Handbook of mathematics and computational science:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1998
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVIII, 1028 S. Ill., graph. Darst. |
| ISBN: | 0387947469 |
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MARC
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| 100 | 1 | |a Harris, John W. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Handbook of mathematics and computational science |c John W. Harris ; Horst Stocker |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1998 | |
| 300 | |a XXVIII, 1028 S. |b Ill., graph. Darst. | ||
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JOHN W. HARRIS HORST STOCKER HANDBOOK OF MATHEMATICS AND COMPUTATIONAL
SCIENCE WITH 545 ILLUSTRATIONS SPRINGER CONTENTS INTRODUCTION V 1
NUMERICAL COMPUTATION (ARITHMETICS AND NUMERICS) 1 1.1 SETS 1 1.1.1
REPRESENTATION OF SETS 1 1.1.2 OPERATIONS ON SETS 2 1.1.3 LAWS OF THE
ALGEBRA OF SETS 4 1.1.4 MAPPING AND FUNCTION 4 1.2 NUMBER SYSTEMS 4
1.2.1 DECIMAL NUMBER SYSTEM 5 1.2.2 OTHER NUMBER SYSTEMS 6 1.2.3
COMPUTER REPRESENTATION . . . ." 6 1.2.4 HOMER'S SCHEME FOR THE
REPRESENTATION OF NUMBERS 7 1.3 NATURAL NUMBERS ' 7 1.3.1 MATHEMATICAL
INDUCTION 8 1.3.2 VECTORS AND FIELDS, INDEXING 8 1.3.3 CALCULATING WITH
NATURAL NUMBERS 9 1.4 INTEGERS 11 1.5 RATIONAL NUMBERS (FRACTIONAL
NUMBERS) 11 1.5.1 DECIMAL FRACTIONS 11 1.5.2 FRACTIONS 13 1.5.3
CALCULATING WITH FRACTIONS 13 1.6 CALCULATING WITH QUOTIENTS 14 1.6.1
PROPORTION 14 1.6.2 RULE OF THREE 15 1.7 MATHEMATICS OF FINANCE 15 1.7.1
CALCULATIONS OF PERCENTAGE 16 1.7.2 INTEREST AND COMPOUND INTEREST 16
1.7.3 AMORTIZATION 17 VIII CONTENTS 1.7.4 ANNUITIES 18 1.7.5
DEPRECIATION 19 1.8 IRRATIONAL NUMBERS 20 1.9 REAL NUMBERS 20 1.10
COMPLEX NUMBERS 20 1.10.1 FIELD OF COMPLEX NUMBERS 21 1.11 CALCULATING
WITH REAL NUMBERS 22 1.11.1 SIGN AND ABSOLUTE VALUE 22 1.11.2 ORDERING
RELATIONS 23 1.11.3 INTERVALS 23 1.11.4 ROUNDING AND TRUNCATING 24
1.11.5 CALCULATING WITH INTERVALS 25 1.11.6 BRACKETS 25 1.11.7 ADDITION
AND SUBTRACTION 26 1.11.8 SUMMATION SIGN 27 1.11.9 MULTIPLICATION AND
DIVISION 28 1.11.10 PRODUCT SIGN 29 1.11.11 POWERS AND ROOTS 30 1.11.12
EXPONENTIATION AND LOGARITHMS 32 1.12 BINOMIAL THEOREM 33 1.12.1
BINOMIAL FORMULAS 33 1.12.2 BINOMIAL COEFFICIENTS 34 1.12.3 PASCAL'S
TRIANGLE 34 1.12.4 PROPERTIES OF BINOMIAL COEFFICIENTS 35 1.12.5
EXPANSION OF POWERS OF SUMS 36 2 EQUATIONS AND INEQUALITIES (ALGEBRA) 37
2.1 FUNDAMENTAL ALGEBRAIC LAWS 37 2.1.1 NOMENCLATURE 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 ALGEBRA
.'. 40 2.2 EQUATIONS WITH ONE UNKNOWN . 41 2.2.1 ELEMENTARY
EQUIVALENCE TRANSFORMATIONS 41 2.2.2 OVERVIEW OF THE DIFFERENT KINDS OF
EQUATIONS 41 2.3 LINEAR EQUATIONS 42 2.3.1 ORDINARY LINEAR EQUATIONS 42
2.3.2 LINEAR EQUATIONS IN FRACTIONAL FORM 42 2.3.3 LINEAR EQUATIONS IN
IRRATIONAL FORM 43 2.4 QUADRATIC EQUATIONS 43 2.4.1 QUADRATIC EQUATIONS
IN FRACTIONAL FORM 44 2.4.2 QUADRATIC EQUATIONS IN IRRATIONAL FORM 44
2.5 CUBIC EQUATIONS 44 2.6 QUARTIC EQUATIONS 46 2.6.1 GENERAL QUARTIC
EQUATIONS 46 2.6.2 BIQUADRATIC EQUATIONS 46 2.6.3 SYMMETRIC QUARTIC
EQUATIONS 46 2.7 EQUATIONS OF ARBITRARY DEGREE 47 2.7.1 POLYNOMIAL
DIVISION 47 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 CONTENTS 3.5.13 ANGULAR 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 MOLLWEIDE'S FORMULAS 77 3.5.19
THEOREMS OF SIDES 78 3.5.20 ISOSCELES TRIANGLE 78 3.5.21 EQUILATERAL
TRIANGLE 79 3.5.22 RIGHT TRIANGLE 80 3.5.23 THEOREM OF THALES 81 3.5.24
PYTHAGOREAN THEOREM 81 3.5.25 THEOREM OF EUCLID 81 3.5.26 ALTITUDE
THEOREM 81 3.6 QUADRILATERALS 82 3.6.1 GENERAL QUADRILATERAL 82 3.6.2
TRAPEZOID 82 3.6.3 PARALLELOGRAM 83 3.6.4 RHOMBUS 83 3.6.5 RECTANGLE 84
3.6.6 SQUARE 84 3.6.7 QUADRILATERAL OF CHORDS 85 3.6.8 QUADRILATERAL OF
TANGENTS 86 3.6.9 KITE 86 3.7 REGULAR N-GONS (POLYGONS) 86 3.7.1 GENERAL
REGULAR N-GONS 87 3.7.2 PARTICULAR REGULAR N-GONS (POLYGONS) 87 3.8
CIRCULAR OBJECTS 89 3.8.1 CIRCLE 89 3.8.2 CIRCULAR AREAS 90 3.8.3
ANNULUS, CIRCULAR RING 91 3.8.4 SECTOR OF A CIRCLE 91 3.8.5 SECTOR OF AN
ANNULUS . \ 92 3.8.6 SEGMENT OF A CIRCLE 92 3.8.7 ELLIPSE - 93 SOLID
GEOMETRY 95 4.1 GENERAL THEOREMS . . 95 4.1.1 CAVALIERI'S THEOREM 95
4.1.2 SIMPSON'S RULE 95 4.1.3 GULDIN'S RULES 96 4.2 PRISM 96 4.2.1
OBLIQUE PRISM 96 4.2.2 RIGHT PRISM 97 4.2.3 CUBOID 97 4.2.4 CUBE 97
4.2.5 OBLIQUELY TRUNCATED N-SIDED PRISM 98 4.3 PYRAMID 98 4.3.1
TETRAHEDRON 98 4.3.2 FRUSTUM OF A PYRAMID 99 4.4 REGULAR POLYHEDRON 99
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 ILL 4.10.1 ELLIPSOID ILL 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 XII CONTENTS 5.1.4 CLASSIFICATION OF FUNCTIONS 122
5.1.5 LIMIT AND CONTINUITY 123 5.2 DISCUSSION OF CURVES 124 5.2.1 DOMAIN
OF DEFINITION 124 5.2.2 SYMMETRY 124 5.2.3 BEHAVIOR AT INFINITY 125
5.2.4 GAPS OF DEFINITION AND POINTS OF DISCONTINUITY 126 5.2.5 ZEROS 127
5.2.6 BEHAVIOR OF SIGN 127 5.2.7 BEHAVIOR OF SLOPE, EXTREMES 128 5.2.8
CURVATURE 129 5.2.9 POINT OF INFLECTION 129 5.3 BASIC PROPERTIES OF
FUNCTIONS 130 SIMPLE 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 CUBIC EQUATION 162 5.12 POWER FUNCTION OF
HIGHER DEGREE 166 5.13 POLYNOMIALS OF HIGHER DEGREE 170 5.14
REPRESENTATION OF POLYNOMIALS AND PARTICULAR POLYNOMIALS 174 5.14.1
REPRESENTATION BY SUMS AND PRODUCTS 174 5.14.2 TAYLOR SERIES . . 175
5.14.3 HOMER'S SCHEME 176 5.14.4 NEWTON'S INTERPOLATION POLYNOMIAL 179
5.14.5 LAGRANGE POLYNOMIALS 180 5.14.6 BEZIER POLYNOMIALS AND SPLINES
181 5.14.7 PARTICULAR POLYNOMIALS 187 FRACTIONAL RATIONAL FUNCTIONS 189
5.15 HYPERBOLA 189 5.16 RECIPROCAL QUADRATIC FUNCTION 192 5.17 POWER
FUNCTIONS WITH A NEGATIVE EXPONENT 196 5.18 QUOTIENT OF TWO POLYNOMIALS
200 5.18.1 POLYNOMIAL DIVISION AND PARTIAL FRACTION DECOMPOSITION . . .
203 5.18.2 PADE'S APPROXIMATION 205 IRRATIONAL ALGEBRAIC FUNCTIONS 209
5.19 SQUARE-ROOT FUNCTION 209 5.20 ROOT FUNCTION 212 5.21 POWER
FUNCTIONS WITH FRACTIONAL EXPONENTS .216 CONTENTS XIII 5.22 ROOTS OF
RATIONAL FUNCTIONS 219 TRANSCENDENTAL FUNCTIONS 228 5.23 LOGARITHMIC
FUNCTION 228 5.24 EXPANSION FUNCTION 233 5.25 EXPONENTIAL FUNCTIONS OF
POWERS 239 HYPERBOLIC FUNCTIONS 245 5.26 HYPERBOLIC SINE AND COSINE
FUNCTIONS 247 5.27 HYPERBOLIC TANGENT AND COTANGENT FUNCTION 252 5.28
HYPERBOLIC SECANT AND HYPERBOLIC COSECANT FUNCTIONS 258 AREA HYPERBOLIC
FUNCTIONS 263 5.29 AREA HYPERBOLIC SINE AND HYPERBOLIC COSINE 264 5.30
AREA-HYPERBOLIC TANGENT AND HYPERBOLIC COTANGENT 267 5.31
AREA-HYPERBOLIC SECANT AND HYPERBOLIC COSECANT 271 TRIGONOMETRIC
FUNCTIONS 274 5.32 SINE AND COSINE FUNCTIONS 278 5.32.1 SUPERPOSITIONS
OF OSCILLATIONS 287 5.32.2 PERIODIC FUNCTIONS 292 5.33 TANGENT AND
COTANGENT FUNCTIONS 294 5.34 SECANT AND COSECANT 300 INVERSE
TRIGONOMETRIC FUNCTIONS 306 5.35 INVERSE SINE AND COSINE FUNCTIONS 307
5.36 INVERSE TANGENT AND COTANGENT FUNCTIONS 311 5.37 INVERSE SECANT AND
COSECANT 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 ANALYSIS 331 6.1 VECTOR ALGEBRA 331 6.1.1 VECTOR AND SCALAR 331
6.1.2 PARTICULAR VECTORS 332 6.1.3 MULTIPLICATION OF A VECTOR BY A
SCALAR 332 6.1.4 VECTOR ADDITION 333 6.1.5 VECTOR SUBTRACTION 333 6.1.6
CALCULATING LAWS 333 6.1.7 LINEAR DEPENDENCE/INDEPENDENCE OF VECTORS 334
XIV CONTENTS 6.1.8 BASIS 335 6.2 SCALAR PRODUCT OR INNER PRODUCT 338
6.2.1 CALCULATING LAWS 339 6.2.2 PROPERTIES AND APPLICATIONS OF THE
SCALAR PRODUCT 339 6.2.3 SCHMIDT'S ORTHONORMALIZATION METHOD 341 6.2.4
DIRECTION COSINE 341 6.2.5 APPLICATION HYPERCUBES OF VECTOR ANALYSIS 342
6.3 VECTOR PRODUCT OF TWO VECTORS 343 6.3.1 PROPERTIES OF THE VECTOR
PRODUCT 344 6.4 MULTIPLE PRODUCTS OF VECTORS 345 6.4.1 SCALAR TRIPLE
PRODUCT 345 7 COORDINATE SYSTEMS 349 7.1 COORDINATE SYSTEMS IN TWO
DIMENSIONS 349 7.1.1 CARTESIAN COORDINATES 349 7.1.2 POLAR COORDINATES
350 7.1.3 CONVERSIONS BETWEEN TWO-DIMENSIONAL COORDINATE SYSTEMS . 350
7.2 TWO-DIMENSIONAL COORDINATE TRANSFORMATION 350 7.2.1 PARALLEL
DISPLACEMENT (TRANSLATION) 351 7.2.2 ROTATION 352 7.2.3 REFLECTION 353
7.2.4 SCALING 353 7.3 COORDINATE SYSTEMS IN THREE DIMENSIONS 354 7.3.1
CARTESIAN COORDINATES 354 7.3.2 CYLINDRICAL COORDINATES 354 7.3.3
SPHERICAL COORDINATES 355 7.3.4 CONVERSIONS BETWEEN THREE-DIMENSIONAL
COORDINATE SYSTEMS . 355 7.4 COORDINATE TRANSFORMATION IN THREE
DIMENSIONS 356 7.4.1 PARALLEL DISPLACEMENT (TRANSLATION) 356 7.4.2
ROTATION 357 7.5 APPLICATION IN COMPUTER GRAPHICS 357 7.6
TRANSFORMATIONS 358 7.6.1 OBJECT REPRESENTATION AND OBJECT DESCRIPTION
358 7.6.2 HOMOGENEOUS COORDINATES 359 7.6.3 TWO-DIMENSIONAL TRANSLATIONS
WITH HOMOGENEOUS COORDINATES 360 7.6.4 TWO-DIMENSIONAL SCALING WITH
HOMOGENEOUS COORDINATES . . 360 7.6.5 THREE-DIMENSIONAL TRANSLATION WITH
HOMOGENEOUS COORDINATES 361 7.6.6 THREE-DIMENSIONAL SCALING WITH
HOMOGENEOUS COORDINATES . 361 7.6.7 THREE-DIMENSIONAL ROTATION OF POINTS
WITH HOMOGENEOUS 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
COORDINATE SYSTEMS 367 7.6.13 TRANSLATION OF A COORDINATE SYSTEM 367
7.6.14 ROTATION OF A COORDINATE SYSTEM ABOUT A PRINCIPAL AXIS . . . .
368 7.7 PROJECTIONS 370 7.7.1 FUNDAMENTAL PRINCIPLES 370 CONTENTS XV
7.7.2 PARALLEL PROJECTION 370 7.7.3 CENTRAL PROJECTION 373 7.7.4 GENERAL
FORMULATION OF PROJECTIONS 374 7.8 WINDOW/VIEWPORT TRANSFORMATION 376 8
ANALYTIC GEOMETRY 377 8.1 ELEMENTS OF THE PLANE 377 8.1.1 DISTANCE
BETWEEN TWO POINTS 377 8.1.2 DIVISION OF A SEGMENT 377 8.1.3 AREA OF A
TRIANGLE 378 8.1.4 EQUATION OF A CURVE 378 8.2 STRAIGHT LINE 378 8.2.1
FORMS OF STRAIGHT-LINE EQUATIONS 379 8.2.2 HESSIAN NORMAL FORM 380 8.2.3
POINT OF INTERSECTION OF STRAIGHT LINES 381 8.2.4 ANGLE BETWEEN STRAIGHT
LINES 381 8.2.5 PARALLEL AND PERPENDICULAR STRAIGHT LINES 382 8.3 CIRCLE
,. 382 8.3.1 EQUATIONS OF A CIRCLE 382 8.3.2 CIRCLE AND STRAIGHT LINE
383 8.3.3 INTERSECTION 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
FOCAL PROPERTIES OF THE ELLIPSE 385 8.4.3 DIAMETERS OF THE ELLIPSE 385
8.4.4 TANGENT AND NORMAL 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 EQUATIONS OF THE PARABOLA 387 8.5.2 FOCAL PROPERTIES
OF THE PARABOLA 388 8.5.3 DIAMETERS OF THE PARABOLA 388 8.5.4 TANGENT
AND NORMAL OF THE PARABOLA , 388 8.5.5 CURVATURE OF A PARABOLA 389 8.5.6
AREAS AND ARC LENGTHS OF THE PARABOLA 389 8.5.7 PARABOLA AND STRAIGHT
LINE 389 8.6 HYPERBOLA 390 8.6.1 EQUATIONS OF THE HYPERBOLA 390 8.6.2
FOCAL PROPERTIES OF THE HYPERBOLA 391 8.6.3 TANGENT AND NORMAL TO THE
HYPERBOLA 392 8.6.4 CONJUGATE HYPERBOLAS AND DIAMETER 392 8.6.5
CURVATURE OF A HYPERBOLA 392 8.6.6 AREAS OF HYPERBOLA 393 8.6.7
HYPERBOLA AND STRAIGHT LINE 393 8.7 GENERAL EQUATION OF CONIES 393 8.7.1
FORM OF CONIES 394 8.7.2 TRANSFORMATION TO PRINCIPAL AXES 394 8.7.3
GEOMETRIC CONSTRUCTION (CONIC SECTION) 395 8.7.4 DIRECTRIX PROPERTY 395
8.7.5 POLAR EQUATION 396 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 C 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 CONTENTS XVII 9.3.8 MATRIX MULTIPLICATION ACCORDING TO FALK'S SCHEME
424 9.3.9 CHECKING OF ROW AND COLUMN SUMS 425 9.4 DETERMINANTS 426 9.4.1
TWO-ROW DETERMINANTS 427 9.4.2 GENERAL COMPUTATIONAL RULES FOR
DETERMINANTS 427 9.4.3 ZERO VALUE OF THE DETERMINANT 429 9.4.4 THREE-ROW
DETERMINANTS 430 9.4.5 DETERMINANTS OF HIGHER (N-TH) ORDER 432 9.4.6
CALCULATION OF N-ROW DETERMINANTS 433 9.4.7 REGULAR AND INVERSE MATRIX
434 9.4.8 CALCULATION OF THE INVERSE MATRIX IN TERMS OF DETERMINANTS . .
435 9.4.9 RANK OF A MATRIX 436 9.4.10 DETERMINATION OF THE RANK BY MEANS
OF MINOR DETERMINANTS . 437 9.5 SYSTEMS OF LINEAR EQUATIONS 437 9.5.1
SYSTEMS OF TWO EQUATIONS WITH TWO UNKNOWNS 439 9.6 NUMERICAL SOLUTION
METHODS 441 9.6.1 GAUSSIAN ALGORITHM FOR SYSTEMS OF LINEAR EQUATIONS 441
9.6.2 FORWARD ELIMINATION 441 9.6.3 PIVOTING 443 9.6.4 BACKSUBSTITUTION
444 9.6.5 LU-DECOMPOSITION 445 9.6.6 SOLVABILITY OF (M X N) SYSTEMS OF
EQUATIONS 448 9.6.7 GAUSS-JORDAN METHOD FOR MATRIX INVERSION 450 9.6.8
CALCULATION OF THE INVERSE MATRIX A" 1 452 9.7 ITERATIVE SOLUTION OF
SYSTEMS OF LINEAR EQUATIONS 454 9.7.1 TOTAL-STEP METHODS (JACOBI) 456
9.7.2 SINGLE-STEP METHODS (GAUSS-SEIDEL) 456 9.7.3 CRITERIA OF
CONVERGENCE FOR ITERATIVE METHODS 457 9.7.4 STORAGE OF THE COEFFICIENT
MATRIX 458 9.8 TABLE OF SOLUTION METHODS 459 9.9 EIGENVALUE EQUATIONS
461 9.10 TENSORS 463 9.10.1 ALGEBRAIC OPERATIONS WITH TENSORS 465 10
BOOLEAN ALGEBRA-APPLICATION IN SWITCHING ALGEBRA 467 10.1 BASIC NOTIONS
467 10.1.1 PROPOSITIONS AND TRUTH VALUES 467 10.1.2 PROPOSITION
VARIABLES ' 468 10.2 BOOLEAN CONNECTIVES 468 10.2.1 NEGATION: NOT 469
10.2.2 CONJUNCTION: AND 469 10.2.3 DISJUNCTION (INCLUSIVE): OR 469
10.2.4 CALCULATING RULES 470 10.3 BOOLEAN FUNCTIONS 471 10.3.1 OPERATOR
BASIS 472 10.4 NORMAL FORMS 472 10.4.1 DISJUNCTIVE NORMAL FORMS 472
10.4.2 CONJUNCTIVE NORMAL FORM 473 10.4.3 REPRESENTATION OF FUNCTIONS BY
NORMAL FORMS 473 10.5 KARNAUGH-VEITCH DIAGRAMS 475 10.5.1 PRODUCING A
KV-DIAGRAM 476 XVIII CONTENTS 10.5.2 ENTERING A FUNCTION IN A KV-DIAGRAM
476 10.5.3 MINIMIZATION WITH THE HELP OF KV-DIAGRAMS 477 10.6
MINIMIZATION ACCORDING TO QUINE AND MCCLUSKEY 478 10.7 MULTI-VALUED
LOGIC AND FUZZY LOGIC 481 10.7.1 MULTI-VALUED LOGIC 481 - 10.7.2 FUZZY
LOGIC 481 11 GRAPHS AND ALGORITHMS 483 11.1 GRAPHS 483 11.1.1 BASIC
DEFINITIONS 483 11.1.2 REPRESENTATION OF GRAPHS 485 11.1.3 TREES 485
11.2 MATCHINGS 486 11.3 NETWORKS 487 11.3.1 FLOWS IN NETWORKS 487 11.3.2
EULERIAN LINE AND HAMILTONIAN CIRCUIT 487 12 DIFFERENTIAL CALCULUS 489
12.1 DERIVATIVE OF A FUNCTION 489 12.1.1 DIFFERENTIAL 490 12.1.2
DIFFERENTIABILITY 491 12.2 DIFFERENTIATION RULES 492 12.2.1 DERIVATIVES
OF ELEMENTARY FUNCTIONS 492 12.2.2 DERIVATIVES OF TRIGONOMETRIC
FUNCTIONS 492 12.2.3 DERIVATIVES OF HYPERBOLIC FUNCTIONS 492 12.2.4
CONSTANT RULE 493 12.2.5 FACTOR RULE 493 12.2.6 POWER RULE 493 12.2.7
SUM RULE 493 12.2.8 PRODUCT RULE 493 12.2.9 QUOTIENT RULE 494 12.2.10
CHAIN RULE 494 12.2.11 LOGARITHMIC DIFFERENTIATION OF FUNCTIONS 495
12.2.12 DIFFERENTIATION OF FUNCTIONS IN PARAMETRIC REPRESENTATION . . .
495 12.2.13 DIFFERENTIATION OF FUNCTIONS IN POLAR COORDINATES 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 THEOREMS 499 12.3.1 ROLLE'S
THEOREM 499 12.3.2 MEAN VALUE THEOREM OF DIFFERENTIAL CALCULUS 499
12.3.3 EXTENDED MEAN VALUE THEOREM OF DIFFERENTIAL CALCULUS . . . . 500
12.4 HIGHER DERIVATIVES 500 12.4.1 SLOPE, EXTREMES 502 12.4.2 CURVATURE
503 12.4.3 POINT OF INFLECTION 503 12.5 APPROXIMATION METHOD OF
DIFFERENTIATION 504 12.5.1 GRAPHICAL DIFFERENTIATION 504 12.5.2
NUMERICAL DIFFERENTIATION 505 12.6 DIFFERENTIATION OF FUNCTIONS WITH
SEVERAL VARIABLES 506 12.6.1 PARTIAL DERIVATIVE 506 CONTENTS XIX 12.6.2
TOTAL DIFFERENTIAL 508 12.6.3 EXTREMES OF FUNCTIONS IN TWO DIMENSIONS
508 12.6.4 EXTREMES WITH CONSTRAINTS 509 12.7 APPLICATION OF
DIFFERENTIAL CALCULUS 510 12.7.1 CALCULATION OF INDEFINITE EXPRESSIONS
510 12.7.2 DISCUSSION OF CURVES 511 12.7.3 EXTREME VALUE PROBLEMS 512
12.7.4 CALCULUS OF ERRORS 513 12.7.5 DETERMINATION OF ZEROS ACCORDING TO
NEWTON'S METHOD . 514 13 DIFFERENTIAL GEOMETRY 517 13.1 PLANE CURVES
517 13.1.1 REPRESENTATION OF CURVES 517 13.1.2 DIFFERENTIATION BY
IMPLICIT REPRESENTATION 517 13.1.3 DIFFERENTIATION BY PARAMETRIC
REPRESENTATION 518 13.1.4 DIFFERENTIATION BY POLAR COORDINATES 518
13.1.5 DIFFERENTIAL OF ARC OF A CURVE 518 13.1.6 TANGENT, NORMAL 519
13.1.7 CURVATURE OF A CURVE 520 13.1.8 EVOLUTES AND EVOLVENTS 522 13.1.9
POINTS OF INFLECTION, VERTICES 522 13.1.10 SINGULAR POINTS 522 13.1.11
ASYMPTOTES 523 13.1.12 ENVELOPE OF A FAMILY OF CURVES 524 13.2 SPACE
CURVES 524 13.2.1 REPRESENTATION OF SPACE CURVES 524 13.2.2 MOVING
TRIHEDRAL 525 13.2.3 CURVATURE 527 13.2.4 TORSION OF A CURVE 527 13.2.5
FRENET FORMULAS 528 13.3 SURFACES 528 13.3.1 REPRESENTATION OF A SURFACE
528 13.3.2 TANGENT PLANE AND NORMAL TO THE SURFACE 529 13.3.3 SINGULAR
POINTS OF THE SURFACE 530 14 INFINITE SERIES 531 14.1 SERIES 531 14.2
CRITERIA OF CONVERGENCE 532 14.2.1 SPECIAL NUMBER SERIES 535 14.3 TAYLOR
AND MACLAURIN SERIES 535 14.3.1 TAYLOR'S FORMULA 535 14.3.2 TAYLOR
SERIES 536 14.4 POWER SERIES 537 14.4.1 TEST OF CONVERGENCE FOR POWER
SERIES 537 14.4.2 PROPERTIES OF CONVERGENT POWER SERIES 538 14.4.3
INVERSION OF POWER SERIES 540 14.5 SPECIAL EXPANSIONS OF SERIES AND
PRODUCTS 540 14.5.1 BINOMIAL SERIES 540 14.5.2 SPECIAL BINOMIAL SERIES
540 14.5.3 SERIES OF EXPONENTIAL FUNCTIONS 541 14.5.4 SERIES OF
LOGARITHMIC FUNCTIONS 542 XX CONTENTS 14.5.5 SERIES OF TRIGONOMETRIC
FUNCTIONS 542 14.5.6 SERIES OF INVERSE TRIGONOMETRIC FUNCTIONS 543
14.5.7 SERIES OF HYPERBOLIC FUNCTIONS 544 14.5.8 SERIES OF AREA
HYPERBOLIC FUNCTIONS 544 14.5.9 PARTIAL FRACTION EXPANSIONS 544 14.5.10
INFINITE PRODUCTS 545 15 INTEGRAL CALCULUS 547 15.1 DEFINITION AND
INTEGRABILITY 547 15.1.1 PRIMITIVE 547 15.1.2 DEFINITE AND INDEFINITE
INTEGRALS 548 15.1.3 GEOMETRICAL INTERPRETATION 549 15.1.4 RULES FOR
INTEGRABILITY 550 15.1.5 IMPROPER INTEGRALS 551 15.2' INTEGRATION RULES
552 15.2.1 RULES FOR INDEFINITE INTEGRALS 552 15.2.2 RULES FOR DEFINITE
INTEGRALS 553 15.2.3 TABLE OF INTEGRATION RULES 554 15.2.4 INTEGRALS OF
SOME ELEMENTARY FUNCTIONS 555 15.3 INTEGRATION METHODS 557 15.3.1
INTEGRATION BY SUBSTITUTION 557 15.3.2 INTEGRATION BY PARTS 560 15.3.3
INTEGRATION BY PARTIAL FRACTION DECOMPOSITION 562 15.3.4 INTEGRATION BY
SERIES EXPANSION 565 15.4 NUMERICAL INTEGRATION 567 15.4.1 RECTANGULAR
RULE 567 15.4.2 TRAPEZOIDAL RULE 568 15.4.3 SIMPSON'S RULE 568 15.4.4
ROMBERG INTEGRATION 569 15.4.5 GAUSSIAN QUADRATURE 570 15.4.6 TABLE OF
NUMERICAL INTEGRATION METHODS 572 15.5 MEAN VALUE THEOREM OF INTEGRAL
CALCULUS 574 15.6 LINE, SURFACE, AND VOLUME INTEGRALS 574 15.6.1 ARC
LENGTH (RECTIFICATION) 574 15.6.2 AREA 575 15.6.3 SOLID OF ROTATION
(SOLID OF REVOLUTION) 576 15.7 FUNCTIONS IN PARAMETRIC REPRESENTATION
577 15.7.1 ARC LENGTH IN PARAMETRIC REPRESENTATION 577 15.7.2 SECTOR
FORMULA 578 15.7.3 SOLIDS OF ROTATION IN PARAMETRIC REPRESENTATION 578
15.8 MULTIPLE INTEGRALS AND THEIR APPLICATIONS 579 15.8.1 DEFINITION OF
MULTIPLE INTEGRALS 579 15.8.2 CALCULATION OF AREAS 580 15.8.3 CENTER OF
MASS OF ARCS 581 15.8.4 MOMENT OF INERTIA OF AN AREA 581 15.8.5 CENTER
OF MASS OF AREAS 582 15.8.6 MOMENT OF INERTIA OF PLANES 582 15.8.7
CENTER OF MASS OF A BODY 582 O 15.8.8 MOMENT OF INERTIA OF A BODY 583
15.8.9 CENTER OF MASS OF ROTATIONAL SOLIDS 583 15.8.10 MOMENT OF INERTIA
OF ROTATIONAL SOLIDS 583 CONTENTS XXI 15.9 TECHNICAL APPLICATIONS OF
INTEGRAL CALCULUS 584 15.9.1 STATIC MOMENT, CENTER OF MASS 584 15.9.2
MASS MOMENT OF INERTIA 585 15.9.3 STATICS 588 15.9.4 CALCULATION OF WORK
588 15.9.5 MEAN VALUES 589 16 VECTOR ANALYSIS 591 16.1 FIELDS 591 16.1.1
SYMMETRIES OF FIELDS 592 16.2 DIFFERENTIATION AND INTEGRATION OF VECTORS
594 16.2.1 SCALE FACTORS IN GENERAL ORTHOGONAL COORDINATES 596 16.2.2
DIFFERENTIAL OPERATORS 597 16.3 GRADIENT AND POTENTIAL 598 16.4
DIRECTIONAL DERIVATIVE AND VECTOR GRADIENT 600 16.5 DIVERGENCE AND
GAUSSIAN INTEGRAL THEOREM 601 16.6 ROTATION AND STOKES'S THEOREM 604
16.7 LAPLACE OPERATOR AND GREEN'S FORMULAS 607 16.7.1 COMBINATIONS OF
DIV, ROT, AND GRAD; CALCULATION OF FIELDS . . . 609 16.8 SUMMARY 610 17
COMPLEX VARIABLES AND FUNCTIONS 613 17.1 COMPLEX NUMBERS 613 17.1.1
IMAGINARY NUMBERS 613 17.1.2 ALGEBRAIC REPRESENTATION OF COMPLEX NUMBERS
614 17.1.3 CARTESIAN REPRESENTATION OF COMPLEX NUMBERS 614 17.1.4
CONJUGATE COMPLEX NUMBERS 615 17.1.5 ABSOLUTE VALUE OF A COMPLEX NUMBER
615 17.1.6 TRIGONOMETRIC REPRESENTATION OF COMPLEX NUMBERS 616 17.1.7
EXPONENTIAL REPRESENTATION OF COMPLEX NUMBERS 616 17.1.8 TRANSFORMATION
FROM CARTESIAN TO TRIGONOMETRIC REPRESENTATION 617 17.1.9 RIEMANN SPHERE
618 17.2 ELEMENTARY ARITHMETICAL OPERATIONS WITH COMPLEX NUMBERS 619
17.2.1 ADDITION AND SUBTRACTION OF COMPLEX NUMBERS 619 17.2.2
MULTIPLICATION AND DIVISION OF COMPLEX NUMBERS 619 17.2.3 EXPONENTIATION
IN THE COMPLEX DOMAIN 622 17.2.4 TAKING THE ROOT IN THE COMPLEX DOMAIN
623 17.3 ELEMENTARY FUNCTIONS OF A COMPLEX VARIABLE 623 17.3.1 SEQUENCES
IN THE COMPLEX DOMAIN 624 17.3.2 SERIES IN THE COMPLEX DOMAIN 625 17.3.3
EXPONENTIAL FUNCTION IN THE COMPLEX DOMAIN 626 17.3.4 NATURAL LOGARITHM
IN THE COMPLEX DOMAIN 626 17.3.5 GENERAL POWER IN THE COMPLEX DOMAIN 627
17.3.6 TRIGONOMETRIC FUNCTIONS IN THE COMPLEX DOMAIN 627 17.3.7
HYPERBOLIC FUNCTIONS IN THE COMPLEX DOMAIN 629 17.3.8 INVERSE
TRIGONOMETRIC, INVERSE HYPERBOLIC FUNCTIONS IN THE COMPLEX DOMAIN 630
17.4 APPLICATIONS OF COMPLEX FUNCTIONS 631 17.4.1 REPRESENTATION OF
OSCILLATIONS IN THE COMPLEX PLANE 631 17.4.2 SUPERPOSITION OF
OSCILLATIONS OF EQUAL FREQUENCY 632 XXII CONTENTS 17.4.3 LOCI 633 17.4.4
INVERSION OF LOCI 634 17.5 DIFFERENTIATION OF FUNCTIONS OF A COMPLEX
VARIABLE 635 17.5.1 DEFINITION OF THE DERIVATIVE IN THE COMPLEX DOMAIN
635 17.5.2 DIFFERENTIATION RALES IN THE COMPLEX DOMAIN 636 17.5.3
CAUCHY-RIEMANN DIFFERENTIABILITY CONDITIONS 637 17.5.4 CONFORMAL MAPPING
637 17.6 INTEGRATION IN THE COMPLEX PLANE 639 17.6.1 COMPLEX CURVILINEAR
INTEGRALS 639 17.6.2 CAUCHY'S INTEGRAL THEOREM 640 17.6.3 PRIMITIVE
FUNCTIONS IN THE COMPLEX DOMAIN 641 17.6.4 CAUCHY'S INTEGRAL FORMULAS
641 17.6.5 TAYLOR SERIES OF AN ANALYTIC FUNCTION 642 17.6.6 LAURENT
SERIES 643 17.6.7 CLASSIFICATION OF SINGULAR POINTS 643 17.6.8 RESIDUE
THEOREM 644 17.6.9 INVERSE LAPLACE TRANSFORMATION 645 18 DIFFERENTIAL
EQUATIONS 647 18.1 GENERAL DEFINITIONS 647 18.2 GEOMETRIC INTERPRETATION
649 18.3 SOLUTION METHODS FOR FIRST-ORDER DIFFERENTIAL EQUATIONS 650
18.3.1 SEPARATION OF VARIABLES 650 18.3.2 SUBSTITUTION 651 18.3.3 EXACT
DIFFERENTIAL EQUATIONS 651 18.3.4 INTEGRATING FACTOR 651 18.4 LINEAR
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 652 18.4.1 VARIATION OF THE
CONSTANTS 652 18.4.2 GENERAL SOLUTION 653 18.4.3 DETERMINATION OF A
PARTICULAR SOLUTION 653 18.4.4 LINEAR DIFFERENTIAL EQUATIONS OF THE
FIRST ORDER WITH CONSTANT COEFFICIENTS 653 18.5 SOME SPECIFIC EQUATIONS
654 18.5.1 BERNOULLI DIFFERENTIAL EQUATION 654 18.5.2 RICCATI
DIFFERENTIAL EQUATION 654 18.6 DIFFERENTIAL EQUATIONS OF THE SECOND
ORDER 655 18.6.1 SIMPLE SPECIAL CASES 655 18.7 LINEAR DIFFERENTIAL
EQUATIONS OF THE SECOND ORDER 656 18.7.1 HOMOGENEOUS LINEAR DIFFERENTIAL
EQUATION OF THE SECOND ORDER 657 18.7.2 INHOMOGENEOUS LINEAR
DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 657 18.7.3 REDUCTION OF
SPECIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER TO DIFFERENTIAL
EQUATIONS OF THE FIRST ORDER 659 18.7.4 LINEAR DIFFERENTIAL EQUATIONS OF
THE SECOND ORDER WITH CONSTANT COEFFICIENTS 659 18.8 DIFFERENTIAL
EQUATIONS OFTHEW-TH ORDER 662 18.9 SYSTEMS OF COUPLED DIFFERENTIAL
EQUATIONS OF THE FIRST ORDER 668 18.10 SYSTEMS OF LINEAR HOMOGENEOUS
DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 670 18.11 PARTIAL
DIFFERENTIAL EQUATIONS 672 CONTENTS XXIII 18.11.1 SOLUTION BY SEPARATION
673 18.12 NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS 676 18.12.1
EULER METHOD 676 18.12.2 HEUN METHOD 677 18.12.3 MODIFIED EULER METHOD
679 18.12.4~ RUNGE-KUTTA METHODS 679 18.12.5 RUNGE-KUTTA METHOD FOR
SYSTEMS OF DIFFERENTIAL EQUATIONS . . 685 18.12.6 DIFFERENCE METHOD FOR
THE SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 685 18.12.7 METHOD OF
FINITE ELEMENTS 688 19 FOURIER TRANSFORMATION 691 19.1 FOURIER SERIES
691 19.1.1 INTRODUCTION 691 19.1.2 DEFINITION AND COEFFICIENTS 691
19.1.3 CONDITION OF CONVERGENCE 693 19.1.4 EXTENDED INTERVAL 694 19.1.5
SYMMETRIES 696 19.1.6 FOURIER SERIES IN COMPLEX AND SPECTRAL
REPRESENTATION . . . . 698 19.1.7 FORMULAS FOR THE CALCULATION OF
FOURIER SERIES 699 19.1.8 FOURIER EXPANSION OF SIMPLE PERIODIC FUNCTIONS
699 19.1.9 FOURIER SERIES (TABLE) 705 19.2 FOURIER INTEGRALS 707 19.2.1
INTRODUCTION 707 19.2.2 DEFINITION AND COEFFICIENTS 707 19.2.3
CONDITIONS FOR CONVERGENCE 708 19.2.4 COMPLEX REPRESENTATION, FOURIER
SINE AND COSINE TRANSFORMATION 708 19.2.5 SYMMETRIES 710 19.2.6
CONVOLUTION AND SOME CALCULATING RALES 710 19.3 DISCRETE FOURIER
TRANSFORMATION (DFT) 712 19.3.1 DEFINITION AND COEFFICIENTS 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 TRANSFORMS 720 19.3.6 FOURIER TRANSFORMS
(TABLE) 720 19.3.7 PARTICULAR FOURIER SINE TRANSFORMS 722 19.3.8
PARTICULAR FOURIER COSINE TRANSFORMS 723 19.4 WAVELET TRANSFORMATION 724
19.4.1 SIGNALS 724 19.4.2 LINEAR SIGNAL ANALYSIS 725 19.4.3 SYMMETRY
TRANSFORMATIONS 726 19.4.4 TIME-FREQUENCY ANALYSIS AND GABOR
TRANSFORMATION 727 19.4.5 WAVELET TRANSFORMATION 728 19.4.6 DISCRETE
WAVELET-TRANSFORMATION 732 20 LAPLACE AND Z TRANSFORMATIONS 735 20.1
INTRODUCTION 735 20.2 DEFINITION OF THE LAPLACE TRANSFORMATION 736 20.3
TRANSFORMATION THEOREMS 737 XXIV CONTENTS 20.4 PARTIAL FRACTION
SEPARATION 745 20.4.1 PARTIAL FRACTION SEPARATION WITH SIMPLE REAL ZEROS
745 20.4.2 PARTIAL FRACTION DECOMPOSITION WITH MULTIPLE REAL ZEROS . . .
746 20.4.3 PARTIAL FRACTION DECOMPOSITION WITH COMPLEX ZEROS 747 20.5
LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 748 20.5.1
LAPLACE TRANSFORMATION: LINEAR DIFFERENTIAL EQUATION OF THE FIRST ORDER
WITH CONSTANT COEFFICIENTS 749 20.5.2 LAPLACE TRANSFORMATION: LINEAR
DIFFERENTIAL EQUATION OF THE SECOND ORDER WITH CONSTANT COEFFICIENTS 751
20.5.3 EXAMPLE: LINEAR DIFFERENTIAL EQUATIONS 753 20.5.4 LAPLACE
TRANSFORMS (TABLE) 756 20.6 Z TRANSFORMATION 764 20.6.1 DEFINITION OF
THE Z TRANSFORMATION 764 20.6.2 CONVERGENCE CONDITIONS FOR THE Z
TRANSFORMATION 766 20.6.3 INVERSION OF THE Z TRANSFORMATION 767 20.6.4
CALCULATING RALES 767 20.6.5 CALCULATING RALES FOR THE Z TRANSFORMATION
770 20.6.6 TABLE OF Z TRANSFORMS 770 21 PROBABILITY THEORY AND
MATHEMATICAL STATISTICS 773 21.1 COMBINATORICS 773 21.2 RANDOM EVENTS
774 21.2.1 BASIC NOTIONS 774 21.2.2 EVENT 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 METHODS TO
CALCULATE PROBABILITIES 778 21.3.3 CONDITIONAL PROBABILITIES 779 21.3.4
CALCULATING WITH PROBABILITIES 779 21.4 RANDOM VARIABLES AND THEIR
DISTRIBUTIONS 781 21.4.1 INDIVIDUAL PROBABILITY, DENSITY FUNCTION AND
DISTRIBUTION FUNCTION X 782 21.4.2 PARAMETERS OF DISTRIBUTIONS 783
21.4.3 SPECIAL DISCRETE DISTRIBUTION 785 21.4.4 SPECIAL CONTINUOUS
DISTRIBUTIONS 793 21.5 LIMIT THEOREMS 800 21.5.1 LAWS OF LARGE NUMBERS
800 21.5.2 LIMIT THEOREMS 801 21.6 MULTIDIMENSIONAL RANDOM VARIABLES 802
21.6.1 DISTRIBUTION FUNCTIONS OF TWO-DIMENSIONAL RANDOM VARIABLES . 802
21.6.2 TWO-DIMENSIONAL DISCRETE RANDOM VARIABLES 803 21.6.3
TWO-DIMENSIONAL CONTINUOUS RANDOM VARIABLES 804 21.6.4 INDEPENDENCE OF
RANDOM VARIABLES 805 21.6.5 PARAMETERS OF TWO-DIMENSIONAL RANDOM
VARIABLES 806 21.6.6 TWO-DIMENSIONAL NORMAL DISTRIBUTION 807 21.7 BASICS
OF MATHEMATICAL STATISTICS 808. 21.7.1 DESCRIPTION OF MEASUREMENTS . .
809 21.7.2 TYPES OF ERROR 810 21.8 PARAMETERS FOR DESCRIBING
DISTRIBUTIONS OF MEASURED VALUES 812 21.8.1 POSITION PARAMETER, MEANS OF
SERIES OF MEASUREMENTS . . . . 812 CONTENTS XXV 21.8.2 DISPERSION
PARAMETER 814 21.9 SPECIAL DISTRIBUTIONS 815 21.9.1 FREQUENCY
DISTRIBUTIONS 815 21.9.2 DISTRIBUTION OF RANDOM SAMPLE FUNCTIONS 816
21.10 ANALYSIS BY MEANS OF RANDOM SAMPLING (THEORY OF TESTING AND
ESTIMATING) 820 21.10.1 ESTIMATION METHODS 821 21.10.2 CONSTRUCTION
PRINCIPLES FOR ESTIMATORS 823 21.10.3 METHOD OF MOMENTS 823 21.10.4
MAXIMUM LIKELIHOOD METHOD 824 21.10.5 METHOD OF LEAST SQUARES 824
21.10.6 X 2 -MINIMUM METHOD 825 21.10.7 METHOD OF QUANTILES, PERCENTILES
825 21.10.8 INTERVAL ESTIMATION 826 21.10.9 INTERVAL BOUNDS FOR NORMAL
DISTRIBUTION 828 21.10.10 PREDICTION AND CONFIDENCE INTERVAL BOUNDS FOR
BINOMIAL AND HYPERGEOMETRIC DISTRIBUTIONS 829 21.10.11 INTERVAL BOUNDS
FOR A POISSON DISTRIBUTION 830 21.10.12 DETERMINATION OF SAMPLE SIZES N
830 21.10.13 TEST METHODS 831 21.10.14 PARAMETER TESTS 834 21.10.15
PARAMETER TESTS FOR A NORMAL DISTRIBUTION 834 21.10.16 HYPOTHESES ABOUT
THE MEAN VALUE OF ARBITRARY DISTRIBUTIONS 836 21.10.17 HYPOTHESES ABOUT
P OF BINOMIAL AND HYPERGEOMETRIC DISTRIBUTIONS 837 21.10.18 TESTS OF
GOODNESS OF FIT 837 21.10.19 APPLICATION: ACCEPTANCE/REJECTION TEST 838
21.11 RELIABILITY 839 21.12 COMPUTATION OF ADJUSTMENT, REGRESSION 841
21.12.1 LINEAR REGRESSION, LEAST SQUARES METHOD 843 21.12.2 REGRESSION
OF THE N-TH ORDER 844 22 FUZZY LOGIC 847 22.1 FUZZY SETS 847 22.2 FUZZY
CONCEPT 848 22.3 FUNCTIONAL GRAPHS FOR THE MODELING OF FUZZY SETS 849
22.4 COMBINATION OF FUZZY SETS 852 22.4.1 ELEMENTARY OPERATIONS 852
22.4.2 CALCULATING RALES FOR FUZZY SETS 855 22.4.3 RULES 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 CONORMS) 858 22.4.6 PARAMETRIZED T AND
S NORMS 859 22.4.7 COMPENSATORY OPERATORS 860 22.5 FUZZY RELATIONS 861
22.6 FUZZY INFERENCE 863 22.7 DENAZIFICATION METHODS 864 22.8 EXAMPLE:
ERECT PENDULUM 866 22.9 FUZZY REALIZATIONS 870 XXVI CONTENTS 23 NEURAL
NETWORKS 871 23.1 FUNCTION AND STRUCTURE 871 23.1.1 FUNCTION 871 23.1.2
STRUCTURE 872 23.2 IMPLEMENTATION OF THE NEURON MODEL 873 ~ 23.2.1
TIME-INDEPENDENT SYSTEMS 873 23.2.2 TIME-DEPENDENT SYSTEMS 873 23.2.3
APPLICATION 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 METHODS 878
23.3.5 HOPFIELD NETWORK 879 23.4 UNSUPERVISED LEARNING 881 23.4.1
PRINCIPLE OF UNSUPERVISED LEARNING 881 23.4.2 KOHONEN MODEL 881 24
COMPUTERS 883 24.1 OPERATING SYSTEMS 883 24.1.1 INTRODUCTION TO MS-DOS
885 24.1.2 INTRODUCTION TO UNIX 886 24.2 HIGH-LEVEL PROGRAMMING
LANGUAGES 889 24.2.1 PROGRAM STRUCTURES 890 24.2.2 OBJECT-ORIENTED
PROGRAMMING (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 NUMBERS
. . . 895 24.4.3 BOOLEAN VALUES 895 24.4.4 ARRAYS \ 895 24.4.5
CHARACTERS AND CHARACTER STRINGS 896 24.4.6 RECORD ' 897 24.4.7 POINTERS
898 24.4.8 SELF-DEFINED TYPES 899 24.5 STATEMENTS 900 24.5.1 ASSIGNMENTS
AND EXPRESSIONS 900 24.5.2 INPUT AND OUTPUT 901 24.5.3 COMPOUND
STATEMENTS 902 24.5.4 CONDITIONAL STATEMENTS IF AND CASE 903 24.5.5
LOOPS FOR, WHILE, AND REPEAT 904 24.6 PROCEDURES 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 DYNAMIC DATA STRUCTURES 909 CONTENTS XXVII 24.8.2
SEARCH 910 24.8.3 SORTING 911 24.9 COMPUTER GRAPHICS 913 24.9.1 BASIC
FUNCTIONS 913 INTRODUCTION TO C 914 24.9.2 BASIC STRUCTURES 914 24.9.3
OPERATORS 916 24.9.4 DATA STRUCTURES 918 24.9.5 LOOPS AND BRANCHES 921
INTRODUCTION TO C++ 924 24.9.6 VARIABLES AND CONSTANTS 924 24.9.7
OVERLOADING OF FUNCTIONS 924 24.9.8 OVERLOADING OF OPERATORS 924 24.9.9
CLASSES 925 24.9.10 INSTANTIATION OF CLASSES 926 24.9.11 FRIEND
FUNCTIONS 926 24.9.12 OPERATORS AS MEMBER FUNCTIONS 926 24.9.13
CONSTRUCTORS 927 24.9.14 DERIVED CLASSES (INHERITANCE) 928 24.9.15 CLASS
LIBRARIES 929 INTRODUCTION TO FORTRAN 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 SUBPROGRAMS 934
COMPUTER ALGEBRA 937 24.9.22 STRUCTURAL ELEMENTS OF MATHEMATICA 937
24.9.23 STRUCTURAL ELEMENTS OF MAPLE 940 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
PROGRAMMING 947 24.9.29 FITTING CURVES AND INTERPOLATION WITH
MATHEMATICA 948 24.9.30 GRAPHICS 949 25 TABLES OF INTEGRALS 951 25.1
INTEGRALS OF RATIONAL FUNCTIONS 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 . .
952 25.1.3 INTEGRALS WITH 1/(X N (AX + B) M ), P = AX + B B ^ 0 . . .
953 25.1.4 INTEGRALS WITH AX + B AND EX + D C^0 955 XXVIII CONTENTS
25.1.5 INTEGRALS WITH A + X AND B + X A = B 955 25.1.6 INTEGRALS WITH P
= AX 2 + BX + C (A / 0) 956 25.1.7 INTEGRALS WITH X N /(AX 2 + BX + C) M
, P = AX 2 + BX+C A / 0 956 25.1.8 INTEGRALS WITH L/(X N (AX 2 + BX + C)
M ), P = AX 2 + BX + C C/0 957 25.1.9 INTEGRALS WITH P = A 2 X 2 958
25.1.10 INTEGRALS WITH L/(A 2 X 2 )", P = A 2 X 2 A / 0 . . . . 9 5
8 25.1.11 INTEGRALS WITH X" / {A 2 X 2 ) M , P = A 2 X 2 A / 0 .
. . 9 5 8 25.1.12 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 WITH A 4 + X A (A 0) 962' 25.1.15 INTEGRALS WITH A 4 - X 4
(A 0) 962 25.2 INTEGRALS OF IRRATIONAL FUNCTIONS 963 25.2.1 INTEGRALS
WITH X 1/2 AND P = AX + B A,B^0 963 25.2.2 INTEGRALS 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 INTEGRALS WITH R = (A 2 + X 2 ) 1 ' 2 A
/ 0 966 25.2.5 INTEGRALS WITH S = (X 2 - A 2 ) Y ' 2 A # 0 968 25.2.6
INTEGRALS WITH T = {A 2 - X 2 ) X ' 2 A / 0 970 25.2.7 INTEGRALS WITH
(AX 2 + BX + C) L/2 X = AX 2 + BX + C A / 0 972 25.3 INTEGRALS OF
TRANSCENDENTAL FUNCTIONS 973 25.3.1 INTEGRALS WITH EXPONENTIAL FUNCTIONS
973 25.3.2 INTEGRALS WITH LOGARITHMIC FUNCTIONS (X 0) 975 25.3.3
INTEGRALS WITH HYPERBOLIC 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 TANGENT OR COTANGENT FUNCTIONS (A / 0)
. 989 25.3.8 INTEGRALS WITH INVERSE TRIGONOMETRIC FUNCTIONS (A / 0) .
. . 990 25.4 DEFINITE INTEGRALS 992 25.4.1 DEFINITE INTEGRALS WITH
ALGEBRAIC FUNCTIONS 992 25.4.2 DEFINITE INTEGRALS WITH EXPONENTIAL
FUNCTIONS 992 25.4.3 DEFINITE INTEGRALS WITH LOGARITHMIC FUNCTIONS 994
25.4.4 DEFINITE INTEGRALS WITH TRIGONOMETRIC FUNCTIONS 995 INDEX 999 |
| any_adam_object | 1 |
| author | Harris, John W. Stöcker, Horst 1952- |
| author_GND | (DE-588)131640909 |
| author_facet | Harris, John W. Stöcker, Horst 1952- |
| author_role | aut aut |
| author_sort | Harris, John W. |
| author_variant | j w h jw jwh h s hs |
| building | Verbundindex |
| bvnumber | BV011570063 |
| callnumber-first | Q - Science |
| callnumber-label | QA40 |
| callnumber-raw | QA40.H38 1998 |
| callnumber-search | QA40.H38 1998 |
| callnumber-sort | QA 240 H38 41998 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SH 300 SC 464 SH 500 SK 110 |
| classification_tum | MAT 001b |
| ctrlnum | (OCoLC)247166082 (DE-599)BVBBV011570063 |
| dewey-full | 510 51021 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics |
| dewey-raw | 510 510 21 |
| dewey-search | 510 510 21 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| genre_facet | Formelsammlung |
| id | DE-604.BV011570063 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T08:46:40Z |
| institution | BVB |
| isbn | 0387947469 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007790071 |
| oclc_num | 247166082 |
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| physical | XXVIII, 1028 S. Ill., graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Springer |
| record_format | marc |
| spelling | Harris, John W. Verfasser aut Handbook of mathematics and computational science John W. Harris ; Horst Stocker New York [u.a.] Springer 1998 XXVIII, 1028 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Informatik Mathematik Mathematics Handbooks, manuals, etc Computer science Handbooks, manuals, etc Informatik (DE-588)4026894-9 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf (DE-588)4155008-0 Formelsammlung gnd-content Mathematik (DE-588)4037944-9 s DE-604 Informatik (DE-588)4026894-9 s Stöcker, Horst 1952- Verfasser (DE-588)131640909 aut GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007790071&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Harris, John W. Stöcker, Horst 1952- Handbook of mathematics and computational science Informatik Mathematik Mathematics Handbooks, manuals, etc Computer science Handbooks, manuals, etc Informatik (DE-588)4026894-9 gnd Mathematik (DE-588)4037944-9 gnd |
| subject_GND | (DE-588)4026894-9 (DE-588)4037944-9 (DE-588)4155008-0 |
| title | Handbook of mathematics and computational science |
| title_auth | Handbook of mathematics and computational science |
| title_exact_search | Handbook of mathematics and computational science |
| title_full | Handbook of mathematics and computational science John W. Harris ; Horst Stocker |
| title_fullStr | Handbook of mathematics and computational science John W. Harris ; Horst Stocker |
| title_full_unstemmed | Handbook of mathematics and computational science John W. Harris ; Horst Stocker |
| title_short | Handbook of mathematics and computational science |
| title_sort | handbook of mathematics and computational science |
| topic | Informatik Mathematik Mathematics Handbooks, manuals, etc Computer science Handbooks, manuals, etc Informatik (DE-588)4026894-9 gnd Mathematik (DE-588)4037944-9 gnd |
| topic_facet | Informatik Mathematik Mathematics Handbooks, manuals, etc Computer science Handbooks, manuals, etc Formelsammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007790071&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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