hand book of mathematics and computational science

8/12/2019 Hand Book of Mathematics and Computational Science

1/23

JohnW.Harris Horst Stocker

Handbook of Mathematics and

Computational Science

With 545 Illustrations

Springer


2/23

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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viii Contents

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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Contents ix

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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Contents

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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Contents xi

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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xii Contents

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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Contents xiii

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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xiv Contents

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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Contents xv

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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xvi Contents

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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Contents xvii

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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xviii Contents

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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Contents xix

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


15/23

xx Contents

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


16/23

Contents xxi

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


17/23

xxii Contents

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


18/23

Contents xxiii

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


19.2.3 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


19.3.2 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


19/23

xxiv Contents

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


20/23

Contents xxv

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


21/23

xxvi Contents

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


22/23

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


23/23

xxviii Contents

25.1.5 Integrals with

a + x

an d

b + x a = b 955


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


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


(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

hand book of mathematics and computational science

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