Hand Book of Mathematics and Computational Science
Transcript of Hand Book of Mathematics and Computational Science
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JohnW.Harris Horst Stocker
Handbook of Mathematics and
Computational Science
With 545 Illustrations
Springer
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Contents
Introduction v
1 Numerical comp utation arithmetics and num erics) 1
1.1
Sets 1
1.1.1 Re presenta tion of sets 1
1.1.2 O pera tions on sets 2
1.1.3 La ws of the alge bra of sets 4
1.1.4 M app ing and function 4
1.2 Num ber system s 4
1.2.1 De cimal num ber system 5
1.2.2 Other num ber system s 6
1.2.3 Co mp uter rep resentatio n . . . . 6
1.2.4 H om er's schem e for the representation of num bers 7
1.3 Natural num bers ' 7
1.3.1 M athema tical induction 8
1.3.2 Vectors and fields, index ing 8
1.3.3 Calculating with natural num bers 9
1.4 Integers 11
1.5 Rational num bers (fractional num bers) 11
1.5.1 D ecim al fractions 11
1.5.2 Fractions 13
1.5.3 Ca lculatin g with fractions 13
1.6 Calculating with quo tients 14
1.6.1 Proportion 14
1.6.2 Ru le of three 15
1.7 Mathematics of finance 15
1.7.1 Ca lculations of percen tage 16
1.7.2 Interes t and com po und interest 16
1.7.3 A mo rtization 17
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1.7.4 Annuitie s 18
1.7.5 Depreciation 19
1.8 Irrational num bers 20
1.9 Real num bers 20
1.10 Com plex num bers 20
1.10.1
Field of com plex num bers 21
1.11 Calculating with real num bers 22
1.11.1
Sign and absolute value 22
1.11.2 Ord ering relations 23
1.11.3
Intervals 23
1.11.4 Rou nding and truncating 24
1.11.5 Calculating with intervals 25
1.11.6
Brackets 25
1.11.7
Add ition and subtraction 26
1.11.8
Sum mation sign 27
1.11.9 M ultiplication and division 28
1.11.10 Produc t sign 29
1.11.11 Pow ers and roots 30
1.11.12 Exp onentiation and logarithms 32
1.12 Binom ial theore m 33
1.12.1
Bino mial formulas 33
1.12.2
Bin om ial coefficients 34
1.12.3
Pas cal's triangle 34
1.12.4 Properties of binom ial coefficients 35
1.12.5
Exp ansion of pow ers of sums 36
2 Equa tions and inequalities algebra) 37
2.1 Fund am ental algebraic laws 37
2.1.1 No me nclature 37
2.1.2 Group 39
2.1.3 Ring 39
2.1.4 Field 39
2.1.5 Vector space . 40
2.1.6 A lgebra . . . . ' . 40
2.2 Equations with one unknow n . 41
2.2.1 Elem entary equivalence transformations 41
2.2.2 Overview of the different kind s of equations 41
2.3 Linear equations 42
2.3.1 Ordinary linear equations 42
2.3.2 Linear equa tions in fractional form 42
2.3.3 Linea r equations in irrational form 43
2.4 Quadratic equations 43
2.4.1 Qu adratic equations in fractional form 44
2.4.2 Qu adratic equations in irrational form 44
2.5 Cub ic equations 44
2.6 Quartic equations 46
2.6.1 Gen eral quartic equations 46
2.6.2 Biqua dratic equations 46
2.6.3 Sym metric quartic equations 46
2.7 Equations of arbitrary degree 47
2.7.1 Polyno mial division 47
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2.8 Fractional rational equations 48
2.9 Irrational equations 48
2.9.1 Radical equations 48
2.9.2 Power equations 49
2.10 Transcendental equations 49
2.10.1 Exponential equations 49
2.10.2 Logarithmic equations 50
2.10.3 Trigonometric (goniometric) equations 51
2.11 Equations with absolute values 51
2.11.1 Equations with one absolute value 51
2.11.2 Equations with several absolute values 52
2.12 Inequalities 53
2.12.1 Equivalence transformations for inequalities 53
2.13 Numerical solution of equations 54
2.13.1 Graphical solution 54
2.13.2 Nesting of intervals 54
2.13.3 Secant methods and method of false position 55
2.13.4 Newton s method 56
2.13.5 Successive approximation 57
Geometry and trigonometry in the plane 59
3.1 Point curves 60
3.2 Basic constructions 60
3.2.1 Construction of the midpoint of a segment 60
3.2.2 Construction of the bisector of an angle 61
3.2.3 Construction of perpendiculars 61
3.2.4 To drop a perpendicular 61
3.2.5 Construction of parallels at a given distance 61
3.2.6 Parallels through a given point 62
3.3 Angles 62
3.3.1 Specification of angles 62
3.3.2 Types of angles 63
3.3.3 Angles between two parallels . .-. 64
3.4 Similarity and intercept theorems 64
3.4.1 Intercept theorems * 64
3.4.2 Division of a segment 65
3.4.3 Mean values 66
3.4.4 Golden Section 66
3.5 Triangles 67
3.5.1 Congruence theorems 67
3.5.2 Similarity of triangles 68
3.5.3 Construction of triangles 68
3.5.4 Calculation of a right triangle 70
3.5.5 Calculation of an arbitrary triangle 70
3.5.6 Relations between angles and sides of a triangle 72
3.5.7 Altitude 73
3.5.8 Angle-bisectors 74
3.5.9 Medians 74
3.5.10 Mid-perpendiculars, incircle, circumcircle, excircle 75
3.5.11 Area of a triangle 76
3.5.12 Generalized Pythagorean theorem 76
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3.5.13 An gular relations 76
3.5.14 Sine theorem 76
3.5.15 Cosine theorem 77
3.5.16 Tangent theorem 77
3.5.17 Half-angle theorems 77
3.5.18 M ollw eide's formulas 77
3.5.19 The orem s of sides 78
3.5.20 Isosceles triangle 78
3.5.21 Equilateral triangle 79
3.5.22 Right triangle 80
3.5.23 The orem of Tha les 81
3.5.24 Pythag orean theorem 81
3.5.25 Theorem of Euclid 81
3.5.26 Altitude theorem 81
3.6 Qua drilaterals 82
3.6.1 Gen eral quadrilateral 82
3.6.2 Trapezoid 82
3.6.3 Parallelogram 83
3.6.4 Rhom bus 83
3.6.5 Rectan gle 84
3.6.6 Square 84
3.6.7 Qu adrilateral of chords 85
3.6.8 Qu adrilateral of tangents 86
3.6.9 Kite 86
3.7 Regular n-go ns (polygons) 86
3.7.1 Ge neral regular n-gon s 87
3.7.2 Particular regular n-go ns (polygons) 87
3.8 Circu lar objects 89
3.8.1 Circle 89
3.8.2 Circular areas 90
3.8.3 A nnu lus, circular ring 91
3.8.4 Sec tor of a circle 91
3.8.5 Sector of an annulus .\ 92
3.8.6 Segm ent of a circle 92
3.8.7 Ellipse - 93
Solid geometry 95
4.1 Ge neral theorem s . . 95
4.1.1 Cav alieri's theorem 95
4.1.2 Sim pson's rule 95
4.1.3 Gu ldin's rules 96
4.2 Prism 96
4.2.1 Ob lique prism 96
4.2.2 Right prism 97
4.2.3 Cuboid 97
4.2.4 Cube 97
4.2.5 Ob liquely truncated n-sided prism 98
4.3 Pyramid 98
4.3.1 Tetrahedron 98
4.3.2 Frustum of a pyram id 99
4.4 Regular polyhedron 99
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4.4.1 Euler s theorem for polyhedrons 99
4.4.2 Tetrahedron 99
4.4.3 Cube (hexahedron) 100
4.4.4 Octahedron 100
4.4.5 Dodecahedron 101
4.4.6 Icosahedron 101
4.5 Other solids 102
4.5.1 Prismoid, prismatoid 102
4.5.2 Wedge 102
4.5.3 Obelisk 102
4.6 Cylinder 102
4.6.1 General cylinder 103
4.6.2 Right circular cylinder 103
4.6.3 Obliquely cut circular cylinder 103
4.6.4 Segment of a cylinder 104
4.6.5 Hollow cylinder (tube) 104
4.7 Cone 104
4.7.1 Right circular cone 105
4.7.2 Frustum of a right circular cone 105
4.8 Sphere 106
4.8.1 Solid sphere 106
4.8.2 Hollow sphere 106
4.8.3 Spherical sector 106
4.8.4 Spherical segment (spherical cap) 107
4.8.5 Spherical zone (spherical layer) 107
4.8.6 Spherical wedge 108
4.9 Spherical geometry 108
4.9.1 General spherical triangle (Euler s triangle) 108
4.9.2 Right-angled spherical triangle 109
4.9.3 Oblique spherical triangle 110
4.10 Solids of rotation I l l
4.10.1 Ellipsoid I l l
4.10.2 Paraboloid of revolution 112
4.10.3 Hyperboloid of revolution 112
4.10.4 Barrel 112
4.10.5 Torus 113
4.11 Fractal geometry 113
4.11.1 Scaling invariance and self-similarity 113
4.11.2 Construction of self-similar objects 113
4.11.3 Hausdorff dimension 113
4.11.4 Cantor set 114
4.11.5 Koch s curve 114
4.11.6 Koch s snowflake 115
4.11.7 Sierpinski gasket 115
4.11.8 Box-counting algorithm 116
5 Functions 117
5.1 Sequences, series, and functions 117
5.1.1 Sequences and series 117
5.1.2 Properties of
sequences,
limits 119
5.1.3 Functions 120
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5.1.4 Classification of functions 122
5.1.5 Lim it and continu ity 123
5.2 Discussion of curves 124
5.2.1 Doma in of definition 124
5.2.2 Sym metry 124
5.2.3 Behav ior at infinity 125
5.2.4 Gaps of definition and points of discon tinuity 126
5.2.5 Zero s 127
5.2.6 Behav ior of sign 127
5.2.7 Beh avior of slope, extremes 128
5.2.8 Curvature 129
5.2.9 Po int of inflection 129
5.3 Basic properties of functions 130
Sim ple functions 137
5.4 Constant function 137
5.5 Step function 139
5.6 Absolute value function 143
5.7 Delta function 147
5.8 Integer-part function, fractional-part function 150
Integral rational functions 155
5.9 Linear function straight line 155
5.10 Quadratic function parabola 158
5.11 Cub ic equation 162
5.12 Power function of highe r degree 166
5.13 Polyn om ials of higher degree 170
5.14 Representation of polynom ials and particular polynom ials 174
5.14.1 Represen tation by sums and produc ts 174
5.14.2 Taylor series . . 175
5.14.3 Ho m er's scheme 176
5.14.4
Ne wto n's interpolation polynom ial 179
5.14.5
Lagran ge polynom ials 180
5.14.6 Bezier polyno mials and splines 181
5.14.7 Particular polynom ials 187
Frac tional rational functions 189
5.15 Hyperbo la 189
5.16 Rec iprocal quad ratic function 192
5.17 Power functions with a negative exponen t 196
5.18 Quotient of two polynom ials 200
5.18.1 Polynom ial division and partial fraction decom position . . . 20 3
5.18.2
Pad e's approximation 205
Irrational algebraic functions 209
5.19 Square-root function 209
5.20 Roo t function 212
5.21 Power functions with fractional exponents . . . 2 1 6
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5.22 Roots of rationa l functions 219
Transcendental functions 228
5.23 Log arithmic function 228
5.24 Expansion function 23 3
5.25 Exponen tial functions of pow ers 239
Hyperbolic functions 245
5.26 Hyperbolic sine and cosin e functions 247
5.27 Hyperbolic tangent and cotange nt function 252
5.28 Hyperbolic secant and hyperb olic cosecant functions 258
Area hyperbolic functions 26 3
5.29 Area hype rbolic sine and hyperb olic cosin e 264
5.30 Area-hyperbo lic tangent and hyperbolic cotangent 267
5.31 Area-hyperbo lic secant and hyperbolic cosecant 271
Trigonometric functions 274
5.32 Sine and cos ine functions 278
5.32.1 Superpo sitions of oscillations 287
5.32.2
Periodic functions 292
5.33 Tangent and cotang ent functions 294
5.34 Secant and cosecant 30 0
Inverse trigono metric functions 306
5.35 Inverse sine and cos ine functions 307
5.36 Inverse tange nt and cotan gen t functions 311
5.37 Inverse secan t and cos ecant functions 315
Plane curves 319
5.38 Algebraic curves of the n-th order 319
5.38.1 Curves of the second order 319
5.38.2 Curves of the third order 321
5.38.3 Curves of the fourth and higher order 323
5.39 Cycloidal curves 324
5.40 Spirals : 327
5.41 Other curves 328
Vector ana lysis 33 1
6.1 Vector alge bra 331
6.1.1 Vector and scalar 331
6.1.2 Particular vectors 332
6.1.3 Mu ltiplication of a vector by a scalar 332
6.1.4 Vector addition 33 3
6.1.5 Vector subtraction 33 3
6.1.6 Calculating laws 333
6.1.7 Linear depe nden ce/indep ende nce of vectors 334
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6.1.8 Basis 335
6.2 Scalar produc t or inner produc t 338
6.2.1 Calculating laws 339
6.2.2 Prope rties and applications of the scalar produ ct 339
6.2.3 Sch mid t's orthonorm alization metho d 341
6.2.4 Direction cosine 341
6.2.5 Ap plication hype rcubes of vector analysis 342
6.3 Vector produc t of two vectors 343
6.3.1 Properties of the vector produc t 344
6.4 Mu ltiple produ cts of vectors 345
6.4.1 Scalar triple prod uct 345
7 Coordinate systems 349
7.1 Coo rdinate system s in two dimen sions 349
7.1.1 Cartesian coordinates 349
7.1.2 Polar coordinates 350
7.1.3 Conv ersions betw een two-d imension al coordinate systems . 350
7.2 Tw o-dimensional coordinate transformation 350
7.2.1 Parallel displacem ent (translation) 351
7.2.2 Rotation 352
7.2.3 Reflection 353
7.2.4 Scaling 353
7.3 Coo rdinate system s in three dimen sions 354
7.3.1 Cartesian coordinates 354
7.3.2 Cylindrical coordinates 354
7.3.3 Sphe rical coordinates 355
7.3.4 Conv ersions betw een three-dim ensional coordinate systems . 355
7.4 Coo rdinate transformation in three dimen sions 356
7.4.1 Parallel displacem ent (translation) 356
7.4.2 Rotation 357
7.5 Ap plication in com puter graph ics 357
7.6 Transformations 358
7.6.1 Object representation and object description 358
7.6.2 Hom ogeneou s coordinates 359
7.6.3 Tw o-dimensional translations with hom ogen eous
coordinates 360
7.6.4 Tw o-dimensional scaling with hom ogen eous coordina tes . . 360
7.6.5 Three -dimen sional translation with hom ogen eous
coordinates 361
7.6.6 Three -dimen sional scaling with hom ogen eous coordina tes . 361
7.6.7 Three -dimen sional rotation of points with hom ogen eous
coordinates 362
7.6.8 Positioning of an object in space 363
7.6.9 Rotation of objects about an arbitrary axis in space 364
7.6.10 Animation 366
7.6.11 Reflections 366
7.6.12 Transformation of coordina te systems 367
7.6.13 Translation of a coordina te system 367
7.6.14 Rotation of a coordinate system about a principal axis . . . . 368
7.7 Projections 370
7.7.1 Fun dam ental principles 370
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7.7.2 Parallel projection 370
7.7.3 Central projection 373
7.7.4 Gen eral formulation of projections 374
7.8 W indow/viewport transformation 376
8 Analytic geom etry 377
8.1 Elements of the plane 377
8.1.1 Distance betwe en two points 377
8.1.2 Division of a segm ent 377
8.1.3 Are a of a triang le 378
8.1.4 Equation of a curve 378
8.2 Straight line 378
8.2.1 Form s of straight-line equations 379
8.2.2 Hessian norm al form 380
8.2.3 Po int of intersec tion of straight lines 381
8.2.4 An gle betw een straight lines 381
8.2.5 Parallel and perp end icular straight lines 382
8.3 Circle ,. 38 2
8.3.1 Equations of a circle 382
8.3.2 Circle and straight line 383
8.3.3 Interse ction of two circles 383
8.3.4 Equation of the tangent to a circle 384
8.4 Ellipse 384
8.4.1 Equations of the ellipse 384
8.4.2 Foca l prop erties of the ellipse 385
8.4.3 Diam eters of the ellipse 385
8.4.4 Tangent and norm al to the ellipse 385
8.4.5 Curvature of the ellipse 386
8.4.6 Areas and circumference of the ellipse 386
8.5 Parabola 387
8.5.1 Equ ations of the parabola 387
8.5.2 Foc al properties of the parab ola 388
8.5.3 Diam eters of the parabola 388
8.5.4 Tangent and norm al of the parabola , 388
8.5.5 Curvature of a parabola 389
8.5.6 Areas and arc lengths of the parab ola 389
8.5.7 Parab ola and straight line 389
8.6 Hyperbola 390
8.6.1 Equations of the hyperbola 390
8.6.2 Focal prop erties of the hyp erbo la 391
8.6.3 Tangent and norm al to the hyperbola 392
8.6.4 Conjugate hyperb olas and diam eter 392
8.6.5 Curvature of a hyperbo la 392
8.6.6 Areas of hype rbola 393
8.6.7 Hy perbo la and straight line 393
8.7 General equation of con ies 393
8.7.1 Form of con ies 394
8.7.2 Transfo rma tion to principa l axes 394
8.7.3 Geo metric construction (conic section) 395
8.7.4 Direc trix prop erty 395
8.7.5 Polar equation 396
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8.8 Elements in space 396
8.8.1 Distance between two points 396
8.8.2 Division of a segment 396
8.8.3 Volume of a tetrahedron 396
8.9 Straight lines in space 397
8.9.1 Parametric representation of a straight line 397
8.9.2 Point of intersection of two straight lines 397
8.9.3 Angle of intersection between two intersecting straight
lines 398
8.9.4 Foot of a perpendicular (perpendicular line) 398
8.9.5 Distance between a point and a straight line 398
8.9.6 Distance between two lines 399
8.10 Planes in space 399
8.10.1 Parametric representation of the plane 399
8.10.2
Coordinate representation of the plane 399
8.10.3 Hessian normal form of the plane 400
8.10.4 Conversions 400
8.10.5 Distance between a point and a plane 401
8.10.6 Point of intersection of a line and a plane 401
8.10.7 Angle of intersection between two intersecting planes . . . . 401
8.10.8 Foot of the perpendicular (perpendicular line) 401
8.10.9 Reflection 402
8.10.10 Distance between two parallel planes 402
8.10.11
Cut set of two planes 402
8.11 Plane of the second order in normal form 403
8.11.1 Ellipsoid 403
8.11.2 Hyperboloid 403
8.11.3 Cone 404
8.11.4 Paraboloid 404
8.11.5 Cylinder 405
8.12 General plane of the second order 406
8.12.1 General equation 406
8.12.2
Transformation to principal axes 406
8.12.3 Shape of a surface of the second order 407
9 Matrices determinants and systems of linear equations 409
9.1 Matrices 409
9.1.1 Row and column vectors 411
9.2 Special matrices 412
9.2.1 Transposed, conjugate, and adjoint matrices 412
9.2.2 Square matrices 412
9.2.3 Triangular matrices 414
9.2.4 Diagonal matrices 415
9.3 Operations with matrices 418
9.3.1 Addition and subtraction of matrices 418
9.3.2 Multiplication of a matrix by a scalar factor
418
9.3.3 Multiplication of
vectors,
scalar product 419
9.3.4 Multiplication of a matrix by a vector 421
9.3.5 Multiplication of matrices 421
9.3.6 Calculating rules of matrix multiplication 422
9.3.7 Multiplication by a diagonal matrix 424
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9.3.8 M atrix mu ltiplication according to Fa lk's schem e 42 4
9.3.9 Ch ecking of row and column sum s 425
9.4 Determinants 42 6
9.4.1 Two-row determ inants 427
9.4.2 Gen eral com putational rules for determ inants 427
9.4.3 Zero value of the determinan t 429
9.4.4 Three-row determ inants 43 0
9.4.5 Determ inants of higher (n-th) order 432
9.4.6 Calculation of n-row determ inants 433
9.4.7 Regu lar and inverse matrix 434
9.4.8 Calculation of the inverse matrix in terms of determ inants . . 435
9.4.9 Ran k of a matrix 436
9.4.10
Determ ination of the rank by me ans of minor determ inants . 437
9.5 Systems of linear equ ations 437
9.5.1 System s of two equation s with two unkno wns 439
9.6 Num erical solution m ethods 441
9.6.1 Gau ssian algorithm for systems of linear equation s 441
9.6.2 Forw ard elimination 441
9.6.3 Pivoting 44 3
9.6.4 Bac ksubstitution 44 4
9.6.5 LU -decom position 445
9.6.6 Solvability of (m x n) systems of equations 448
9.6.7 Gau ss-Jordan me thod for matrix inversion 45 0
9.6.8 Calculation of the inverse matrix A
1
452
9.7 Iterative solution of syste ms of linear equ ation s 45 4
9.7.1 Total-step m ethod s (Jacobi) 45 6
9.7.2 Single-step me thods (Gauss-Se idel) 456
9.7.3 Criteria of convergence for iterative metho ds 457
9.7.4 Sto rage of the coefficient matrix 458
9.8 Table of solution methods 45 9
9.9 Eigenvalue equations 46 1
9.10 Tensors 463
9.10.1
Algeb raic operations with tensors 465
10 Boolean algebr a-app lication in sw itching algebr a 467
10.1 Basic notions 467
10.1.1 Propositions and truth values 467
10.1.2 Proposition variables ' 468
10.2 Boolean connectives 46 8
10.2.1 Negation: not 46 9
10.2.2 Conjunction: and 469
10.2.3 Disjunction (inclusiv e): or 469
10.2.4 Calculating rules 470
10.3 Boo lean functions 47 1
10.3.1 Ope rator basis 47 2
10.4 Norm al forms 47 2
10.4.1 Disjunctive norm al forms 472
10.4.2 Conjunctive norm al form 47 3
10.4.3 Represen tation of functions by norm al forms 47 3
10.5 Karnaugh-Veitch diagram s 475
10.5.1 Produ cing a KV -diagram 476
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10.5.2 Entering a function in a KV-diagram 476
10.5.3 Min imization with the help of KV -diagrams 477
10.6 Minim ization according to Q uine and Mc Cluskey 478
10.7 Mu lti-valued logic and fuzzy logic 481
10.7.1 Mu lti-valued logic 481
- 10.7.2 Fuzzy logic 481
11 Graph s and Algorithm s 483
11.1 Grap hs 483
11.1.1 Basic definitions 48 3
11.1.2 Representation of graphs 485
11.1.3 Trees 485
11.2 Matching s 486
11.3 Networks 487
11.3.1 Flow s in networks 487
11.3.2 Eulerian line and Ham iltonian circuit 487
12 Differential calculu s 489
12.1 Deriva tive of a function 489
12.1.1 Differential 490
12.1.2 Differentiability 491
12.2 Differentiation rules 492
12.2.1 Derivatives of elem entary functions 492
12.2.2 Derivatives of trigonom etric functions 492
12.2.3 Derivatives of hyperbo lic functions 492
12.2.4 Con stant rule 493
12.2.5 Factor rule 493
12.2.6 Pow er rule 493
12.2.7 Sum rule 493
12.2.8 Prod uct rule 493
12.2.9 Qu otient rule 494
12.2.10 Chain rule 494
12.2.11 Logarith mic differentiation of functions 495
12.2.12 Differentiation of functions in para me tric representatio n . . . 49 5
12.2.13 Differentiation of functions in polar coo rdinates 496
12.2.14 Differentiation of an implicit function 496
12.2.15 Differentiation of the inverse function 497
12.2.16 Table of differentiation rules 498
12.3 Mean value theorem s 499
12.3.1 Ro lle's theorem 499
12.3.2 M ean value theorem of differential calculus 499
12.3.3 Extended mean value theorem of differential calculus . . . . 500
12.4 Highe r derivatives 500
12.4.1 Slop e, extremes 502
12.4.2 Curvature 503
12.4.3 Po int of inflection 503
12.5 Ap proxim ation method of differentiation 504
12.5.1 Gra phical differentiation 504
12.5.2 N um erical differentiation 505
12.6 Differentiation of functions with several variables 506
12.6.1 Partial derivative 506
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12.6.2 Total differential 508
12.6.3 Extrem es of functions in two dim ension s 508
12.6.4 Extrem es with cons traints 509
12.7 App lication of differential calculu s 510
12.7.1 Calcu lation of indefinite expression s 510
12.7.2 Discussion of curves 511
12.7.3 Ex trem e value prob lem s 512
12.7.4 Calcu lus of errors 513
12.7.5 Determination of zeros according to New ton's method . . . 514
13 Differential geo m etry 517
13.1 Plane curves 517
13.1.1 Rep resentation of curv es 517
13.1.2 Differentiation by imp licit represen tation 517
13.1.3 Differentiation by param etric represen tation 518
13.1.4 Differentiation by polar coo rdinates 518
13.1.5 Differential of arc of a curve 518
13.1.6 Tangen t, norma l 519
13.1.7 Cu rvature of a curve 520
13.1.8 Evo lutes and evolve nts 522
13.1.9 Po ints of inflection, vertices 522
13.1.10 Singular points 522
13.1.11 Asy m ptotes 523
13.1.12 Enve lope of a family of curves 524
13.2 Space curves 524
13.2.1 Rep resentation of space curves 524
13.2.2 M oving trihedral 525
13.2.3 Cu rvature 527
13.2.4 Torsion of a curv e 527
13.2.5 Frenet formulas 528
13.3 Surfaces 528
13.3.1 Rep resentation of a surface 528
13.3.2 Tangent plane and norm al to the surface 529
13.3.3 Singu lar points of the surface 530
14 Infinite series 531
14.1 Series 531
14.2 Criteria of con verg ence 53 2
14.2.1 Special num ber series 535
14.3 Taylor and M acL aur in series 535
14.3.1 Tay lor's formula 535
14.3.2 Tay lor series 536
14.4 Power series 537
14.4.1 Test of converg ence for pow er series 537
14.4.2 Prope rties of converge nt pow er series 538
14.4.3 Inversion of pow er series 540
14.5 Special expan sions of series and prod ucts 540
14.5.1 Bino m ial series 540
14.5.2 Special binom ial series 540
14.5.3 Serie s of exp onential functions 541
14.5.4 Series of loga rithm ic functions 54 2
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14.5.5 Series of trigono me tric functions 542
14.5.6 Series of inverse trigonom etric functions 543
14.5.7 Series of hyp erbolic functions 544
14.5.8 Series of area hype rbolic functions 544
14.5.9 Partial fraction expa nsions 544
14.5.10 Infinite produ cts 545
15 Integral calculus 547
15.1 Definition and integrab ility 547
15.1.1 Primitive 547
15.1.2 Definite and indefinite integrals 548
15.1.3 Ge om etrical interpretation 549
15.1.4 Rules for integrability 550
15.1.5 Imp rope r integrals 551
15 .2 '
Integration rules 552
15.2.1 Rules for indefinite integrals 552
15.2.2 Ru les for definite integ rals 553
15.2.3 Table of integration rules 554
15.2.4 Integrals of some eleme ntary functions 555
15.3 Integration m ethod s 557
15.3.1 Integration by substitution 557
15.3.2 Integration by parts 560
15.3.3 Integration by partial fraction deco mp osition 562
15.3.4 Integration by series expa nsion 565
15.4 Num erical integration 567
15.4.1 Rectan gular rule 567
15.4.2 Trape zoidal rule 568
15.4.3 Sim pson 's rule 568
15.4.4 Rom berg integration 569
15.4.5 Gau ssian qua drature 570
15.4.6 Table of num erical integration me thods 572
15.5 Me an value theorem of integral calculus 574
15.6 Lin e, surface, and volum e integrals 574
15.6.1 Arc length (rectification) 57 4
15.6.2 Are a 575
15.6.3 Solid of rotation (solid of revolution ) 576
15.7 Fun ctions in param etric representation 577
15.7.1 Arc length in param etric representation 577
15.7.2 Sector formula 578
15.7.3 Solids of rotation in param etric represen tation 578
15.8 M ultiple integrals and their applications 579
15.8.1 Definition of m ultiple integra ls 579
15.8.2 Calcu lation of areas 580
15.8.3 Cen ter of mass of arcs 581
15.8.4 M om ent of inertia of an area 581
15.8.5 Cen ter of mass of areas 582
15.8.6 M om ent of inertia of planes 582
15.8.7 Cen ter of mass of a body 582
o 15.8.8 M om ent of inertia of a body 583
15.8.9 Cen ter of mass of rotational solids 583
15.8.10 M om ent of inertia of rotational solids 583
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15.9 Technical app lications of integra l calcu lus 58 4
15.9.1 Static mo men t, center of mass 58 4
15.9.2 M ass mo men t of inertia 585
15.9.3 Statics 588
15.9.4 Calculation of work 588
15.9.5 M ean values 589
16 Vector ana lysis 59 1
16.1 Fields 591
16.1.1 Sym metries of fields 592
16.2 Differentiation and integration of vecto rs 594
16.2.1 Scale factors in general orthogon al coordinates 596
16.2.2 Differential operators 597
16.3 Gradient and po tential 598
16.4 Directional derivative and vector grad ient 60 0
16.5 Divergence and Gau ssian integral theorem 601
16.6 Rotation and Stok es's theorem 60 4
16.7 Laplace operator and G ree n's formulas 607
16.7.1 Co m bina tions of div, rot, and grad; calcu lation of fields . . . 60 9
16.8 Summ ary 610
17 Complex variables and functions 613
17.1 Complex num bers 613
17.1.1 Imag inary num bers 613
17.1.2 Alge braic representation of com plex num bers 614
17.1.3 Cartesian representation of com plex num bers 614
17.1.4 Conjuga te com plex num bers 615
17.1.5 Ab solute value of a comp lex num ber 615
17.1.6 Trigono me tric representation of com plex num bers 616
17.1.7 Exp one ntial representation of com plex num bers 616
17.1.8 Transformation from Cartesian to trigonom etric
representation 617
17.1.9 Rieman n sphere 618
17.2 Elementary arithm etical ope rations with com plex num bers 619
17.2.1 Ad dition and subtraction of com plex num bers 619
17.2.2 M ultiplication and division of com plex num bers 619
17.2.3 Exp onen tiation in the comp lex dom ain 622
17.2.4 Taking the root in the comp lex dom ain 623
17.3 Elementary functions of a com plex variable 623
17.3.1 Sequ ence s in the com plex dom ain 624
17.3.2 Series in the com plex dom ain 625
17.3.3 Exp onen tial function in the com plex dom ain 626
17.3.4 Natural logarithm in the comp lex dom ain 626
17.3.5 Gen eral pow er in the com plex dom ain 627
17.3.6 Trigono metric functions in the com plex dom ain 627
17.3.7 Hy perbolic functions in the com plex domain 629
17.3.8 Inverse trigonom etric, inverse hyperb olic functions in the
complex domain 630
17.4 Applications of com plex functions 631
17.4.1 Rep resentation of oscillations in the com plex plane 631
17.4.2 Sup erposition of osc illations of equal frequency 63 2
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17.4.3 Loci 633
17.4.4 Inversion of loci 634
17.5 Differentiation of functions of a com plex variab le 635
17.5.1 Definition of the derivative in the com plex dom ain 635
17.5.2 Differentiation rales in the com plex dom ain 636
17.5.3 Cauch y-Riem ann differentiability conditions 637
17.5.4 Conformal ma pping 637
17.6 Integration in the com plex plane 639
17.6.1 Com plex curvilinear integrals 639
17.6.2 Ca uch y's integral theorem 640
17.6.3 Primitive functions in the com plex dom ain 641
17.6.4 Ca uch y's integral formulas 641
17.6.5 Taylor series of an analytic function 642
17.6.6 Lau rent series 64 3
17.6.7 Classification of singular poin ts 643
17.6.8 Residu e theorem 644
17.6.9 Inverse Laplace transformation 645
18 Differential equ ations 647
18.1 Ge neral definitions 647
18.2 Geom etric interpretation 649
18.3 Solution metho ds for first-order differential equa tions 650
18.3.1 Separation of variables 650
18.3.2 Substitution 651
18.3.3 Ex act differential equ ation s 651
18.3.4 Integrating factor 651
18.4 Linear differential equa tions of the first orde r 652
18.4.1 Variation of the constants 652
18.4.2 Gen eral solution 653
18.4.3 Determ ination of a particular solution 653
18.4.4 Linear differential equ ations of the first order with con stant
coefficients 653
18.5 Som e specific equations 654
18.5.1 Be rnou lli differential equation 654
18.5.2 Ricc ati differential equation 654
18.6 Differential equation s of the second order 655
18.6.1 Sim ple special cases 655
18.7 Linear differential equa tions of the second orde r 656
18.7.1 Ho m ogeneo us linear differential equa tion of the
second order 657
18.7.2 Inho m oge neo us linear differential equ ations of the second
order 657
18.7.3 Red uction of special differential equations of the second
order to differential equa tions of the first order 659
18.7.4 Linear differential equations of the second order with
con stant coefficients 659
18.8 Differential equations oft he w -th order 662
18.9 System s of coupled differential equ ations of the first order 668
18.10 Systems of linear hom ogen eous differential equations with constant
coefficients 670
18.11 Partial differential equations 672
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18.11.1 Solution by sepa ration 67 3
18.12 Nu merical integration of differential equ ations 676
18.12.1 Euler method 676
18.12.2 Heun m ethod 677
18.12.3 Mod ified Euler method 679
18.12.4~ Run ge-Ku tta m ethod s 679
18.12.5 Run ge-Ku tta m ethod for system s of differential equa tions . . 68 5
18.12.6 Difference m ethod for the solution of partial differential
equations 685
18.12.7 M ethod of finite elem ents 68 8
19 Fourier transformation 691
19.1 Fourier series 691
19.1.1 Introduc tion 691
19.1.2 Definition and coefficients 69 1
19.1.3 Cond ition of convergen ce 693
19.1.4 Extend ed interval 694
19.1.5 Sym m etries 696
19.1.6 Fourier series in complex and spectral representation . . . . 698
19.1.7 Form ulas for the calculation of Fourier series 699
19.1.8 Fou rier expa nsion of simple perio dic functions 699
19.1.9 Fou rier series (table) 705
19.2 Fourier integrals 707
19.2.1 Introduc tion 707
19.2.2 Definition and coefficients 70 7
19.2.3 Con ditions for convergence 708
19.2.4 Com plex representation, Fourier sine and cosine
transformation 708
19.2.5 Sym m etries 71 0
19.2.6 Convolution and som e calculating rales 71 0
19.3 Discrete Fou rier transform ation (DFT) 712
19.3.1 Definition and coefficients 71 2
19.3.2 Shannon scanning theorem 713
19.3.3 Discrete sine and cosine transformation 714
19.3.4 Fast Fourier transformation (FFT ) 715
19.3.5 Particular pairs of Fourier transform s 72 0
19.3.6 Fou rier transform s (table) 720
19.3.7 Particular Fourier sine transform s 722
19.3.8 Particular Fourier cosine transforms 723
19.4 Wavelet transform ation 72 4
19.4.1 Signa ls 724
19.4.2 Linea r signal analy sis 725
19.4.3 Sym metry transformations 726
19.4.4 Time-frequency analysis and Gab or transformation 727
19.4.5 Wavelet transform ation 728
19.4.6 Discre te wav elet-transforma tion 73 2
20 Laplace and z tran sform ations 735
20.1 Introduction 735
20.2 Definition of the Laplace transform ation 73 6
20.3 Transformation theorem s 73 7
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20.4 Partial fraction separation 745
20.4.1 Partial fraction separation with simple real zeros 745
20.4 .2 Partial fraction deco mposition with multiple real zeros . . . 746
20.4.3 Partial fraction decom position with com plex zeros 747
20.5 Lin ear differential equations with constant coefficients 748
20.5 .1 La place transform ation: linear differential equa tion of the
first orde r with constant coefficients 749
20.5.2 La place transform ation: linear differential equa tion of the
second order with con stant coefficients 751
20.5.3 Ex am ple: linear differential equations 753
20.5.4 Lap lace transforms (table) 756
20.6 z transformation 764
20.6.1 Definition of the z transforma tion 764
20.6.2 Con vergence cond itions for the z transformation 766
20.6.3 Inversion of the z transform ation 767
20.6.4 Calculating rales 767
20.6.5 C alculating rales for the z transformation 770
20.6.6 Table of z transforms 770
21 Probab ility theory and mathem atical statistics 773
21.1 Co mb inatorics 773
21.2 Rand om events 774
21.2.1 Basic notions 774
21.2.2 Even t relations and event operations 775
21.2.3 Structural representation of events 777
21.3 Probability of events 778
21.3.1 Properties of probabilities 778
21.3.2 M ethods to calculate probabilities 778
21.3.3 Co nditional probab ilities 779
21.3.4 Calculating with probabilities 779
21.4 Rand om variables and their distributions 781
21.4.1 Individual probability, density function and distribution
function x 782
21.4.2 Param eters of distributions 783
21.4.3 Special discrete distribution 785
21.4.4 Special continuous distributions 793
21.5 Lim it theorem s 800
21.5.1 Law s of large num bers 800
21.5.2 Lim it theorem s 801
21.6 M ultidimensional random variables 802
21.6.1 Distribution functions of two-dim ensional random variables . 802
21.6.2 Tw o-dimensional discrete random variables 803
21.6.3 Tw o-dimensional continuou s random variables 804
21.6.4 Independence of random variables 805
21.6.5 Param eters of two-dim ensional rand om variables 806
21.6.6 Tw o-dimensional norm al distribution 807
21.7 Basics of ma thematical statistics 8 0 8 .
21.7.1 Description of m easurem ents . . 809
21.7.2 Types of error 810
21.8 Parameters for describing distributions of mea sured values 812
21.8.1 Posit ion parameter, means of series of measurements . . . . 812
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21.8.2 Dispersion parameter 814
21.9 Special distribu tions 815
21.9.1 Freque ncy distributions 815
21.9.2 Distribution of rand om sample functions 816
21.10 Analysis by means of random sampling (theory of testing
and estimating) 820
21.10.1 Estimation method s 821
21.10.2 Con struction principle s for estimators 823
21.10.3 Me thod of mom ents 823
21.10.4 Ma ximum likelihood method 824
21.10.5 M ethod of least squares 824
21.10.6 x
2
-minim um method 825
21.10.7 M ethod of qua ntiles, percentiles 825
21.10.8 Interval estimation 82 6
21.10.9 Interval boun ds for norma l distribution 828
21.10.10 Pred iction and confidence interval bou nds for binom ial
and hypergeom etric distributions 829
21.10.11 Interval bou nds for a Poisson distribution 830
21.10.12 Determ ination of samp le sizes n 830
21.10.13 Test me thods 831
21.10.14 Param eter tests 834
21.10.15 Param eter tests for a norm al distribution 834
21.10.16 Hy poth eses about the mean value of arbitrary
distributions 836
21.10.17 Hy potheses about p of binomial and hypergeometric
distributions 837
21.10.18 Tests of goo dne ss of fit 837
21.10.19 Ap plication: acceptance /rejection test 838
21.11 Reliability 839
21.12 Com putation of adjustment, regression 841
21.12.1 Line ar regressio n, least squares me thod 843
21.12.2 Regress ion of the n-th order 844
22 Fuzzy logic 847
22.1 Fuzzy sets 847
22.2 Fuzzy concept 848
22.3 Functional graph s for the mo deling of fuzzy sets 849
22.4 Com bination of fuzzy sets 852
22.4.1 Elem entary operation s 852
22.4.2 Ca lcula ting rales for fuzzy sets 855
22.4.3 Ru les for families of fuzzy sets 856
22.4.4 t norm and t conorm 856
22.4.5 Non -parametrized operators: t norms and s norms
(t conorm s) 858
22.4.6 Parametrized t and s norms 859
22.4.7 Com pensatory operators 860
22.5 Fuzzy relations 861
22.6 Fuzzy inference 863
22.7 De nazification method s 864
22.8 Exam ple: erect pendulum 866
22.9 Fuzzy realizations 870
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23 Neural networks 871
23.1 Fun ction and structure 871
23.1.1 Function 871
23.1.2 Structure 872
23.2 Implementation of the neuron mod el 873
~ 23.2.1 Time-independent systems 873
23.2.2 Time-dependent systems 873
23.2.3 Ap plication 874
23.3 Supervised learning 874
23.3.1 Principle of supervised learning 874
23.3.2 Standard backpropagation 876
23.3.3 Backpropagation through time 877
23.3.4 Improved learning method s 878
23.3.5 Hopfield netw ork 879
23.4 Unsupervised learning 881
23.4.1 Principle of unsupervised learning 881
23.4.2 Kohonen model 881
24 Com puters 883
24.1 Operating systems 883
24.1.1 Introduction to M S-DO S 885
24.1.2 Introduction to UN IX 886
24.2 High-level program ming languages 889
24.2.1 Prog ram structures 890
24.2.2 Object-oriented program ming (OOP) 892
Introduction to PASCAL 893
24.3 Basic structure 894
24.4 Variables and types 894
24.4.1 Integers 895
24.4.2 Real num bers . . . 895
24.4.3 Boo lean values 895
24 .4.4 ARRAYS \ 895
24.4.5 Cha racters and character strings 896
24.4.6 RECORD ' 897
24.4.7 Poin ters 898
24.4.8 Self-defined types 899
24.5 Statements 900
24.5.1 Assignmen ts and expressions 900
24.5.2 Input and output 901
24.5.3 Com pound statements 902
24.5.4 Conditional statements I F and CASE 903
24.5.5 Lo ops FOR, WH ILE, and REPEAT 904
24.6 Proce dures and functions 905
24.6.1 Procedures 905
24.6.2 Functions 906
24.6.3 Local and global variables, parameter passing 906
24.7 Recursion 908
24.8 Basic algorithms 909
24.8.1 Dy nam ic data structures 909
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Contents xxvii
24.8.2 Search 910
24.8.3 Sorting 911
24.9 Com puter graphics 913
24.9.1 Basic functions 91 3
Introduction to C 91 4
24.9.2 Basic structures 914
24.9.3 Op erators 916
24.9.4 Da ta structures 918
24.9.5 Loops and branches 921
Introduction to C++ 92 4
24.9.6 Variables and cons tants 924
24.9.7 Ov erloading of functions 92 4
24.9.8 Overloading of operators 924
24.9.9 Classes 925
24.9.10 Instantiation of classes 92 6
24.9 .11 f r i e n d functions 926
24.9.12 Operators as mem ber functions 926
24.9.13 Constructors 927
24.9.14 Derived classes (inheritance) 928
24.9.15 Class libraries 929
Introduction to FO RTR AN 930
24.9.16 Program structure 930
24.9.17 Data structures 930
24.9.18 Type conversion 931
24.9.19 Operators 933
24.9.20 Loops and branches 933
24.9.21 Subprogram s 934
Computer algebra 937
24.9.22 Structural elemen ts of Ma thematica 937
24.9.23 Structural eleme nts of M aple 94 0
24.9.24 Algebraic expressions 942
24.9.25 Equations and systems of equations 943
24.9.26 Linear algebra 944
24.9.27 Differential and integral calculus 945
24.9.28 Programm ing 947
24.9.29 Fitting curves and interpolation with M athema tica 948
24.9.30 Graphics 949
25 Tables of integ rals 951
25.1 Integrals of rationa l function s 951
25.1.1 Integrals with P - ax + b, a^0 951
25.1.2 Integrals with x
1
/(ax + fc)\ P = ax + b,a ^ 0, F ^ 0 . . 9 52
25.1.3 Integrals with1/ x
n
ax + b)
m
), P = ax + b b ^ 0 . . . 95 3
25.1.4 Integrals withax + band ex + d c ^ 0 955
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25.1.5 Integrals with
a + x
an d
b + x a = b 955
25.1.6 Integrals with
P = ax
2
+ bx + c (a / 0) 956
25.1.7 Integrals withx
n
/ ax
2
+ bx + c)
m
, P = ax
2
+ bx+c
a / 0 956
25.1.8 Integrals withl/ x
n
ax
2
+ bx +c)
m
),P = ax
2
+ bx + c
c / 0 9 5 7
2 5 . 1 . 9 I n t e g r a l s w i t h P =a
2
x
2
9 5 8
2 5 . 1 . 1 0 I n t e g r a l s w i t h l / ( a
2
x
2
) , P = a
2
x
2
a / 0 . . . . 9 5 8
2 5 . 1 . 1 1 I n t e g r a l s w i t h x / { a
2
x
2
m
, P =
a
2
x
2
a / 0 . . . 9 5 8
2 5 . 1 . 1 2 Integrals with 1/(x
n
{a
2
x
2
)
m
) P = a
2
x
2
a/ 0 . . 960
25.1.13 Integrals with P =a
3
x
3
a / 0 961
25.1.14 Integrals witha
4
+ x
A
a> 0) 962 '
25.1.15 Integrals witha
4
- x
4
a >0) 962
25.2 Integralsofirrational functions 963
25.2.1 Integrals with
x
1 / 2
and
P = ax + b a,b^0 963
25.2.2 Integra ls with ax + b)
l/2
P = ax + b a / 0 964
25.2.3 Integrals with
(ax
+ b)
l/2
and (ex
+ d )
1 / 2
, a, c/ 0 . . . . 966
25.2.4 Integra ls withR = a
2
+x
2
)
1
'
2
a / 0 966
25.2.5 Integra ls withS= (x
2
-
a
2
)
y
'
2
a 0 968
25.2.6 Integrals withT = {a
2
- x
2
)
x
'
2
a / 0 970
25.2.7 Integ rals with
( a x
2
+ bx + c)
l/2
X
= ax
2
+ bx + c a / 0 972
25.3 Integralsoftranscend ental functions 973
25.3.1 Integrals with exponential functions
973
25.3.2 Integrals with logarithm ic functions
(x > 0) 975
25.3.3 Integrals with hyperbo lic functions
a/ 0) 977
25.3.4 Integrals with inverse hyperbolic functions 979
25.3.5 Integrals with sine and cosine functions
a/ 0) 979
25.3.6 Integrals with sine and cosine functions
a/ 0) 984
25.3.7 Integrals with tangentorcotang ent functions
a
/ 0) . . . 989
25.3.8 Integrals with inverse trigonom etric functions a/ 0) . . . 990
25 .4 Definite integra ls
992
25.4.1 Definite integrals with algeb raic functions
992
25.4.2 Definite integrals with expo nential functions 992
25.4.3 Definite integrals with logarithmic functions 994
25.4.4 Definite integrals with trigonom etric functions
995
Index