformly Convergent Series......Page 292
9.5 Problems......Page 294
10.1 Power Series......Page 297
10.1.1 Taylor Series......Page 300
10.2 Series for Some Familiar Functions......Page 301
10.3 Helmholtz Coil......Page 305
10.4 Indeterminate Forms and L\'Hôpital\'s Rule......Page 308
10.5 Multipole Expansion......Page 311
10.6 Fourier Series......Page 313
10.7 Multivariable Taylor Series......Page 319
10.8 Application to Differential Equations......Page 321
10.9 Problems......Page 325
11.1 Integrals as Functions......Page 331
11.1.1 Gamma Function......Page 332
11.1.2 The Beta Function......Page 334
11.1.4 Elliptic Functions......Page 336
11.2 Power Series as Functions......Page 341
11.2.1 Hypergeometric Functions......Page 342
11.2.2 Confluent Hypergeometric Functions......Page 346
11.2.3 Bessel Functions......Page 347
11.3 Problems......Page 350
Part IV Analysis of Vectors......Page 354
12 Vectors and Derivatives......Page 355
12.1.1 Ordinary Angle Revisited......Page 356
12.1.2 Solid Angle......Page 359
12.2 Time Derivative of Vectors......Page 362
12.2.1 Equations of Motion in a Central Force Field......Page 364
12.3 The Gradient......Page 367
12.3.1 Gradient and Extremum Problems......Page 371
12.4 Problems......Page 374
13.1 Flux of a Vector Field......Page 376
13.1.1 Flux Through an Arbitrary Surface......Page 381
13.2.1 Flux Density......Page 382
13.2.2 Divergence Theorem......Page 385
13.2.3 Continuity Equation......Page 389
13.3 Problems......Page 394
14.1 The Line Integral......Page 397
14.2 Curl of a Vector Field and Stokes\' Theorem......Page 401
14.3 Conservative Vector Fields......Page 408
14.4 Problems......Page 414
15.1 Double Del Operations......Page 417
15.2 Magnetic Multipoles......Page 419
15.3 Laplacian......Page 421
15.3.1 A Primer of Fluid Dynamics......Page 423
15.4 Maxwell\'s Equations......Page 425
15.4.1 Maxwell\'s Contribution......Page 426
15.4.2 Electromagnetic Waves in Empty Space......Page 427
15.5 Problems......Page 430
16.1 Elements of Length......Page 432
16.2 The Gradient......Page 434
16.3 The Divergence......Page 436
16.4 The Curl......Page 440
16.4.1 The Laplacian......Page 444
16.5 Problems......Page 445
17.1 Vectors and Indices......Page 448
17.1.1 Transformation Properties of Vectors......Page 450
17.1.2 Covariant and Contravariant Vectors......Page 454
17.2 From Vectors to Tensors......Page 456
17.2.1 Algebraic Properties of Tensors......Page 459
17.2.2 Numerical Tensors......Page 461
17.3 Metric Tensor......Page 463
17.3.1 Index Raising and Lowering......Page 466
17.3.2 Tensors and Electrodynamics......Page 468
17.4.1 Covariant Differential and Affine Connection......Page 471
17.4.2 Covariant Derivative......Page 473
17.4.3 Metric Connection......Page 474
17.5 Riemann Curvature Tensor......Page 477
17.6 Problems......Page 480
Part V Complex Analysis......Page 484
18.1 Cartesian Form of Complex Numbers......Page 485
18.2 Polar Form of Complex Numbers......Page 490
18.3 Fourier Series Revisited......Page 496
18.4 A Representation of Delta Function......Page 499
18.5 Problems......Page 501
19.1 Complex Functions......Page 505
19.1.1 Derivatives of Complex Functions......Page 507
19.1.2 Integration of Complex Functions......Page 511
19.1.3 Cauchy Integral Formula......Page 516
19.1.4 Derivatives as Integrals......Page 517
19.2 Problems......Page 519
20 Complex Series......Page 523
20.1 Power Series......Page 524
20.2 Taylor and Laurent Series......Page 526
20.3 Problems......Page 530
21.1 The Residue......Page 532
21.2 Integrals of Rational Functions......Page 536
21.3 Products of Rational and Trigonometric Functions......Page 539
21.4 Functions of Trigonometric Functions......Page 541
21.5 Problems......Page 543
Part VI Differential Equations......Page 545
22 From PDEs to ODEs......Page 546
22.1 Separation of Variables......Page 547
22.2 Separation in Cartesian Coordinates......Page 549
22.3 Separation in Cylindrical Coordinates......Page 552
22.4 Separation in Spherical Coordinates......Page 553
22.5 Problems......Page 555
23.1 Normal Form of a FODE......Page 556
23.2 Integrating Factors......Page 558
23.3 First-Order Linear Differential Equations......Page 561
23.4 Problems......Page 566
24 Second-Order Linear Differential Equations......Page 568
24.1 Linearity, Superposition, and Uniqueness......Page 569
24.2 The Wronskian......Page 571
24.3 A Second Solution to the HSOLDE......Page 572
24.4 The General Solution to an ISOLDE......Page 574
24.5 Sturm Liouville Theory......Page 575
24.5.1 Adjoint Differential Operators......Page 576
24.5.2 Sturm Liouville System......Page 579
24.6 SOLDEs with Constant Coefficients......Page 580
24.6.1 The Homogeneous Case......Page 581
24.6.2 Central Force Problem......Page 584
24.6.3 The Inhomogeneous Case......Page 588
24.7 Problems......Page 592
25 Laplace\'s Equation: Cartesian Coordinates......Page 596
25.1 Uniqueness of Solutions......Page 597
25.2 Cartesian Coordinates......Page 599
25.3 Problems......Page 608
26 Laplace\'s Equation: Spherical Coordinates......Page 611
26.1 Frobenius Method......Page 612
26.2 Legendre Polynomials......Page 614
26.3 Second Solution of the Legendre DE......Page 621
26.4 Complete Solution......Page 623
26.5.2 Recurrence Relation......Page 626
26.5.3 Orthogonality......Page 628
26.5.4 Rodrigues Formula......Page 630
26.6 Expansions in Legendre Polynomials......Page 632
26.7 Physical Examples......Page 635
26.8 Problems......Page 639
27.1 The ODEs......Page 642
27.2 Solutions of the Bessel DE......Page 645
27.3 Second Solution of the Bessel DE......Page 648
27.4.2 Recurrence Relations......Page 649
27.4.3 Orthogonality......Page 650
27.4.4 Generating Function......Page 652
27.5 Expansions in Bessel Functions......Page 656
27.6 Physical Examples......Page 657
27.7 Problems......Page 660
28.1 The Heat Equation......Page 663
28.1.1 Heat-Conducting Rod......Page 664
28.1.2 Heat Conduction in a Rectangular Plate......Page 665
28.1.3 Heat Conduction in a Circular Plate......Page 666
28.2 The Schrödinger Equation......Page 668
28.2.1 Quantum Harmonic Oscillator......Page 669
28.2.2 Quantum Particle in a Box......Page 677
28.2.3 Hydrogen Atom......Page 679
28.3 The Wave Equation......Page 682
28.3.1 Guided Waves......Page 684
28.3.2 Vibrating Membrane......Page 688
28.4 Problems......Page 689
Part VII Special Topics......Page 692
29.1 The Fourier Transform......Page 693
29.1.1 Properties of Fourier Transform......Page 696
29.1.2 Sine and Cosine Transforms......Page 697
29.1.3 Examples of Fourier Transform......Page 698
29.1.4 Application to Differential Equations......Page 702
29.2 Fourier Transform and Green\'s Functions......Page 705
29.2.1 Green\'s Function for the Laplacian......Page 708
29.2.2 Green\'s Function for the Heat Equation......Page 709
29.2.3 Green\'s Function for the Wave Equation......Page 711
29.3 The Laplace Transform......Page 712
29.3.1 Properties of Laplace Transform......Page 713
29.3.2 Derivative and Integral of the Laplace Transform......Page 717
29.3.3 Laplace Transform and Differential Equations......Page 718
29.3.4 Inverse of Laplace Transform......Page 721
29.4 Problems......Page 723
30 Calculus of Variations......Page 727
30.1 Variational Problem......Page 728
30.1.1 Euler-Lagrange Equation......Page 729
30.1.2 Beltrami identity......Page 731
30.1.4 Several Independent Variables......Page 734
30.1.5 Second Variation......Page 735
30.1.6 Variational Problems with Constraints......Page 738
30.2.1 From Newton to Lagrange......Page 740
30.2.2 Lagrangian Densities......Page 744
30.3 Hamiltonian Dynamics......Page 747
30.4 Problems......Page 750
31 Nonlinear Dynamics and Chaos......Page 752
31.1 Systems Obeying Iterated Maps......Page 753
31.1.1 Stable and Unstable Fixed Points......Page 754
31.1.2 Bifurcation......Page 756
31.1.3 Onset of Chaos......Page 760
31.2 Systems Obeying DEs......Page 762
31.2.1 The Phase Space......Page 763
31.2.2 Autonomous Systems......Page 765
31.2.3 Onset of Chaos......Page 769
31.3.1 Feigenbaum Numbers......Page 772
31.3.2 Fractal Dimension......Page 774
31.4 Problems......Page 777
32.1 Basic Concepts......Page 779
32.1.1 A Set Theory Primer......Page 780
32.1.2 Sample Space and Probability......Page 782
32.1.3 Conditional and Marginal Probabilities......Page 784
32.1.4 Average and Standard Deviation......Page 787
32.1.5 Counting: Permutations and Combinations......Page 789
32.2 Binomial Probability Distribution......Page 790
32.3 Poisson Distribution......Page 795
32.4 Continuous Random Variable......Page 799
32.4.1 Transformation of Variables......Page 802
32.4.2 Normal Distribution......Page 804
32.5 Problems......Page 807
Bibliography......Page 812
Index......Page 814




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