INDEX

Abel, Niels Henrik, 79n.6

Abel's formula, 79

Absolute convergence (series):

defined, 674

and uniform convergence, 704

Absolute frequency (probability):

of an event, 1019

cumulative, 1012

of a value, 1012

Absolutely integrable nonperiodic function, 512–513

Absolute value (complex numbers), 613

Acceleration, 386–389

Acceleration of gravity, 8

Acceleration vector, 386

Acceptable lots, 1094

Acceptable quality level (AQL), 1094

Acceptance:

of a hypothesis, 1078

of products, 1092

Acceptance number, 1092

Acceptance sampling, 1092–1096, 1113

errors in, 1093–1094

rectification, 1094–1095

Adams, John Couch, 912n.2

Adams–Bashforth methods, 911–914, 947

Adams–Moulton methods, 913–914, 947

Adaptive integration, 835–836, 843

Addition:

for arbitrary events, 1021–1022

of complex numbers, 609, 610

of matrices and vectors, 126, 259–261

of means, 1057–1058

for mutually exclusive events, 1021

of power series, 687

termwise, 173, 687

of variances, 1058–1059

vector, 309, 357–359

ADI (alternating direction implicit) method, 928–930

Adjacency matrix:

of a digraph, 973

of a graph, 972–973

Adjacent vertices, 971, 977

Airy, Sir George Bidell, 556n.2, 918n.4

Airy equation, 556

RK method, 917–919

RKN method, 919–920

Airy function:

RK method, 917–919

RKN method, 919–920

Algebraic equations, 798

Algebraic multiplicity, 326, 878

Algorithms:

complexity of, 978–979

defined, 796

numeric analysis, 796

numeric methods as, 788

numeric stability of, 796, 842

ALGORITHMS:

BISECT, A46

DIJKSTRA, 982

EULER, 903

FORD–FULKERSON, 998

GAUSS, 849

GAUSS–SEIDEL, 860

INTERPOL, 814

KRUSKAL, 985

MATCHING, 1003

MOORE, 977

NEWTON, 802

PRIM, 989

RUNGE–KUTTA, 905

SIMPSON, 832

Aliasing, 531

Alternating direction implicit (ADI) method, 928–930

Alternating path, 1002

Alternative hypothesis, 1078

Ampère, André Marie, 93n.7

Amplification, 91

Amplitude, 90

Amplitude spectrum, 511

Analytic functions, 172, 201, 641

complex analysis, 623–624

conformal mapping, 737–742

derivatives of, 664–668, 688–689, A95–A96

integration of:

indefinite, 647

by use of path, 647–650

Laurent series:

analytics at infinity, 718–719

zeros of, 717–718

maximum modulus theorem, 782–783

mean value property, 781–782

power series representation of, 688–689

real functions vs., 694

Analyticity, 623

Angle of intersection:

conformal mapping, 738

between two curves, 36

Angular speed (rotation), 372

Angular velocity (fluid flow), 775

AOQ (average outgoing quality), 1095

AOQL (average outgoing quality limit), 1095

Apparent resistance (RLC circuits), 95

Approximation (s):

errors involved in, 794

polynomial, 808

by trigonometric polynomials, 495–498

Approximation theory, 495

A priori estimates, 805

AQL (acceptable quality level), 1094

Arbitrary positive, 191

Arc, of a curve, 383

Archimedes, 391n.4

Arc length (curves), 385–386

Area:

of a region, 428

of region bounded by ellipses, 436

of a surface, 448–450

Argand, Jean Robert, 611n.2

Argand diagram, 611n.2

Argument (complex numbers), 613

Artificial variables, 965–968

Assignment problems (combinatorial optimization), 1001–1006

Associative law, 264

Asymptotically equal, 189, 1027, 1050

Asymptotically normal, 1076

Asymptotically stable critical points, 149

Augmented matrices, 258, 272, 273, 321, 845, 959

Augmenting path, 1002–1003. See also Flow augmenting paths

Autonomous ODEs, 11, 33

Autonomous systems, 152, 165

Auxiliary equation, 54. See also Characteristic equation

Average flow, 458

Average outgoing quality (AOQ), 1095

Average outgoing quality limit (AOQL), 1095

Axioms of probability, 1020

Back substitution (linear systems), 274–276, 846

Backward edges:

cut sets, 994

initial flow, 998

of a path, 992

Backward Euler formula, 909

Backward Euler method (BEM):

first-order ODEs, 909–910

stiff systems, 920–921

Backward Euler scheme, 909

Balance law, 14

Band matrices, 928

Bashforth, Francis, 912n.2

Basic feasible solution:

normal form of linear optimization problems, 957

simplex method, 959

Basic Rule (method of undetermined coefficients):

higher-order homogeneous linear ODEs, 115

second-order nonhomogeneous linear ODEs, 81, 82

Basic variables, 960

Basis:

eigenvectors, 339–340

of solutions:

higher-order linear ODEs, 106, 113, 123

homogeneous linear systems, 290

homogeneous ODEs, 50–52, 75, 104, 106, 113

second-order homogeneous linear ODEs, 50–52, 75, 104

systems of ODEs, 139

standard, 314

vector spaces, 286, 311, 314

Beats (oscillation), 89

Bellman, Richard, 981n.3

Bellman equations, 981

Bellman's principle, 980–981

Bell-shaped curve, 13, 574

BEM, see Backward Euler method

Benoulli, Niklaus, 31n.7

Bernoulli, Daniel, 31n.7

Bernoulli, Jakob, 31n.7

Bernoulli, Johann, 31n.7

Bernoulli distribution, 1040. See also Binomial distributions

Bernoulli equation, 45

defined, 31

linear ODEs, 31–33

Bernoulli's law of large numbers, 1051

Bessel, Friedrich Wilhelm, 187n.6

Bessel functions, 167, 187–191, 202

of the first kind, 189–190

with half-integer v, 193–194

of order 1, 189

of order v, 191

orthogonality of, 506

of the second kind:

general solution, 196–200

of order v, 198–200

table, A97–A98

of the third kind, 200

Bessel's equation, 167, 187–196, 202

Bessel functions, 167, 187–191, 196–200

circular membrane, 587

general solution, 194–200

Bessel's inequality:

for Fourier coefficients, 497

orthogonal series, 508–509

Beta function, formula for, A67

Bezier curve, 827

BFS algorithms, see Breadth First search algorithms

Bijective mapping, 737n.1

Binomial coefficients:

Newton's forward difference formula, 816

probability theory, 1027–1028

Binomial distributions, 1039–1041, 1061

normal approximation of, 1049–1050

sampling with replacement for, 1042

table, A99

Binomial series, 696

Binomial theorem, 1029

Bipartite graphs, 1001–1006, 1008

BISECT, ALGORITHM, A46

Bisection method, 807–808

Bolzano, Bernard, A94n.3

Bolzano–Weierstrass theorem, A94–A95

Bonnet, Ossian, 180n.3

Bonnet's recursion, 180

Borda, J. C., 16n.4

Boundaries:

ODEs, 39

of regions, 426n.2

sets in complex plane, 620

Boundary conditions:

one-dimensional heat equation, 559

PDEs, 541, 605

periodic, 501

two-dimensional wave equation, 577

vibrating string, 545–547

Boundary points, 426n.2

Boundary value problem (BVP), 499

conformal mapping for, 763–767, A96

first, see Dirichlet problem

mixed, see Mixed boundary value problem

second, see Neumann problem

third, see Mixed boundary value problem

two-dimensional heat equation, 564

Bounded domains, 652

Bounded regions, 426n.2

Bounded sequence, A93–A95

Boxplots, 1013

Boyle, Robert, 19n.5

Boyle–Mariotte's law for idea gases, 19

Bragg, Sir William Henry, 938n.5

Bragg, Sir William Lawrence, 938n.5

Branch, of logarithm, 639

Branch cut, of logarithm, 639

Branch point (Riemann surfaces), 755

Breadth First search (BFS) algorithms, 977

defined, 977, 998

Moore's, 977–980

BVP, see Boundary value problem

CAD (computer-aided design), 820

Cancellation laws, 306–307

Canonical form, 344

Cantor, Georg, A72n.3

Cantor–Dedekind axiom, A72n.3, A95n.4

Capacity:

cut sets, 994

networks, 991

Cardano, Girolamo, 608n.1

Cardioid, 391, 437

Cartesian coordinates:

linear element in, A75

transformation law, A86–A87

vector product in, A83–A84

writing, A74

Cartesian coordinate systems:

complex plane, 611

left-handed, 369, 370, A84

right-handed, 368–369, A83–A84

in space, 315, 356

transformation law for vector components, A85–A86

Cartesius, Renatus, 356n.1

Cauchy, Augustin-Louis, 71n.4, 625n.4, 683n.1

Cauchy determinant, 113

Cauchy–Goursat theorem, see Cauchy's integral theorem

Cauchy–Hadamard formula, 683

Cauchy principal value, 727, 730

Cauchy–Riemann equations, 38, 642

complex analysis, 623–629

proof of, A90–A91

Cauchy–Schwarz inequality, 363, 871–782

Cauchy's convergence principle, 674–675, A93–A94

Cauchy's inequality, 666

Cauchy's integral formula, 660–663, 670

Cauchy's integral theorem, 652–660, 669

existence of indefinite integral, 656–658

Goursat's proof of, A91–A93

independence of path, 655

for multiply connected domains, 658–659

principle of deformation of path, 656

Cayley, Arthur, 748n.2

c-charts, 1092

Center:

as critical point, 144, 165

of a graph, 991

of power series, 680

Center control line (CL), 1088

Center of gravity, of mass in a region, 429

Central difference notation, 819

Central limit theorem, 1076

Central vertex, 991

Centrifugal force, 388

Centripetal acceleration, 387–388

Chain rules, 392–394

Characteristics, 555

Characteristics, method of, 555

Characteristic determinant, of a

matrix, 129, 325, 326, 353, 877

Characteristic equation:

matrices, 129, 325, 326, 353, 877

PDEs, 555

second-order homogeneous linear ODEs, 54

Characteristic matrix, 326

Characteristic polynomial, 325, 353, 877

Characteristic values, 87, 324, 353. See also Eigenvalues

Characteristic vectors, 324, 877. See also eigenvectors

Chebyshev, Pafnuti, 504n.6

Chebyshev equation, 504

Chebyshev polynomials, 504

Checkerboard pattern (determinants), 294

Chi-square (x²) distribution, 1074–1076, A104

Chi-square (x²) test, 1096–1097, 1113

Choice of numeric method, for matrix eigenvalue problems, 879

Cholesky, André-Louis, 855n.3

Cholesky's method, 855–856, 898

Chopping, error caused by, 792

Chromatic number, 1006

Circle, 386

Circle of convergence (power series), 682

Circulation, of flow, 467, 774

CL (center control line), 1088

Clairaut equation, 35

Clamped condition (spline interpolation), 823

Class intervals, 1012

Class marks, 1012

Closed annulus, 619

Closed circular disk, 619

Closed integration formulas, 833, 838

Closed intervals, A72n.3

Closed Newton–Cotes formulas, 833

Closed paths, 414, 645, 975–976

Closed regions, 426n.2

Closed sets, 620

Closed trails, 975–976

Closed walks, 975–976

CN (Crank–Nicolson) method, 938–941

Coefficients:

binomial:

Newton's forward difference formula, 816

probability theory, 1027–1028

constant:

higher-order homogeneous linear ODEs, 111–116

second-order homogeneous linear ODEs, 53–60

second-order nonhomogeneous linear ODEs, 81

systems of ODEs, 140–151

correlation, 1108–1111, 1113

Fourier, 476, 484, 538, 582–583

of kinetic friction, 19

of linear systems, 272, 845

of ODEs, 47

higher-order homogeneous linear ODEs, 105

second-order homogeneous linear ODEs, 53–60, 73

second-order nonhomogeneous linear ODEs, 81–85

series of ODEs, 168, 174

variable, 167, 240–241

of power series, 680

regression, 1105, 1107–1108

variable:

Frobenius method, 180–187

Laplace transforms ODEs with, 240–241

of ODEs, 167, 240–241

power series method, 167–175

second-order homogeneous linear ODEs, 73

Coefficient matrices, 257, 273

Hermitian or skew-Hermitian forms, 351

linear systems, 845

quadratic form, 343

Cofactor (determinants), 294

Collatz, Lothar, 883n.9

Collatz inclusion theorem, 883–884

Columns:

determinants, 294

matrix, 125, 257, 320

Column “sum” norm, 861

Column vectors, 126

matrices, 257, 284–285, 320

rank in terms of, 284–285

Combinations (probability theory), 1024, 1026–1027

of n things taken k at a time without repetitions, 1026

of n things taken k at a time with repetitions, 1026

Combinatorial optimization, 970, 975–1008

assignment problems, 1001–1006

flow problems in networks, 991–997

cut sets, 994–996

flow augmenting paths, 992–993

paths, 992

Ford–Fulkerson algorithm for maximum flow, 998–1001

shortest path problems, 975–980

Bellman's principle, 980–981

complexity of algorithms, 978–980

Dijkstra's algorithm, 981–983

Moore's BFS algorithm, 977–980

shortest spanning trees:

Greedy algorithm, 984–988

Prim's algorithm, 988–991

Commutation (matrices), 271

Complements:

of events, 1016

of sets in complex plane, 620

Complementation rule, 1020–1021

Complete bipartite graphs, 1005

Complete graphs, 974

Complete matching, 1002

Completeness (orthogonal series), 508–509

Complete orthonormal set, 508

Complex analysis, 607

analytic functions, 623–624

Cauchy–Riemann equations, 623–629

circles and disks, 619

complex functions, 620–623

exponential, 630–633

general powers, 639–640

hyperbolic, 635

logarithm, 636–639

trigonometric, 633–635

complex integration, 643–670

Cauchy's integral formula, 660–663, 670

Cauchy's integral theorem, 652–660, 669

derivatives of analytic

functions, 664–668

Laurent series, 708–719

line integrals, 643–652, 669

power series, 671–707

residue integration, 719–733

complex numbers, 608–619

addition of, 609, 610

conjugate, 612

defined, 608

division of, 610

multiplication of, 609, 610

polar form of, 613–618

subtraction of, 610

complex plane, 611

conformal mapping, 736–757

geometry of analytic functions, 737–742

linear fractional transformations, 742–750

Riemann surfaces, 754–756

by trigonometric and hyperbolic analytic functions, 750–754

half-planes, 619–620

harmonic functions, 628–629

Laplace's equation, 628–629

Laurent series, 708–719, 734

analytic or singular at infinity, 718–719

point at infinity, 718

Riemann sphere, 718

singularities, 715–717

zeros of analytic functions, 717

power series, 168, 671–707

convergence behavior of, 680–682

convergence tests, 674–676, A93–A94

functions given by, 685–690

Maclaurin series, 690

in powers of x, 168

radius of convergence, 682–684

ratio test, 676–678

root test, 678–679

sequences, 671–673

series, 673–674

Taylor series, 690–697

uniform convergence, 698–705

residue integration, 719–733

formulas for residues, 721–722

of real integrals, 725–733

several singularities inside contour, 723–725

Taylor series, 690–697, 707

Complex conjugate numbers, 612

Complex conjugate roots, 72–73

Complex Fourier integral, 523

Complex functions, 620–623

exponential, 630–633

general powers, 639–640

hyperbolic, 635

logarithm, 636–639

trigonometric, 633–635

Complex heat potential, 767

Complex integration, 643–670

Cauchy's integral formula, 660–663, 670

Cauchy's integral theorem, 652–660, 669

existence of indefinite integral, 656–658

independence of path, 655

for multiply connected domains, 658–659

principle of deformation of path, 656

derivatives of analytic functions, 664–668

Laurent series, 708–719

analytic or singular at infinity, 718–719

point at infinity, 718

Riemann sphere, 718

singularities, 715–717

zeros of analytic functions, 717–718

line integrals, 643–652, 669

basic properties of, 645

bounds for, 650–651

definition of, 643–645

existence of, 646

indefinite integration and substitution of limits, 646–647

representation of a path, 647–650

power series, 671–707

convergence behavior of, 680–682

convergence tests, 674–676

functions given by, 685–690

Maclaurin series, 690

radius of convergence of, 682–684

ratio test, 676–678

root test, 678–679

sequences, 671–673

series, 673–674

Taylor series, 690–697

uniform convergence, 698–705

residue integration, 719–733

formulas for residues, 721–722

of real integrals, 725–733

several singularities inside contour, 723–725

Complexity, of algorithms, 978–979

Complex line integrals, see Line integrals

Complex matrices and forms, 346–352

Complex numbers, 608–619, 641

addition of, 609, 610

conjugate, 612

defined, 608

division of, 610

multiplication of, 609, 610

polar form of, 613–618

subtraction of, 610

Complex plane, 611

extended, 718, 744–745

sets in, 620

Complex potential, 786

electrostatic fields, 760–761

of fluid flow, 771, 773–774

Complex roots:

higher-order homogeneous linear ODEs:

multiple, 115

simple, 113–114

second-order homogeneous linear ODEs, 57–59

Complex trigonometric polynomials, 529

Complex variables, 620–621

Complex vector space, 309, 310, 349

Components (vectors), 126, 356, 365

Composition, of linear transformations, 316–317

Computer-aided design (CAD), 820

Condition:

of incompressibility, 405

spline interpolation, 823

Conditionally convergent series, 675

Conditional probability, 1022–1023, 1061

Condition number, 868–870, 899

Confidence intervals, 1063, 1068–1077, 1113

interval estimates, 1065

for mean of normal distribution:

with known variance, 1069–1071

with unknown variance, 1071–1073

for parameters of distributions other than normal, 1076

in regression analysis, 1107–1108

for variance of a normal distribution, 1073–1076

Confidence level, 1068

Conformality, 738

Conformal mapping, 736–757

boundary value problems, 763–767, A96

defined, 738

geometry of analytic functions, 737–742

linear fractional transformations, 742–750

extended complex plane, 744–745

mapping standard domains, 747–750

Riemann surfaces, 754–756

by trigonometric and hyperbolic analytic functions, 750–754

Connected graphs, 977, 981, 984

Connected set, in complex plane, 620

Conservative physical systems, 422

Conservative vector fields, 400, 408

Consistent linear systems, 277

Constant coefficients:

higher-order homogeneous linear ODEs, 111–116

distinct real roots, 112–113

multiple real roots, 114–115

simple complex roots, 113–114

second-order homogeneous linear ODEs, 53–60

complex roots, 57–59

real double root, 55–56

two distinct real roots, 54–55

second-order nonhomogeneous linear ODEs, 81

systems of ODEs, 140–151

critical points, 142–146, 148–151

graphing solutions in phase plane, 141–142

Constant of gravity, at the Earth's surface, 63

Constant of integration, 18

Constant revenue, lines of, 954

Constrained (linear) optimization, 951, 954–958, 969

normal form of problems, 955–957

simplex method, 958–968

degenerate feasible solution, 962–965

difficulties in starting, 965–968

Constraints, 951

Consumers, 1092

Consumer's risk, 1094

Consumption matrix, 334

Continuity equation (compressible fluid flow), 405

Continuous complex functions, 621

Continuous distributions, 1029, 1032–1034

marginal distribution of, 1055

two-dimensional, 1053

Continuous random variables, 1029, 1032–1034, 1061

Continuous vector functions, 378–379

Contour integral, 653

Contour lines, 21, 36

Control charts, 1088

for mean, 1088–1089

for range, 1090–1091

for standard deviation, 1090

for variance, 1089–1090

Controlled variables, in regression analysis, 1103

Control limits, 1088, 1089

Control variables, 951

Convergence:

absolute:

defined, 674

and uniform convergence, 704

of approximate and exact solutions, 936

circle of, 682

defined, 861

Gauss–Seidel iteration, 861–862

mean square (orthogonal series), 507–508

in the norm, 507

power series, 680–682

convergence tests, 674–676, A93–A94

radius of convergence of, 682–684, 706

uniform convergence, 698–705

radius of, 172

defined, 172

power series, 682–684, 706

sequence of vectors, 378

speed of (numeric analysis), 804–805

superlinear, 806

uniform:

and absolute convergence, 704

power series, 698–705

Convergence interval, 171, 683

Convergence tests, 674–676

power series, 674–676, A93–A94

uniform convergence, 698–705

Convergent iteration processes, 800

Convergent sequence of functions, 507–508, 672

Convergent series, 171, 673

Convolution:

defined, 232

Fourier transforms, 527–528

Laplace transforms, 232–237

Convolution theorem, 232–233

Coriolis, Gustave Gaspard, 389n.3

Coriolis acceleration, 388–389

Corrector (improved Euler method), 903

Correlation analysis, 1063, 1108–1111, 1113

defined, 1103

test for correlation coefficient, 1110–1111

Correlation coefficient, 1108–1111, 1113

Cosecant, formula for, A65

Cosine function:

conformal mapping by, 752

formula for, A63–A65

Cosine integral:

formula for, A69

table, A98

Cosine series, 781

Cotangent, formula for, A65

Coulomb, Charles Augustin de, 19n.6, 93n.7, 401n.6

Coulomb's law, 19, 401

Covariance:

in correlation analysis, 1109

defined, 1058

Cramer, Gabriel, 31n.7, 298n.2

Cramer's rule, 292, 298–300, 321

for three equations, 293

for two equations, 292

Cramer's Theorem, 298

Crank, John, 938n.5

Crank–Nicolson (CN) method, 938–941

Critical damping, 65, 66

Critical points, 33, 165

asymptotically stable, 149

and conformal mapping, 738, 757

constant-coefficient systems of ODEs, 142–146

center, 144

criteria for, 148–151

degenerate node, 145–146

improper node, 142

proper node, 143

saddle point, 143

spiral point, 144–145

stability of, 149–151

isolated, 152

nonlinear systems, 152

stable, 140, 149

stable and attractive, 140, 149

unstable, 140, 149

Critical region, 1079

Cross product, 368, 410. See also Vector product

Crout, Prescott Durand, 853n.2

Crout's method, 853, 898

Cubic spline, 821

Cumulative absolute frequencies (of values), 1012

Cumulative distribution functions, 1029

Cumulative relative frequencies (of values), 1012

Curl, A76

invariance of, A85–A88

of vector fields, 406–409, 412

Curvature, of a curve, 389–390

Curves:

arc of, 383

bell-shaped, 13, 574

Bezier, 827

deflection, 120

elastic, 120

equipotential, 36, 759, 761

one-parameter family of, 36–37

operating characteristic, 1081, 1092, 1095

oriented, 644

orthogonal coordinate, A74

parameter, 442

plane, 383

regression, 1103

simple, 383

simple closed, 646

smooth, 414, 644

solution, 4–6

twisted, 383

vector differential calculus, 381–392, 411

arc length of, 385–386

length of, 385

in mechanics, 386–389

tangents to, 384–385

and torsion, 389–390

Curve fitting, 872–876

method of least squares, 872–874

by polynomials of degree m, 874–875

Curvilinear coordinates, 354, 412, A74

Cut sets, 994–996, 1008

Cycle (paths), 976, 984

Cylindrical coordinates, 593–594, A74–A76

D'Alembert, Jean le Rond, 554n.1

D'Alembert's solution, 553–556

Damped oscillations, 67

Damping constant, 65

Dantzig, George Bernard, 959

Data processing:

frequency distributions, 1011–1012

and randomness, 1064

Data representation:

frequency distributions, 1011–1015

Empirical Rule, 1014

graphic, 1012

mean, 1013–1014

standard deviation, 1014

variation, 1014

and randomness, 1064

Decisions:

false, risks of making, 1080

statistics for, 1077–1078

Dedekind, Richard, A72n.3

Defect (eigenvalue), 328

Defectives, 1092

Definite integrals, complex, see Line integrals

Deflection curve, 120

Deformation of path, principle of, 656

Degenerate feasible solution (simplex method), 962–965

Degenerate node, 145–146

Degrees of freedom (d.f.), number of, 1071, 1074

Degree of incidence, 971

Degree of precision (DP), 833

Deleted neighborhood, 720

Demand vector, 334

De Moivre, Abraham, 616n.3

De Moivre–Laplace limit theorem, 1050

De Moivre's formula, 616

De Morgan's laws, 1018

Density, 1061

continuous two-dimensional distributions, 1053

of a distribution, 1033

Dependent random variables, 1055, 1056

Dependent variables, 393, 1055, 1056

Depth First Search (DFS) algorithms, 977

Derivatives:

of analytic functions, 664–668, 688–689, A95–A96

of complex functions, 622, 641

Laplace transforms of, 211–212

of matrices or vectors, 127

of vector functions, 379–380

Derived series, 687

Descartes, René, 356n.1, 391n.4

Determinants, 293–301, 321

Cauchy, 113

Cramer's rule, 298–300

defined, A81

general properties of, 295–298

of a matrix, 128

of matrix products, 307–308

of order n, 293

proof of, A81–A83

second-order, 291–292

second-order homogeneous linear ODEs, 76

third-order, 292–293

Vandermonde, 113

Wronski:

second-order homogeneous linear ODEs, 75–78

systems of ODEs, 139

Developed, in a power series, 683

D.f. (degrees of freedom), number of, 1071, 1074

DFS (Depth First Search) algorithms, 977

DFTs (discrete Fourier transforms), 528–531

Diagonalization of matrices, 341–342

Diagonally dominant matrices, 881

Diagonal matrices, 268

inverse of, 305–306

scalar, 268

Diameter (graphs), 991

Difference:

complex numbers, 610

scalar multiplication, 260

Difference equations (elliptic PDEs), 923–925

Difference quotients, 923

Difference table, 814

Differentiable complex functions, 622–623

Differentiable vector functions, 379

Differential (total differential), 20, 45

Differential equations:

applications of, 3

defined, 2

Differential form, 422

exact, 21, 470

first fundamental form, of S, 451

floating-point, of numbers, 791–792

path independence and exactness of, 422, 470

Differential geometry, 381

Differential operators:

second-order, 60

for second-order homogeneous linear ODEs, 60–62

Differentiation:

of Laplace transforms, 238–240

matrices or vectors, 127

numeric, 838–839

of power series, 687–688, 703

termwise, 173, 687–688, 703

Diffusion equation, 459–460, 558. See also Heat equation

Digraphs (directed graphs), 971–972, 1007

computer representation of, 972–974

defined, 972

incidence matrix of, 975

subgraphs, 972

Dijkstra, Edsger Wybe, 981n.4

Dijkstra's algorithm, 981–983, 1008

DIJKSTRA, ALGORITHM, 982

Dimension of vector spaces, 286, 311, 359

Diocles, 391n.4

Dirac, Paul, 226n.2

Dirac delta function, 226–228, 237

Directed graphs, see Digraphs (directed graphs)

Directed path, 1000

Directional derivatives (scalar functions), 396–397, 411

Direction field (slope field), 9–10, 44

Direct methods (linear system solutions), 858, 898. See also iteration

Dirichlet, Peter Gustav LeJeune, 462n.8

Dirichlet boundary condition, 564

Dirichlet problem, 605, 923

ADI method, 929

heat equation, 564–566

Laplace equation, 593–596, 925–928, 934–935

Poisson equation, 925–928

two-dimensional heat equation, 564–565

uniqueness theorem for, 462, 784

Dirichlet's discontinuous factor, 514

Discharge (flow modeling), 776

Discrete distributions, 1029–1032

marginal distributions of, 1053–1054

two-dimensional, 1052–1053

Discrete Fourier transforms (DFTs), 528–531

Discrete random variables, 1029, 1030–1032, 1061

defined, 1030

marginal distributions of, 1054

Discrete spectrum, 525

Disjoint events, 1016

Disks:

circular, open and closed, 619

mapping, 748–750

Poisson's integral formula, 779–780

Dissipative physical systems, 422

Distance:

graphs, 991

vector norms, 866

Distinct real roots:

higher-order homogeneous linear ODEs, 112–113

second-order homogeneous linear ODEs, 54–55

Distinct roots (Frobenius method), 182

Distributions, 226n.2. See also

Frequency distributions;

Probability distributions

Distribution-free tests, 1100

Distribution function, 1029–1032

cumulative, 1029

normal distributions, 1046–1047

of random variables, 1056, A109

sample, 1096

two-dimensional probability distributions, 1051–1052

Distributive laws, 264

Distributivity, 363

Divergence, A75

fluid flow, 775

of vector fields, 402–406

of vector functions, 411, 453

Divergence theorem of Gauss, 405, 470

applications, 458–463

vector integral calculus, 453–457

Divergent sequence, 672

Divergent series, 171, 673

Division, of complex numbers, 610, 615–616

Domain (s), 393

bounded, 652

doubly connected, 658, 659

of f, 620

holes of, 653

mapping, 737, 747–750

multiply connected:

Cauchy's integral formula, 662–663

Cauchy's integral theorem, 658–659

p-fold connected, 652–653

sets in complex plane, 620

simply connected, 423, 646, 652, 653

triply connected, 653, 658, 659

Dominant eigenvalue, 883

Doolittle, Myrick H., 853n.1

Doolittle's method, 853–855, 898

Dot product, 312, 410. See also Inner product

Double Fourier series:

defined, 582

rectangular membrane, 577–585

Double integrals (vector integral calculus), 426–432, 470

applications of, 428–429

change of variables in, 429–431

evaluation of, by two successive integrations, 427–428

Double precision, floating-point standard for, 792

Double root (Frobenius method), 183

Double subscript notation, 125

Doubly connected domains, 658, 659

DP (degree of precision), 833

Driving force, see Input (driving force)

Duffing equation, 160

Duhamel, Jean-Marie Constant, 603n.4

Duhamel's formula, 603

Eccentricity, of vertices, 991

Edges:

backward:

cut sets, 994

initial flow, 998

of a path, 992

forward:

cut sets, 994

initial flow, 998

of a path, 992

graphs, 971, 1007

incident, 971

Edge chromatic number, 1006

Edge condition, 991

Edge incidence list (graphs), 973

Efficient algorithms, 979

Eigenbases, 339–341

Eigenfunctions, 605

circular membrane, 588

one-dimensional heat equation, 560

Sturm–Liouville Problems, 499–500

two-dimensional heat equation, 565

two-dimensional wave equation, 578, 580

vibrating string, 547

Eigenfunction expansion, 504

Eigenspaces, 326, 878

Eigenvalues, 129–130, 166, 353, 605, 877, 899. See also Matrix eigenvalue problems

circular membrane, 588

complex matrices, 347–351

and critical points, 149

defined, 324

determining, 323–329

dominant, 883

finding, 324–328

one-dimensional heat equation, 560

Sturm–Liouville Problems, 499–500, A89

two-dimensional wave equation, 580

vibrating string, 547

Eigenvalues of A, 322

Eigenvalue problem, 140

Eigenvectors, 129–130, 166, 353, 877, 899

basis of, 339–340

convergent sequence of, 886

defined, 324

determining, 323–329

finding, 324–328

Eigenvectors of A, 322

EISPACK, 789

Elastic curve, 120

Electric circuits:

analogy of electrical and mechanical quantities, 97–98

second-order nonhomogeneous linear ODEs, 93–99

Electrostatic fields (potential theory), 759–763

complex potential, 760–761

superposition, 761–762

Electrostatic potential, 759

Electrostatics (Laplace's equation), 593

Elementary matrix, 281

Elementary row operations (linear systems), 277

Ellipses, area of region bounded by, 436

Elliptic PDEs:

defined, 923

numeric analysis, 922–936

ADI method, 928–930

difference equations, 923–925

Dirichlet problem, 925–928

irregular boundary, 933–935

mixed boundary value problems, 931–933

Neumann problem, 931

Empirical Rule, 1014

Energies, 157

Entire function, 630, 642, 707, 718

Entries:

determinants, 294

matrix, 125, 257

Equal complex numbers, 609

Equality:

of matrices, 126, 259

of vectors, 355

Equally likely events, 1018

Equal spacing (interpolation):

Newton's backward difference formula, 818–819

Newton's forward difference formula, 815–818

Equilibrium harvest, 36

Equilibrium solutions (equilibrium points), 33–34

Equipotential curves, 36, 759, 761

Equipotential lines, 38

electrostatic fields, 759, 761

fluid flow, 771

Equipotential surfaces, 759

Equivalent vector norms, 871

Error (s):

in acceptance sampling, 1093–1094

of approximations, 495

in numeric analysis, 842

basic error principle, 796

error propagation, 795

errors of numeric results, 794–795

roundoff, 792

in statistical tests, 1080–1081

and step size control, 906–907

trapezoidal rule, 830

vector norms, 866

Error bounds, 795

Error estimate, 908

Error function, 828, A67–A68, A98

Essential singularity, 715–716

Estimation of parameters, 1063

EULER, ALGORITHM, 903

Euler, Leonhard, 31n.7, 71n.4

Euler–Cauchy equations, 71–74, 104

higher-order nonhomogeneous linear ODEs, 119–120

Laplace's equation, 595

third-order, IVP for, 108

Euler–Cauchy method, 901

Euler constant, 198

Euler formulas, 58

complex Fourier integral, 523

derivation of, 479–480

exponential function, 631

Fourier coefficients given by, 476, 484

generalized, 582

Taylor series, 694

trigonometric function, 634

Euler graph, 980

Euler's method:

defined, 10

error of, 901–902, 906, 908

first-order ODEs, 10–11, 901–902

backward method, 909–910

improved method, 902–904

higher order ODEs, 916–917

Euler trail, 980

Even functions, 486–488

Even periodic extension, 488–490

Events (probability theory), 1016–1017, 1060

addition rule for, 1021–1022

arbitrary, 1021–1022

complements of, 1016

defined, 1015

disjoint, 1016

equally likely, 1018

independent, 1022–1023

intersection, 1016, 1017

mutually exclusive, 1016, 1021

simple, 1015

union, 1016–1017

Exact differential equation, 21

Exact differential form, 422, 470

Exact ODEs, 20–27, 45

defined, 21

integrating factors, 23–26

Existence, problem of, 39

Existence theorems:

cubic splines, 822

first-order ODEs, 39–42

homogeneous linear ODEs:

higher-order, 108

second-order, 74

of the inverse, 301–302

Laplace transforms, 209–210

linear systems, 138

power series solutions, 172

systems of ODEs, 137

Expectation, 1035, 1037–1038, 1057

Experiments:

defined, 1015, 1060

in probability theory, 1015–1016

random, 1011, 1015–1016, 1060

Experimental error, 794

Explicit formulas, 913

Explicit method:

heat equation, 937, 940–941

wave equation, 943

Explicit solution, 21

Exponential decay, 5, 7

Exponential function, 630–633, 642

formula for, A63

Taylor series, 694

Exponential growth, 5

Exponential integral, formula for, A69

Exposed vertices, 1001, 1003

Extended complex plane:

conformal mapping, 744–745

defined, 718

Extended method (separable ODEs), 17–18

Extended problems, 966

Extrapolation, 808

Extrema (unconstrained optimization), 951

Factorial function, 1027, A66, A98. See also Gamma functions

Failing to reject a hypothesis, 1081

Fair die, 1018, 1019

False decisions, risks of making, 1080

False position, method of, 807–808

Family of curves, one-parameter, 36–37

Family of solutions, 5

Faraday, Michael, 93n.7

Fast Fourier transforms (FFTs), 531–532

F-distribution, 1086, A105–A108

Feasibility region, 954

Feasible solutions, 954–955

basic, 957, 959

degenerate, 962–965

normal form of linear optimization problems, 957

Fehlberg, E., 907

Fehlberg's fifth-order RK method, 907–908

Fehlberg's fourth-order RK method, 907–908

FFTs (fast Fourier transforms), 531–532

Fibonacci (Leonardo of Pisa), 690n.2

Fibonacci numbers, 690

Fibonacci's rabbit problem, 690

Finite complex plane, 718. See also Complex plane

Finite jumps, 209

First boundary value problem, see Dirichlet problem

First fundamental form, of S, 451

First-order method, Euler method as, 902

First-order ODEs, 2–45, 44

defined, 4

direction fields, 9–10

Euler's method, 10–11

exact, 20–27, 45

defined, 21

integrating factors, 23–26

explicit form, 4

geometric meanings of, 9–12

implicit form, 4

initial value problem, 38–43

linear, 27–36

Bernoulli equation, 31–33

homogeneous, 28

nonhomogeneous, 28–29

population dynamics, 33–34

modeling, 2–8

numeric analysis, 901–915

Adams–Bashforth methods, 911–914

Adams–Moulton methods, 913–914

backward Euler method, 909–910

Euler's method, 901–902

improved Euler's method, 902–904

multistep methods, 911–915

Runge–Kutta–Fehlberg method, 906–908

Runge–Kutta methods, 904–906

orthogonal trajectories, 36–38

separable, 12–20, 44

extended method, 17–18

modeling, 13–17

systems of, 165

transformation of systems to, 157–159

First (first order) partial derivatives, A71

First shifting theorem (s-shifting), 208–209

First transmission line equation, 599

Fisher, Sir Ronald Aylmer, 1086

Fixed points:

defined, 799

of a mapping, 745

Fixed-point iteration (numeric analysis), 798–801, 842

Fixed-point systems, numbers in, 791

Floating, 793

Floating-point form of numbers, 791–792

Flow augmenting paths, 992–993, 998, 1008

Flow problems in networks (combinatorial optimization), 991–997

cut sets, 994–996

flow augmenting paths, 992–993

paths, 992

Fluid flow:

Laplace's equation, 593

potential theory, 771–777

Fluid state, 404

Flux (motion of a fluid), 404

Flux integral, 444, 450

Forced motions, 68, 86

Forced oscillations:

Fourier analysis, 492–495

second-order nonhomogeneous linear ODEs, 85–92

damped, 89–90

resonance, 88–91

undamped, 87–89

Forcing function, 86

Ford, Lester Randolph, Jr., 998n.7

FORD–FULKERSON, ALGORITHM, 998

Ford–Fulkerson algorithm for maximum flow, 998–1001, 1008

Forest (graph), 987

Form (s):

canonical, 344

complex, 351

differential, 422

exact, 21, 470

path independence and exactness of, 422

Hesse's normal, 366

Lagrange's, 812

normal (linear optimization problems), 955–957, 959, 969

Pfaffian, 422

polar, of complex numbers, 613–618, 631

quadratic, 343–344, 346

reduced echelon, 279

row echelon, 279–280

skew-Hermitian and Hermitian, 351

standard:

first-order ODEs, 27

higher-order homogeneous linear ODEs, 105

higher-order linear ODEs, 123

power series method, 172

second-order linear ODEs, 46, 103

triangular (Gauss elimination), 846

Forward edge:

cut sets, 994

initial flow, 998

of a path, 992

Four-color theorem, 1006

Fourier, Jean-Baptiste Joseph, 473n.1

Fourier analysis, 473–539

approximation by trigonometric polynomials, 495–498

forced oscillations, 492–495

Fourier integral, 510–517

applications, 513–515

complex form of, 522–523

sine and cosine, 515–516

Fourier series, 474–483

convergence and sum of, 480–481

derivation of Euler formulas, 479–480

even and odd functions, 486–488

half-range expansions, 488–490

from period 2 to 2L, 483–486

Fourier transforms, 522–536

complex form of Fourier integral, 522–523

convolution, 527–528

cosine, 518–522, 534

discrete, 528–531

fast, 531–532

and its inverse, 523–524

linearity, 526–527

sine, 518–522, 535

spectrum representation, 525

orthogonal series (generalized Fourier series), 504–510

completeness, 508–509

mean square convergence, 507–508

Sturm–Liouville Problems, 498–504

eigenvalues, eigenfunctions, 499–500

orthogonal functions, 500–503

Fourier–Bessel series, 506–507, 589

Fourier coefficients, 476, 484, 538, 582–583

Fourier constants, 504–505

Fourier cosine integral, 515–516

Fourier cosine series, 484, 486, 538

Fourier cosine transforms, 518–522, 534

Fourier cosine transform method, 518

Fourier integrals, 510–517, 539

applications, 513–515

complex form of, 522–523

heat equation, 568–571

residue integration, 729–730

sine and cosine, 515–516

Fourier–Legendre series, 505–506, 596–598

Fourier matrix, 530

Fourier series, 473–483, 538

convergence and sum or, 480–481

derivation of Euler formulas, 479–480

double, 577–585

even and odd functions, 486–488

half-range expansions, 488–490

heat equation, 558–563

from period 2 to 2L, 483–486

Fourier sine integral, 515–516

Fourier sine series, 477, 486, 538

one-dimensional heat equation, 561

vibrating string, 548

Fourier sine transforms, 518–522, 535

Fourier transforms, 522–536, 539

complex form of Fourier integral, 522–523

convolution, 527–528

cosine, 518–522, 534, 539

defined, 522, 523

discrete, 528–531

fast, 531–532

heat equation, 571–574

and its inverse, 523–524

linearity of, 526–527

sine, 518–522, 535, 539

spectrum representation, 525

Fourier transform method, 524

Four-point formulas, 841

Fraction defective chars, 1091–1092

Francis, J. G. F., 892

Fredholm, Erik Ivar, 198n.7, 263n.3

Free condition (spline interpolation), 823

Free oscillations of mass–spring system (second-order ODEs), 62–70

critical damping, 65, 66

damped system, 64–65

overdamping, 65–66

undamped system, 63–64

underdamping, 65, 67

Frenet, Jean-Frédéric, 392

Frenet formulas, 392

Frequency (in statistics):

absolute, 1012, 1019

cumulative absolute, 1012

cumulative relative, 1012

relative class, 1012

Frequency (of vibrating string), 547

Frequency distributions, mean and variance of:

expectation, 1037–1038

moments, 1038

transformation of, 1036–1037

Fresnel, Augustin, 697n.4, A68n.1

Fresnel integrals, 697, A68

Frobenius, Georg, 180n.4

Frobenius method, 167, 180–187, 201

indicial equation, 181–183

proof of, A77–A81

typical applications, 183–185

Frobenius norm, 861

Fulkerson, Delbert Ray, 998n.7

Function, of complex variable, 620–621

Function spaces, 313

Fundamental matrix, 139

Fundamental period, 475

Fundamental region (exponential function), 632

Fundamental system, 50, 104. See also Basis, of solutions

Fundamental Theorem:

higher-order homogeneous linear

ODEs, 106

for linear systems, 288

PDEs, 541–542

second-order homogeneous linear ODEs, 48

Galilei, Galileo, 16n.4

Gamma functions, 190–191, 208

formula for, A66–A67

incomplete, A67

table, A98

GAMS (Guide to Available Mathematical Software), 789

GAUSS, ALGORITHM, 849

Gauss, Carl Friedrich, 186n.5, 608n.1, 1103

Gauss distribution, 1045. See also Normal distributions

Gauss “Double Ring,” 451

Gauss elimination, 320, 849

linear systems, 274–280, 844–852, 898

back substitution, 274–276, 846

elementary row operations, 277

if infinitely many solutions exist, 278

if no solution exists, 278–279

operation count, 850–851

row echelon form, 279–280

operation count, 850–851

Gauss integration formulas, 807, 836–838, 843

Gauss–Jordan elimination, 302–304, 856–857

GAUSS–SEIDEL, ALGORITHM, 860

Gauss–Seidel iteration, 858–863, 898

Gauss's hypergeometric ODE, 186, 202

Geiger, H., 1044, 1100

Generalized Euler formula, 582

Generalized Fourier series, see Orthogonal series

Generalized solution (vibrating string), 550

Generalized triangle inequality, 615

General powers, 639–640, 642

General solution:

Bessel's equation, 194–200

first-order ODEs, 6, 44

higher-order linear ODEs, 106, 110–111, 123

nonhomogeneous linear systems, 160

second-order linear ODEs:

homogeneous, 49–51, 77–78, 104

nonhomogeneous, 80–81

systems of ODEs, 131–132, 139

Generating functions, 179, 241

Geometric interpretation:

partial derivatives, A70

scalar triple product, 373, 374

Geometric multiplicity, 326, 878

Geometric series, 168, 675

Taylor series, 694

uniformly convergent, 698

Gerschgorin, Semyon Aranovich, 879n.6

Gerschgorin's theorem, 879–881, 899

Gibbs phenomenon, 515

Global error, 902

Golden Rule, 15, 24

Gompertz model, 19

Goodness of fit, 1096–1100

Gosset, William Sealy, 1086n.4

Goursat, Édouard, 654n.1

Goursat's proof, 654

Gradient, A75

fluid flow, 771

of a scalar field, 395–402

directional derivatives, 396–397

maximum increase, 398

as surface normal vector, 398–399

vector fields that are, 400–401

of a scalar function, 396, 411

unconstrained optimization, 952

Gradient method, 952. See also Method of steepest descent

Graphs, 970–971, 1007

bipartite, 1001–1006, 1008

center of, 991

complete, 974

complete bipartite, 1005

computer representation of, 972–974

connected, 977, 981, 984

diameter of, 991

digraphs (directed graphs), 971–974, 1007

computer representation of, 972–974

defined, 972

incidence matrix of, 975

subgraphs, 972

Euler, 980

forest, 987

incidence matrix of, 975

planar, 1005

radius of, 991

sparse, 974

subgraphs, 972

trees, 984

vertices, 971, 977, 1007

adjacent, 971, 977

central, 991

coloring, 1005–1006

double labeling of, 986

eccentricity of, 991

exposed, 1001, 1003

four-color theorem, 1006

scanning, 998

weighted, 976

Graphic data representation, 1012

Gravitation (Laplace's equation), 593

Gravity, acceleration of, 8

Gravity constant, at the Earth's surface, 63

Greedy algorithm, 984–988

Green, George, 433n.4

Green's first formula, 461, 470

Green's second formula, 461, 470

Green's theorem:

first and second forms of, 461

in the plane, 433–438, 470

Gregory, James, 816n.2

Gregory–Newton's (Newton's) backward difference interpolation formula, 818–819

Gregory–Newton's (Newton's) forward difference interpolation formula, 815–818

Growth restriction, 209

Guidepoints, 827

Guide to Available Mathematical Software (GAMS), 789

Guldin, Habakuk, 452n.7

Guldin's theorem, 452n.7

Hadamard, Jacques, 683n.1

Half-planes:

complex analysis, 619–620

mapping, 747–749

Half-range expansions (Fourier series), 488–490, 538

Hamilton, William Rowan, 976n.1

Hamiltonian cycle, 976

Hankel, Hermann, 200n.8

Hankel functions, 200

Harmonic conjugate function (Laplace's equation), 629

Harmonic functions, 460, 462, 758

complex analysis, 628–629

under conformal mapping, 763

defined, 758

Laplace's equation, 593, 628–629

maximum modulus theorem, 783–784

potential theory, 781–784, 786

Harmonic oscillation, 63–64

Heat equation, 459–460, 557–558

Dirichlet problem, 564–566

Laplace's equation, 564

numeric analysis, 936–941, 948

Crank–Nicolson method, 938–941

explicit method, 937, 940–941

one-dimensional, 559

solution:

by Fourier integrals, 568–571

by Fourier series, 558–563

by Fourier transforms, 571–574

steady two-dimensional heat problems, 546–566

two-dimensional, 564–566

unifying power of methods, 566

Heat flow:

Laplace's equation, 593

potential theory, 767–770

Heat flow lines, 767

Heaviside, Oliver, 204n.1

Heaviside calculus, 204n.1

Heaviside expansions, 228

Heaviside function, 217–219

Helix, 386

Henry, Joseph, 93n.7

Hermite, Charles, 510n.8

Hermite interpolation, 826

Hermitian form, 351

Hermitian matrices, 347, 348, 350, 353

Hertz, Heinrich, 63n.3

Hesse, Ludwig Otto, 366n.2

Hesse's normal form, 366

Heun, Karl, 905n.1

Heun's method, 903. See also Improved Euler's method

Higher functions, 167. See also Special functions

Higher-order linear ODEs, 105–123

homogeneous, 105–116, 123

nonhomogeneous, 116–123

systems of, see Systems of ODEs

Higher order ODEs (numeric analysis), 915–922

Euler method, 916–917

Runge–Kutta methods, 917–919

Runge–Kutta–Nyström methods, 919–921

Higher transcendental functions, 920

High-frequency line equations, 600

Hilbert, David, 198n.7, 312n.4

Hilbert spaces, 363

Histograms, 1012

Holes, of domains, 653

Homogeneous first-order linear ODEs, 28

Homogeneous higher-order linear ODEs, 105–111

Homogeneous linear systems, 138, 165, 272, 290–291, 845

constant-coefficient systems, 140–151

matrices and vectors, 124–130, 321

trivial solution, 290

Homogeneous PDEs, 541

Homogeneous second-order linear ODEs, 46–48

basis, 50–52

with constant coefficients, 53–60

complex roots, 57–59

real double root, 55–56

two distinct real-roots, 54–55

differential operators, 60–62

Euler–Cauchy equations, 71–74

existence and uniqueness of solutions, 74–79

general solution, 49–51, 77–78

initial value problem, 49–50

modeling free oscillations of mass–spring system, 62–70

particular solution, 49–51

reduction of order, 51–52

Wronskian, 75–78

Hooke, Robert, 62

Hooke's law, 62

Householder, Alston Scott, 888n.11

Householder's tridiagonalization method, 888–892

Hyperbolic analytic functions (conformal mapping), 750–754

Hyperbolic cosine, 635, 752

Hyperbolic functions, 635, 642

formula for, A65–A66

inverse, 640

Taylor series, 695

Hyperbolic PDEs:

defined, 923

numeric analysis, 942–945

Hyperbolic sine, 635, 752

Hypergeometric distributions, 1042–1044, 1061

Hypergeometric equations, 167, 185–187

Hypergeometric functions, 167, 186

Hypergeometric series, 186

Hypothesis, 1077

Hypothesis testing (in statistics), 1063, 1077–1087

comparison of means, 1084–1085

comparison of variances, 1086

errors in tests, 1080–1081

for mean of normal distribution with known variance, 1081–1083

for mean of normal distribution with unknown variance, 1083–1084

one- and two-sided alternatives, 1079–1080

Idempotent matrices, 270

Identity mapping, 745

Identity matrices, 268

Identity operator (second-order homogeneous linear ODEs), 60

Ill-conditioned equations, 805

Ill-conditioned problems, 864

Ill-conditioned systems, 864, 865, 899

Ill-conditioning (linear systems), 864–872

condition number of a matrix, 868–870

matrix norms, 866–868

vector norms, 866

Image:

conformal mapping, 737

linear transformations, 313

Imaginary axis (complex plane), 611

Imaginary part (complex numbers), 609

Imaginary unit, 609

Impedance (RLC circuits), 95

Implicit formulas, 913

Implicit method:

backward Euler scheme as, 909

for hyperbolic PDEs, 943

Implicit solution, 21

Improper integrals:

defined, 205

residue integration, 726–732

Improper node, 142

Improved Euler's method:

error of, 904, 906, 908

first-order ODEs, 902–904

Impulse, of a force, 225

short impulses, 225–226

unit impulse function, 226

Incidence matrices (graphs and digraphs), 975

Incident edges, 971

Inclusion theorems:

defined, 882

matrix eigenvalue problems, 879–884

Incomplete gamma functions, formula for, A67

Inconsistent linear systems, 277

Indefinite (quadratic form), 346

Indefinite integrals:

defined, 643

existence of, 656–658

Indefinite integration (complex line integral), 646–647

Independence:

of path, 669

of path in domain (integrals), 470, 655

of random variables, 1055–1056

Independent events, 1022–1023, 1061

Independent sample values, 1064

Independent variables:

in calculus, 393

in regression analysis, 1103

Indicial equation, 181–183, 188, 202

Indirect methods (solving linear systems), 858, 898

Inference, statistical, 1059, 1063

Infinite dimensional vector space, 311

Infinite populations, 1044

Infinite sequences:

bounded, A93–A95

monotone real, A72–A73

power series, 671–673

Infinite series, 673–674

Infinity:

analytic of singular at, 718–719

point at, 718

Initial conditions:

first-order ODEs, 6, 7, 44

heat equation, 559, 568, 569

higher-order linear ODEs:

homogeneous, 107

nonhomogeneous, 117

one-dimensional heat equation, 559

PDEs, 541, 605

second-order homogeneous linear ODEs, 49–50, 104

systems of ODEs, 137

two-dimensional wave equation, 577

vibrating string, 545

Initial point (vectors), 355

Initial value problem (IVP):

defined, 6

first-order ODEs, 6, 39, 44, 901

bell-shaped curve, 13

existence and uniqueness of solutions for, 38–43

higher-order linear ODEs, 123

homogeneous, 107–108

nonhomogeneous, 117

Laplace transforms, 213–216

for RLC circuit, 99

second-order homogeneous linear ODEs, 49, 74–75, 104

systems of ODEs, 137

Injective mapping, 737n.1

Inner product (dot product), 312

for complex vectors, 349

invariance of, 336

vector differential calculus, 361–367, 410

applications, 364–366

orthogonality, 361–363

Inner product spaces, 311–313

Input (driving force), 27, 86, 214

Instability, numeric vs. mathematical, 796

Integrals, see Line integrals

Integral equations:

defined, 236

Laplace transforms, 236–237

Integral of a function, Laplace transforms of, 212–213

Integral transforms, 205, 518

Integrand, 414, 644

Integrating factors, 23–26, 45

defined, 24

finding, 24–26

Integration. See also Complex integration

constant of, 18

of Laplace transforms, 238–240

numeric, 827–838

adaptive, 835–836

Gauss integration formulas, 836–838

rectangular rule, 828

Simpson's rule, 831–835

trapezoidal rule, 828–831

termwise, of power series, 687, 688

Intermediate value theorem, 807–808

Intermediate variables, 393

Intermittent harvesting, 36

INTERPOL, ALGORITHM, 814

Interpolation, 529

defined, 808

numeric analysis, 808–820, 842

equal spacing, 815–819

Lagrange, 809–812

Newton's backward difference formula, 818–819

Newton's divided difference, 812–815

Newton's forward difference formula, 815–818

spline, 820–827

Interpolation polynomial, 808, 842

Interquartile range, 1013

Intersection, of events, 1016, 1017

Intervals. See also Confidence intervals

class, 1012

closed, A72n.3

convergence, 171, 683

open, 4, A72n.3

Interval estimates, 1065

Invariance, of curl, A85–A88

Invariant rank, 283

Invariant subspace, 878

Inverse cosine, 640

Inverse cotangent, 640

Inverse Fourier cosine transform, 518

Inverse Fourier sine transform, 519

Inverse Fourier sine transform method, 519

Inverse Fourier transform, 524

Inverse hyperbolic function, 640

Inverse hyperbolic sine, 640

Inverse mapping, 741, 745

Inverse of a matrix, 128, 301–309, 321

cancellation laws, 306–307

determinants of matrix products, 307–308

formulas for, 304–306

Gauss–Jordan method, 302–304, 856–857

Inverse sine, 640

Inverse tangent, 640

Inverse transform, 205, 253

Inverse transformation, 315

Inverse trigonometric function, 640

Irreducible, 883

Irregular boundary (elliptic PDEs), 933–935

Irrotational flow, 774

Isocline, 10

Isolated critical point, 152

Isolated essential singularity, 715

Isolated singularity, 715

Isotherms, 36, 38, 402, 767

Iteration (iterative) methods:

numeric analysis, 798–808

fixed-point iteration, 798–801

Newton's (Newton–Raphson) method, 801–805

secant method, 805–806

speed of convergence, 804–805

numeric linear algebra, 858–864, 898

Gauss–Seidel iteration, 858–862

Jacobi iteration, 862–863

IVP, see Initial value problem

Jacobi, Carl Gustav Jacob, 430n.3

Jacobians, 430, 741

Jacobi iteration, 862–863

Jordan, Wilhelm, 302n.3

Joukowski airfoil, 739–740

Kantorovich, Leonid Vitaliyevich, 959n.1

KCL (Kirchhoff's Current Law), 93n.7, 274

Kernel, 205

Kinetic friction, coefficient of, 19

Kirchhoff, Gustav Robert, 93n.7

Kirchhoff's Current Law (KCL), 93n.7, 274

Kirchhoff's law, 991

Kirchhoff's Voltage Law (KVL), 29, 93, 274

Koopmans, Tjalling Charles, 959n.1

Kreyszig, Erwin, 855n.3

Kronecker, Leopold, 500n.5

Kronecker delta, A85

Kronecker symbol, 500

Kruskal, Joseph Bernard, 985n.5

KRUSKAL, ALGORITHM, 985

Kruskal's Greedy algorithm, 984–988, 1008

k th backward difference, 818

k th central moment, 1038

k th divided difference, 813

k th forward difference, 815–816

k th moment, 1038, 1065

Kublanovskaya, V. N., 892

Kutta, Wilhelm, 905n.1

Kutta's third-order method, 911

KVL, see Kirchhoff's Voltage Law

Lagrange, Joseph Louis, 51n.1

Lagrange interpolation, 809–812

Lagrange's form, 812, 842

Laguerre, Edmond, 504n.7

Laguerre polynomials, 241, 504

Laguerre's equation, 240–241

LAPACK, 789

Laplace, Pierre Simon Marquis de, 204n.1

Laplace equation, 400, 564, 593–600, 642, 923

boundary value problem in spherical coordinates, 594–596

complex analysis, 628–629

in cylindrical coordinates, 593–594

Fourier–Legendre series, 596–598

heat equation, 564

numeric analysis, 922–936, 948

ADI method, 928–930

difference equations, 923–925

Dirichlet problem, 925–928, 934–935

Liebmann's method, 926–928

in spherical coordinates, 594

theory of solutions of, 460, 786. See also Potential theory

two-dimensional heat equation, 564

two-dimensional problems, 759

uniqueness theorem for, 462

Laplace integrals, 516

Laplace operator, 401. See also Laplacian

Laplace transforms, 203–253

convolution, 232–237

defined, 204, 205

of derivatives, 211–212

differentiation of, 238–240

Dirac delta function, 226–228

existence, 209–210

first shifting theorem (s-shifting), 208–209

general formulas, 248

initial value problems, 213–216

integral equations, 236–237

of integral of a function, 212–213

integration of, 238–240

linearity of, 206–208

notation, 205

ODEs with variable coefficients, 240–241

partial differential equations, 600–603

partial fractions, 228–230

second shifting theorem (t-shifting), 219–223

short impulses, 225–226

systems of ODEs, 242–247

table of, 249–251

uniqueness, 210

unit step function (Heaviside function), 217–219

Laplacian, 400, 463, 605, A76

in cylindrical coordinates, 593–594

heat equation, 557

Laplace's equation, 593

in polar coordinates, 585–592

in spherical coordinates, 594

of u in polar coordinates, 586

Lattice points, 925–926

Laurent, Pierre Alphonse, 708n.1

Laurent series, 708–719, 734

analytic or singular at infinity, 718–719

point at infinity, 718

Riemann sphere, 718

singularities, 715–717

zeros of analytic functions, 717

Laurent's theorem, 709

LCL (lower control limit), 1088

Least squares approximation, of a function, 875–876

Least squares method, 872–876, 899

Least squares principle, 1103

Lebesgue, Henri, 876n.5

Left-handed Cartesian coordinate system, 369, 370, A84

Left-hand limit (Fourier series), 480

Left-sided tests, 1079, 1082

Legendre, Adrien-Marie, 175n.1, 1103

Legendre function, 175

Legendre polynomials, 167, 177–179, 202

Legendre's equation, 167, 175–177, 201, 202

Laplace's equation, 595–596

special, 169–170

Leibniz, Gottfried Wilhelm, 15n.3

Leibniz test for real series, A73–A74

Length:

curves, 385

vectors, 355, 356, 410

Leonardo of Pisa, 690n.2

Leontief, Wassily, 334n.1

Leontief input–output model, 334

Leslie model, 331

Level surfaces, 380, 398

LFTs, see Linear fractional transformations

Libby, Willard Frank, 13n.2

Liebmann's method, 926–928

Likelihood function, 1066

Limit (sequences), 672

Limit cycle, 158–159, 621

Limit l, 378

Limit point, A93

Limit vector, 378

Linear algebra, 255. See also Numeric linear algebra

determinants, 293–301

Cramer's rule, 298–300

general properties of, 295–298

of matrix products, 307–308

second-order, 291–292

third-order, 292–293

inverse of a matrix, 301–309

cancellation laws, 306–307

determinants of matrix products, 307–308

formulas for, 304–306

Gauss–Jordan method, 302–304

linear systems, 272–274

back substitution, 274–276

elementary row operations, 277

Gauss elimination, 274–280

homogeneous, 290–291

nonhomogeneous, 291

solutions of, 288–291

matrices and vectors, 257–262

addition and scalar multiplication of, 259–261

diagonal matrices, 268

linear independence and dependence of vectors, 282–283

matrix multiplication, 263–266, 269–279

notation, 258

rank of, 283–285

symmetric and skew-symmetric matrices, 267–268

transposition of, 266–267

triangular matrices, 268

matrix eigenvalue problems, 322–353

applications, 329–334

complex matrices and forms, 346–352

determining eigenvalues and

eigenvectors, 323–329

diagonalization of matrices, 341–342

eigenbases, 339–341

orthogonal matrices, 337–338

orthogonal transformations, 336

quadratic forms, 343–344

symmetric and skew- symmetric matrices, 334–336

transformation to principal axes, 344

vector spaces:

inner product spaces, 311–313

linear transformations, 313–317

real, 309–311

special, 285–287

Linear combination:

homogeneous linear ODEs:

higher-order, 107

second-order, 48

of matrices, 129, 271

of vectors, 129, 282

of vectors in vector space, 311

Linear dependence, of vectors, 282–283

Linear element, 386

Linear equations, systems of, see

Linear systems

Linear fractional transformations (LFTs), 742–750, 757

extended complex plane, 744–745

mapping standard domains, 747–750

Linear independence:

scalar triple product, 373

of vectors, 282–283

Linear inequalities, 954

Linear interpolation, 809–810

Linearity:

Fourier transforms, 526–527

Laplace transforms, 206–208

line integrals, 645

Linearity principle, see Superposition principle

Linearization, 152–155

Linearized system, 153

Linearly dependent functions:

higher-order homogeneous linear ODEs, 106, 109

second-order homogeneous linear ODEs, 50, 75

Linearly dependent sets, 129, 311

Linearly dependent vectors, 282–283, 285

Linearly independent functions:

higher-order homogeneous linear ODEs, 106, 109, 113

second-order homogeneous linear ODEs, 50, 75

Linearly independent sets, 128–129, 311

Linearly independent vectors, 282–283

Linearly related variables, 1109

Linear mapping, 314. See also Linear transformations

Linear ODEs, 45, 46

first order, 27–36

Bernoulli equation, 31–33

homogeneous, 28

nonhomogeneous, 28–29

population dynamics, 33–34

higher-order, 105–123

homogeneous, 105–116

nonhomogeneous, 116–122

higher-order homogeneous, 105

second-order, 46–104

homogeneous, 46–78, 103

nonhomogeneous, 79–102, 103

Linear operations:

Fourier cosine and sine transforms as, 520

integration as, 645

Linear operators (second-order homogeneous linear ODEs), 61

Linear optimization, see Constrained (linear) optimization

Linear PDEs, 541

Linear programming problems, 954–958

normal form of problems, 955–957

simplex method, 958–968

degenerate feasible solution, 962–965

difficulties in starting, 965–968

Linear systems, 138–139, 165, 272–274, 320, 845

back substitution, 274–276

defined, 267, 845

elementary row operations, 277

Gauss elimination, 274–280, 844–852

applications, 277–180

back substitution, 274–276

elementary row operations, 277

operation count, 850–851

row echelon form, 279–280

Gauss–Jordan elimination, 856–857

homogeneous, 138, 165, 272, 290–291

constant-coefficient systems, 140–151

matrices and vectors, 124–130

ill-conditioning, 864–872

condition number of a matrix, 868–870

matrix norms, 866–868

vector norms, 866

iterative methods, 858–864

Gauss–Seidel iteration, 858–882

Jacobi iteration, 862–863

LU-factorization, 852–855

Cholesky's method, 855–856

of m equations in n unknowns, 272

nonhomogeneous, 138, 160–163, 272, 290, 291

solutions of, 288–291, 898

Linear transformations, 320

motivation of multiplication by, 265–266

vector spaces, 313–317

Line integrals, 643–652, 669

basic properties of, 645

bounds for, 650–651

definition of, 414, 643–645

existence of, 646

indefinite integration and substitution of limits, 646–647

path dependence of, and integration around closed curves, 421–425

representation of a path, 647–650

vector integral calculus, 413–419

definition and evaluation of, 414–416

path dependence of, 418–426

work done by a force, 416–417

Lines of constant revenue, 954

Lines of force, 760–762

LINPACK, 789

Liouville, Joseph, 499n.4

Liouville's theorem, 666–667

Lipschitz, Rudolf, 42n.9

Lipschitz condition, 42

Ljapunov, Alexander Michailovich, 149n.2

Local error, 830

Local maximum (unconstrained optimization), 952

Local minimum (unconstrained optimization), 951

Local truncation error, 902

Logarithm, 636–639

natural, 636–638, 642, A63

Taylor series, 695

Logarithmic decrement, 70

Logarithmic integral, formula for, A69

Logarithm of base ten, formula for, A63

Logistic equation, 32–33

Longest path, 976

Loss of significant digits (numeric analysis), 793–794

Lotka, Alfred J., 155n.3

Lotka–Volterra population model, 155–156

Lot tolerance percent defective (LTPD), 1094

Lower confidence limits, 1068

Lower control limit (LCL), 1088

Lower triangular matrices, 268

LTPD (lot tolerance percent defective), 1094

LU-factorization (linear systems), 852–855

Machine numbers, 792

Maclaurin, Colin, 690n.2, 712

Maclaurin series, 690, 694–696

Main diagonal:

determinants, 294

matrix, 125, 258

Malthus, Thomas Robert, 5n.1

Malthus’ law, 5, 33

Maple, 789

Maple Computer Guide, 789

Mapping, 313, 736, 737, 757

bijective, 737n.1

conformal, 736–757

boundary value problems, 763–767, A96

defined, 738

geometry of analytic functions, 737–742

linear fractional transformations, 742–750

Riemann surfaces, 754–756

by trigonometric and hyperbolic analytic functions, 750–754

of disks, 748–750

fixed points of, 745

of half-planes onto half-planes, 748

identity, 745

injective, 737n.1

inverse, 741, 745

linear, 314. See also Linear transformations

one-to-one, 737n.1

spectral mapping theorem, 878

surjective, 737n.1

Marconi, Guglielmo, 63n.3

Marginal distributions, 1053–1055, 1062

of continuous distributions, 1055

of discrete distributions, 1053–1054

Mariotte, Edme, 19n.5

Markov, Andrei Andrejevitch, 270n.1

Markov process, 270, 331

MATCHING, ALGORITHM, 1003

Matching, 1008

assignment problems, 1001

complete, 1002

maximum cardinality, 1001, 1008

Mathcad, 789

Mathematica, 789

Mathematica Computer Guide, 789

Mathematical models, see Models

Mathematical modeling, see Modeling

Mathematical statistics, 1009, 1063–1113

acceptance sampling, 1092–1096

errors in, 1093–1094

rectification, 1094–1095

confidence intervals, 1068–1077

for mean of normal distribution with known variance, 1069–1071

for mean of normal distribution with unknown variance, 1071–1073

for parameters of distributions other than normal, 1076

for variance of a normal distribution, 1073–1076

correlation analysis, 1108–1111

defined, 1103

test for correlation coefficient, 1110–1111

defined, 1063

goodness of fit, 1096–1100

hypothesis testing, 1077–1087

comparison of means, 1084–1085

comparison of variances, 1086

errors in tests, 1080–1081

for mean of normal distribution with known variance, 1081–1083

for mean of normal distribution with unknown variance, 1083–1084

one- and two-sided alternatives, 1079–1080

main purpose of, 1015

nonparametric tests, 1100–1102

point estimation of parameters, 1065–1068

quality control, 1087–1092

for mean, 1088–1089

for range, 1090–1091

for standard deviation, 1090

for variance, 1089–1090

random sampling, 1063–1065

regression analysis, 1103–1108

confidence intervals in, 1107–1108

defined, 1103

Matlab, 789

Matrices, 124–130, 256–262, 320

addition and scalar multiplication of, 259–261

calculations with, 126–127

condition number of, 868–870

definitions and terms, 125–126, 128, 257

diagonal, 268

diagonalization of, 341–342

eigenvalues, 129–130

equality of, 126, 259

fundamental, 139

inverse of, 128, 301–309, 321

cancellation laws, 306–307

determinants of matrix products, 307–308

formulas for, 304–306

Gauss–Jordan method, 302–304, 856–857

matrix multiplication, 127, 263–266, 269–279

applications of, 269–279

cancellation laws, 306–307

determinants of matrix

products, 307–308

scalar, 259–261

normal, 352, 882

notation, 258

orthogonal, 337–338

rank of, 283–285

square, 126

symmetric and skew-symmetric, 267–268

transposition of, 266–267

triangular, 268

unitary, 347–350, 353

Matrix eigenvalue problems, 322–353, 876–896

applications, 329–334

choice of numeric method for, 879

complex matrices and forms, 346–352

determining eigenvalues and eigenvectors, 323–329

diagonalization of matrices, 341–342

eigenbases, 339–341

inclusion theorems, 879–884

orthogonal matrices, 337–338

orthogonal transformations, 336

power method, 885–888

QR-factorization, 892–896

quadratic forms, 343–344

symmetric and skew-symmetric

matrices, 334–336

transformation to principal axes, 344

tridiagonalization, 888–892

Matrix multiplication, 127, 263–266, 269–279

applications of, 269–279

cancellation laws, 306–307

determinants of matrix products, 307–308

scalar, 259–261

Matrix norms, 861, 866–868

Maximum cardinality matching, 1001, 1003–1004, 1008

Maximum flow:

Ford–Fulkerson algorithm, 998–1000

and minimum cut set, 996

Maximum increase:

gradient of a scalar field, 398

unconstrained optimization, 951

Maximum likelihood estimates (MLEs), 1066–1067

Maximum likelihood method, 1066–1067, 1113

Maximum modulus theorem, 782–784

Maximum principle, 783

Mean (s), 1013–1014, 1061

comparison of, 1084–1085

control chart for, 1088–1089

of normal distributions:

confidence intervals for, 1069–1073

hypothesis testing for, 1081–1084

probability distributions, 1035–1039

addition of, 1057–1058

transformation of, 1036–1037

sample, 1064

Mean square convergence (orthogonal series), 507–508

Mean value (fluid flow), 774n.1

Mean value property:

analytic functions, 781–782

harmonic functions, 782

Mean value theorem, 395

for double integrals, 427

for surface integrals, 448

for triple integrals, 456–457

Median, 1013, 1100–1101

Mendel, Gregor, 1100

Meromorphic function, 719

Mesh incidence matrix, 262

Mesh points (lattice points, nodes), 925–926

Mesh size, 924

Method of characteristics (PDEs), 555

Method of least squares, 872–876, 899

Method of moments, 1065

Method of separating variables, 12–13

circular membrane, 587

partial differential equations, 545–553, 605

Fourier series, 548–551

satisfying boundary conditions, 546–548

two ODEs from wave equation, 545–546

vibrating string, 545–546

Method of steepest descent, 952–954

Method of undetermined coefficients:

higher-order homogeneous linear ODEs, 115, 123

nonhomogeneous linear systems of ODEs, 161

second-order nonhomogeneous linear ODEs, 81–85, 104

Method of variation of parameters:

higher-order nonhomogeneous linear ODEs, 118–120, 123

nonhomogeneous linear systems of ODEs, 162–163

second-order nonhomogeneous linear ODEs, 99–102, 104

Minimization (normal form of linear optimization problems), 957

Minimum (unconstrained optimization), 951

Minimum cut set, 996

Minors, of determinants, 294

Mixed boundary condition (two-dimensional heat equation), 564

Mixed boundary value problem, 605, 923. See also Robin problem

elliptic PDEs, 931–933

heat conduction, 768–769

Mixed type PDEs, 555

Mixing problems, 14

MLEs (maximum likelihood estimates), 1066–1067

ML-inequality, 650–651

Möbius, August Ferdinand, 447n.5

Möbius strip, 447

Möbius transformations, 743. See also Linear fractional transformations (LFTs)

Models, 2

Modeling, 1, 2–8, 44

and concept of solution, 4–6

defined, 2

first-order ODEs, 2–8

initial value problem, 6

separable ODEs, 13–17

typical steps of, 6–7

and unifying power of mathematics, 766

Modification Rule (method of undetermined coefficients):

higher-order homogeneous linear ODEs, 115–116

second-order nonhomogeneous linear ODEs, 81, 83

Modulus (complex numbers), 613

Moments, method of, 1065

Moments of inertia, of a region, 429

Moment vector (vector moment), 371

Monotone real sequences, A72–A73

Moore, Edward Forrest, 977n.2

MOORE, ALGORITHM, 977

Moore's BFS algorithm, 977–980, 1008

Morera's theorem, 667

Moulton, Forest Ray, 913n.3

Multinomial distribution, 1045

Multiple complex roots, 115

Multiple points, curves with, 383

Multiplication:

of complex numbers, 609, 610, 615

in conditional probability, 1022–1023

matrix, 127, 263–266

applications of, 269–279

cancellation laws, 306–307

determinants of matrix products, 307–308

scalar, 259–261

of means, 1057–1058

of power series, 687

scalar, 126–127, 259–261, 310

termwise, 173, 687

of transforms, 232. See also Convolution

Multiplicity, algebraic, 326, 878

Multiply connected domains, 652, 653

Cauchy's integral formula, 662–663

Cauchy's integral theorem, 658–659

Multistep methods, 911–915, 947

Adams–Bashforth methods, 911–914

Adams–Moulton methods, 913–914

defined, 908

first-order ODEs, 911

Mutually exclusive events, 1016, 1021

m × n matrix, 258

Nabla, 396

NAG (Numerical Algorithms Group, Inc.), 789

National Institute of Standards and Technology (NIST), 789

Natural condition (spline interpolation), 823

Natural frequency, 63

Natural logarithm, 636–638, 642, A63

Natural spline, 823

n-dimensional vector spaces, 311

Negative (scalar multiplication), 260

Negative definite (quadratic form), 346

Neighborhood, 619, 720

Net flow, through cut set, 994–995

NETLIB, 789

Networks:

defined, 991

flow problems in, 991–997

cut sets, 994–996

flow augmenting paths, 992–993

paths, 992

Neumann, Carl, 198n.7

Neumann, John von, 959n.1

Neumann boundary condition, 564

Neumann problem, 605, 923

elliptic PDEs, 931

Laplace's equation, 593

two-dimensional heat equation, 564

Neumann's function, 198

NEWTON, ALGORITHM, 802

Newton, Sir Isaac, 15n.3

Newton–Cotes formulas, 833, 843

Newton's (Gregory–Newton's) backward difference interpolation formula, 818–819

Newton's divided difference interpolation, 812–815, 842

Newton's divided difference interpolation formula, 814–815

Newton's (Gregory–Newton's) forward difference interpolation formula, 815–818, 842

Newton's law of cooling, 15–16

Newton's law of gravitation, 377

Newton's (Newton–Raphson) method, 801–805, 842

Newton's second law, 11, 63, 245, 544, 576

Neyman, Jerzy, 1068n.1, 1077n.2

Nicolson, Phyllis, 938n.5

Nicomedes, 391n.4

Nilpotent matrices, 270

NIST (National Institute of Standards and Technology), 789

Nodal incidence matrix, 262

Nodal lines, 580–581, 588

Nodes, 165, 925–926

degenerate, 145–146

improper, 142

interpolation, 808

proper, 143

spline interpolation, 820

trapezoidal rule, 829

vibrating string, 547

Nonbasic variables, 960

Nonconservative physical systems, 422

Nonhomogeneous linear ODEs:

convolution, 235–236

first-order, 28–29

higher-order, 106, 116–122

second-order, 79–102

defined, 47

method of undetermined coefficients, 81–85

modeling electric circuits, 93–99

modeling forced oscillations, 85–92

particular solution, 80

solution by variation of parameters, 99–102

Nonhomogeneous linear systems, 138, 160–163, 166, 272, 290, 291, 845

method of undetermined coefficients, 161

method of variation of parameters, 162–163

Nonhomogeneous PDEs, 541

Nonlinear ODEs, 46

first-order, 27

higher-order homogeneous, 105

second-order, 46

Nonlinear PDEs, 541

Nonlinear systems, qualitative methods for, 152–160

linearization, 152–155

Lotka–Volterra population model, 155–156

transformation to first-order equation in phase plane, 157–159

Nonparametric tests (statistics), 1100–1102, 1113

Nonsingular matrices, 128, 301

Norm (s):

matrix, 861, 866–868

orthogonal functions, 500

vector, 312, 355, 410, 866

Normal accelerations, 391

Normal acceleration vector, 387

Normal derivative, 437

defined, 437

mixed problems, 768, 931

Neumann problems, 931

solutions of Laplace's equation, 460

Normal distributions, 1045–1051, 1062

as approximation of binomial distribution, 1049–1050

confidence intervals:

for means of, 1069–1073

for variances of, 1073–1076

distribution function, 1046–1047

means of:

confidence intervals for, 1069–1073

hypothesis testing for, 1081–1084

numeric values, 1047–1048

tables, A101–A102

two-dimensional, 1110

working with normal tables, 1048–1049

Normal equations, 873, 1105–1106

Normal form (linear optimization problems), 955–957, 959, 969

Normalizing, eigenvectors, 326

Normal matrices, 352, 882

Normal mode:

circular membrane, 588

vibrating string, 547–548

Normal plane, 390

Normal random variables, 1045

Normal vectors, 366, 441

Not rejecting a hypothesis, 1081

No trend hypothesis, 1101

n th order linear ODEs, 105, 123

n th-order ODEs, 134–135

n th partial sum, 170

Fourier series, 495

of series, 673

n th roots, 616

n th roots of unity, 617

Null hypothesis, 1078

Nullity, 287, 291

Null space, 287, 291

Numbers:

acceptance, 1092

Bernoulli's law of large numbers, 1051

chromatic, 1006

complex, 608–619, 641

addition of, 609, 610

conjugate, 612

defined, 608

division of, 610

multiplication of, 609, 610

polar form of, 613–618

subtraction of, 610

condition, 868–870, 899

Fibonacci, 690

floating-point form of, 791–792

machine, 792

random, 1064

Number of degrees of freedom, 1071, 1074

Numerics, see Numeric analysis

Numerical Algorithms Group, Inc. (NAG), 789

Numerically stable algorithms, 796, 842

Numerical Recipes, 789

Numeric analysis (numerics), 787–843

algorithms, 796

basic error principle, 796

error propagation, 795

errors of numeric results, 794–795

floating-point form of numbers, 791–792

interpolation, 808–820

equal spacing, 815–819

Lagrange, 809–812

Newton's backward difference formula, 818–819

Newton's divided difference, 812–815

Newton's forward difference formula, 815–818

spline, 820–827

loss of significant digits, 793–794

numeric differentiation, 838–839

numeric integration, 827–838

adaptive, 835–836

Gauss integration formulas, 836–838

rectangular rule, 828

Simpson's rule, 831–835

trapezoidal rule, 828–831

for ODEs, 901–922

first-order, 901–915

higher order, 915–922

for PDEs, 922–945

elliptic, 922–936

hyperbolic, 942–945

parabolic, 936–942

roundoff, 792–793

software for, 788–789

solution of equations by iteration, 798–808

fixed-point iteration, 798–801

Newton's (Newton–Raphson) method, 801–805

secant method, 805–806

speed of convergence, 804–805

spline interpolation, 820–827

Numeric differentiation, 838–839

Numeric integration, 827–838

adaptive, 835–836

Gauss integration formulas, 836–838

rectangular rule, 828

Simpson's rule, 831–835

trapezoidal rule, 828–831

Numeric linear algebra, 844–899

curve fitting, 872–876

least squares method, 872–876

linear systems, 845

Gauss elimination, 844–852

Gauss–Jordan elimination, 856–857

ill-conditioning norms, 864–872

iterative methods, 858–864

LU-factorization, 852–855

matrix eigenvalue problems, 876–896

inclusion theorems, 879–884

power method, 885–888

QR-factorization, 892–896

tridiagonalization, 888–892

Numeric methods:

choice of, 791, 879

defined, 791

n × n matrix, 125

Nyström, E. J., 919

Objective function, 951, 969

OCs (operating characteristics), 1081

OC curve, see Operating characteristic curve

Odd functions, 486–488

Odd periodic extension, 488–490

ODEs, see Ordinary differential equations

Ohm, Georg Simon, 93n.7

Ohm's law, 29

One-dimensional heat equation, 559

One-dimensional wave equation, 544–545

One-parameter family of curves, 36–37

One-sided alternative (hypothesis testing), 1079–1080

One-sided tests, 1079

One-step methods, 908, 911, 947

One-to-one mapping, 737n.1

Open annulus, 619

Open circular disk, 619

Open integration formula, 838

Open intervals, 4, A72n.3

Open Leontief input–output model, 334

Open set, in complex plane, 620

Operating characteristic curve (OC curve), 1081, 1092, 1095

Operating characteristics (OCs), 1081

Operational calculus, 60, 203

Operation count (Gauss elimination), 850

Operators, 60–61, 313

Optimal solutions (normal form of linear optimization problems), 957

Optimization:

combinatorial, 970, 975–1008

assignment problems, 1001–1006

flow problems in networks, 991–997

Ford–Fulkerson algorithm for

maximum flow, 998–1001

shortest path problems, 975–980

constrained (linear), 951, 954–968

normal form of problems, 955–957

simplex method, 958–968

unconstrained:

basic concepts, 951–952

method of steepest descent, 952–954

Optimization methods, 949

Optimization problems, 949, 954–958

normal form of problems, 955–957

objective, 951

simplex method, 958–968

degenerate feasible solution, 962–965

difficulties in starting, 965–968

Order:

and complexity of algorithms, 978

Gauss elimination, 850

of iteration process, 804

of PDE, 540

singularities, 714

Ordering (Greedy algorithm), 987

Order statistics, 1100

Ordinary differential equations (ODEs), 44

autonomous, 11, 33

defined, 1, 3–4

first-order, 2–45

direction fields, 9–10

Euler's method, 10–11

exact, 20–27

geometric meanings of, 9–12

initial value problem, 38–43

linear, 27–36

modeling, 2–8

numeric analysis, 901–915

orthogonal trajectories, 36–38

separable, 12–20

higher-order linear, 105–123

homogeneous, 105–116, 123

nonhomogeneous, 116–123

systems of, see Systems of ODEs

Laplace transforms, 203–253

convolution, 232–237

defined, 204, 205

of derivatives, 211–212

differentiation of, 238–240

Dirac delta function, 226–228

existence, 209–210

first shifting theorem (s-shifting), 208–209

general formulas, 248

initial value problems, 213–216

integral equations, 236–237

of integral of a function, 212–213

integration of, 238–240

linearity of, 206–208

notation, 205

ODEs with variable coefficients, 240–241

partial differential equations, 600–603

partial fractions, 228–230

second shifting theorem (t-shifting), 219–223

short impulses, 225–226

systems of ODEs, 242–247

table of, 249–251

uniqueness, 210

unit step function (Heaviside function), 217–219

linear, 46

nonlinear, 46

numeric analysis, 901–922

first-order ODEs, 901–915

higher order ODEs, 915–922

second-order linear, 46–104

homogeneous, 46–79

nonhomogeneous, 79–102

second-order nonlinear, 46

series solutions of ODEs, 167–202

Bessel functions, 187–194, 196–200

Bessel's equation, 187–200

Frobenius method, 180–187

Legendre polynomials, 177–179

Legendre's equation, 175–179

power series method, 167–175

systems of, 124–166

basic theory, 137–139

constant-coefficient, 140–151

conversion of n th-order ODEs to, 134–135

homogeneous, 138

Laplace transforms, 242–247

linear, 124–130, 138–151, 160–163

matrices and vectors, 124–130

as models of applications, 130–134

nonhomogeneous, 138, 160–163

nonlinear, 152–160

in phase plane, 124, 141–146, 157–159

qualitative methods for nonlinear systems, 152–160

Orientable surfaces, 446–447

Oriented curve, 644

Oriented surfaces, integrals over, 446–447

Origin (vertex), 980

Orthogonal, to a vector, 362

Orthogonal coordinate curves, A74

Orthogonal expansion, 504

Orthogonal functions:

defined, 500

Sturm–Liouville Problems, 500–503

Orthogonality:

trigonometric system, 479–480, 538

vector differential calculus, 361–363

Orthogonal matrices, 335, 337–338, 353, A85n.2

Orthogonal polynomials, 179

Orthogonal series (generalized Fourier series), 504–510

completeness, 508–509

mean square convergence, 507–508

Orthogonal trajectories:

defined, 36

first-order ODEs, 36–38

Orthogonal transformations, 336, A85n.2

Orthogonal vectors, 312, 362, 410

Orthonormal functions, 500, 501, 508

Orthonormal system, 337

Oscillations:

forced, 85–92

free, 62–70

harmonic, 63–64

second-order linear ODEs:

homogeneous, 62–70

nonhomogeneous, 85–92

Osculating plane, 389, 390

Outcomes:

of experiments, 1015, 1060

probability theory, 1015

Outer normal derivative, 460, 931

Outliers, 1013–1015

Output (response to input), 27, 86, 214

Overdamping, 65–66

Overdetermined linear systems, 277

Overflow (floating-point numbers), 792

Overrelaxation factor, 863

Paired comparison, 1084, 1113

Pappus, theorem of, 452

Pappus of Alexandria, 452n.7

Parabolic PDEs:

defined, 923

numeric analysis, 936–942

Parallelogram law, 357

Parallel processing of products (on computer), 265

Parameters, 175, 381, 1112

estimation of, 1063

point estimation of, 1065–1068

probability distributions, 1035

of a sample, 1065

Parameter curves, 442

Parametric representations, 381, 439–441

Parseval, Marc Antoine, 497n.3

Parseval equality, 509

Parseval's identity, 497

Parseval's theorem, 497

Partial derivatives, A69–A71

defined, A69

first (first order), A71

second (second order), A71

third (third order), A71

of vector functions, 380

Partial differential equations (PDEs), 473, 540–605

basic concepts of, 540–543

d'Alembert's solution, 553–556

defined, 540

double Fourier series solution, 577–585

heat equation, 557–558

Dirichlet problem, 564–566

Laplace's equation, 564

solution by Fourier integrals, 568–571

solution by Fourier series, 558–563

solution by Fourier transforms, 571–574

steady two-dimensional heat

problems, 546–566

unifying power of methods, 566

homogeneous, 541

Laplace's equation, 593–600

boundary value problem in spherical coordinates, 594–596

in cylindrical coordinates, 593–594

Fourier–Legendre series, 596–598

in spherical coordinates, 594

Laplace transforms, solution by, 600–603

Laplacian in polar coordinates, 585–592

linear, 541

method of separating variables, 545–553

Fourier series, 548–551

satisfying boundary conditions, 546–548

two ODEs from wave equation, 545–546

nonhomogeneous, 541

nonlinear, 541

numeric analysis, 922–945

elliptic, 922–936

hyperbolic, 942–945

parabolic, 936–942

ODEs vs., 4

wave equation, 544–545

d'Alembert's solution, 553–556

solution by separating variables, 545–553

two-dimensional, 575–584

Partial fractions (Laplace transforms), 228–230

Partial pivoting, 276, 846–848, 898

Partial sums, of series, 477, 478, 495

Particular solution (s):

first-order ODEs, 6, 44

higher-order homogeneous linear ODEs, 106

nonhomogeneous linear systems, 160

second-order linear ODEs:

homogeneous, 49–51, 104

nonhomogeneous, 80

Partitioning, of a path, 645

Pascal, Blaise, 391n.4

Pascal, Étienne, 391n.4

Paths:

alternating, 1002

augmenting, 1002–1003

closed, 414, 645, 975–976

deformation of, 656

directed, 1000

flow augmenting, 992–993, 998, 1008

flow problems in networks, 992

integration by use of, 647–650

longest, 976

partitioning of, 645

principle of deformation of, 656

shortest, 976

shortest path problems, 975–976

simple closed, 652

Path dependence (line integrals), 418–426, 470, 649–650

defined, 418

and integration around closed curves, 421–425

Path independence, 669

Cauchy's integral theorem, 655

in a domain D in space, 419

proof of, A88–A89

Stokes's Theorem applied to, 468

Path of integration, 414, 644

Pauli spin matrices, 351

p-charts, 1091–1092

PDEs, see Partial differential equations

Pearson, Egon Sharpe, 1077n.2

Pearson, Karl, 1077, 1086n.4

Period, 475

Periodic boundary conditions, 501

Periodic extensions, 488–490

Periodic function, 474–475, 538

Periodic Sturm–Liouville problem, 501

Permutations:

of n things taken k at a time, 1025

of n things taken k at a time with repetitions, 1025–1026

probability theory, 1024–1026

Perron, Oskar, 882n.8

Perron–Frobenius Theorem, 883

Perron's theorem, 334, 882–883

Pfaff, Johann Friedrich, 422n.1

Pfaffian form, 422

p-fold connected domains, 652–653

Phase angle, 90

Phase lag, 90

Phase plane, 134, 165

linear systems, 141, 148

nonlinear systems, 152

Phase plane method, 124

linear systems:

critical points, 142–146

graphing solutions, 141–142

nonlinear systems, 152

linearization, 152–155

Lotka–Volterra population model, 155–156

transformation to first-order equation in, 157–159

Phase plane representations, 134

Phase portrait, 165

linear systems, 141–142, 148

nonlinear systems, 152

Picard, Emile, 42n.10

Picard's Iteration Method, 42

Picard's theorem, 716

Piecewise continuous functions, 209

Piecewise smooth path of integration, 414, 645

Piecewise smooth surfaces, 442, 447

Pivot, 276, 898, 960

Pivot equation, 276, 846, 898, 960

Planar graphs, 1005

Plane:

complex, 611

extended, 718, 744–745

finite, 718

sets in, 620

normal, 390

osculating, 389, 390

phase, 134, 165

linear systems, 141, 148

nonlinear systems, 152

rectifying, 390

tangent, 398, 441–442

vectors in, 309

Plane curves, 383

Planimeters, 436

Poincaré, Henri, 141n.1, 510n.8

Points:

boundary, 426n.2, 620

branch, 755

center, 144, 165

critical, 33, 144, 165

asymptotically stable, 149

and conformal mapping, 738, 757

constant-coefficient systems of

ODEs, 142–151

isolated, 152

nonlinear systems, 152

stable, 140, 149

stable and attractive, 140, 149

unstable, 140, 149

equilibrium, 33–34

fixed, 745, 799

guidepoints, 827

at infinity, 718

initial (vectors), 355

lattice, 925–926

limit, A93

mesh, 925–926

regular, 181

regular singular, 180n.4

saddle, 143, 165

sample, 1015

singular, 181, 201

analytic functions, 693

regular, 180n.4

spiral, 144–145, 165

stagnation, 773

stationary, 952

terminal (vectors), 355

Point estimation of parameters (statistics), 1065–1068, 1113

defined, 1065

maximum likelihood method, 1066–1067

Point set, in complex plane, 620

Point source (flow modeling), 776

Point spectrum, 525

Poisson, Siméon Denis, 779n.2

Poisson distributions, 1041–1042, 1061, A100

Poisson equation:

defined, 923

numeric analysis, 922–936

ADI method, 928–930

difference equations, 923–925

Dirichlet problem, 925–928

mixed boundary value problem, 931–933

Poisson's integral formula:

derivation of, 778–778

potential theory, 777–781

series for potentials in disks, 779–780

Polar coordinates, 431

Laplacian in, 585–592

notation for, 594

two-dimensional wave equation in, 586

Polar form, of complex numbers, 613–618, 631

Polar moment of inertia, of a region, 429

Poles (singularities), 714–715

of order m, 735

and zeros, 717

Polynomials, 624

characteristic, 325, 353, 877

Chebyshev, 504

interpolation, 808, 842

Laguerre, 241, 504

Legendre, 167, 177–179, 202

orthogonal, 179

trigonometric:

approximation by, 495–498

complex, 529

of the same degree N, 495

Polynomial approximations, 808

Polynomial interpolation, 808, 842

Polynomially bounded, 979

Polynomial matrix, 334, 878–879

Populations:

infinite, 1044

for statistical sampling, 1063

Population dynamics:

defined, 33

logistic equation, 33–34

Position vector, 356

Positive correlation, 1111

Positive definite (quadratic form), 346

Positive sense, on curve, 644

Possible values (random variables), 1030

Postman problem, 980

Potential (potential function), 400

complex, 760–761

Laplace's equation, 593

Poisson's integral formula for, 777–781

Potential theory, 179, 420, 460, 758–786

conformal mapping for boundary value problems, 763–767

defined, 758

electrostatic fields, 759–763

complex potential, 760–761

superposition, 761–762

fluid flow, 771–777

harmonic functions, 781–784

heat problems, 767–770

Laplace's equation, 593, 628

Poisson's integral formula, 777–781

Power function, of a test, 1081, 1113

Power method (matrix eigenvalue problems), 885–888, 899

Power series, 168, 671–707

convergence behavior of, 680–682

convergence tests, 674–676, A93–A94

functions given by, 685–690

Maclaurin series, 690

in powers of x, 168

radius of convergence, 682–684

ratio test, 676–678

root test, 678–679

sequences, 671–673

series, 673–674

Taylor series, 690–697

uniform convergence, 698–705

and absolute convergence, 704

properties of, 700–701

termwise integration, 701–703

test for, 703–704

Power series method, 167–175, 201

extension of, see Frobenius method

idea and technique of, 168–170

operations on, 173–174

theory of, 170–174

Practical resonance, 90

Predator–prey population model, 155–156

Predictor–corrector method, 913

PRIM, ALGORITHM, 989

Prim, Robert Clay, 988n.6

Prim's algorithm, 988–991, 1008

Principal axes, transformation to, 344

Principal branch, of logarithm, 639

Principal directions, 330

Principal minors, 346

Principal part, 735

of isolated singularities, 715

of singularities, 708, 709

Principal value (complex numbers), 614, 617, 642

complex logarithm, 637

general powers, 639

Principle of deformation of path, 656

Prior estimates, 805

Probability, 1060

axioms of, 1020

basic theorems of, 1020–1022

conditional, 1022–1023

definitions of, 1018–1020

independent events, 1023

Probability distributions, 1029, 1061

binomial, 1039–1042

continuous, 1032–1034

discrete, 1030–1032

hypergeometric, 1042–1044

mean and variance of, 1035–1039

multinomial, 1045

normal, 1045–1051

Poisson, 1041–1042

of several random variables, 1051–1060

addition of means, 1057–1058

addition of variances, 1058–1059

continuous two-dimensional

distributions, 1053

discrete two-dimensional

distributions, 1052–1053

function of random variables, 1056

independence of random

variables, 1055–1056

marginal distributions, 1053–1055

symmetric, 1036

two-dimensional, 1051

continuous, 1053

discrete, 1052–1053

uniform, 1035–1036

Probability function, 1030–1032, 1052, 1061

Probability theory, 1009, 1015–1062

binomial coefficients, 1027–1028

combinations, 1024, 1026–1027

distributions (probability distributions), 1029

binomial, 1039–1042

continuous, 1032–1034

discrete, 1030–1032

hypergeometric, 1042–1044

mean and variance of, 1035–1039

normal, 1045–1051

Poisson, 1041–1042

of several random variables, 1051–1060

events, 1016–1017

experiments, 1015–1016

factorial function, 1027

outcomes, 1015

permutations, 1024–1026

probability:

basic theorems of, 1020–1022

conditional, 1022–1023

definition of, 1018–1020

independent events, 1023

random variables, 1029–1030

continuous, 1032–1034

discrete, 1030–1032

Problem of existence, 39

Problem of uniqueness, 39

Producers, 1092

Producer's risk, 1094

Product:

inner (dot), 312

for complex vectors, 349

invariance of, 336

vector differential calculus, 361–367, 410

of matrix, 260

determinants of, 307–308

inverting, 306

matrix multiplication, 263, 320

parallel processing of (on computer), 265

scalar multiplication, 260

scalar triple, 373–374, 411

vector (cross):

in Cartesian coordinates, A83–A84

vector differential calculus, 368–375, 410

Product method, 605. See also Method of separating variables

Projection (vectors), 365

Proper node, 143

Pseudocode, 796

Pure imaginary complex numbers, 609

QR-factorization, 892–896

Quadrant, of a circle, 604

Quadratic forms (matrix eigenvalue problems), 343–344

Quadratic interpolation, 810–811

Qualitative methods, 124, 141n.1

defined, 152

for nonlinear systems, 152–160

linearization, 152–155

Lotka–Volterra population model, 155–156

transformation to first-order equation in phase plane, 157–159

Quality control (statistics), 1087–1092, 1113

for mean, 1088–1089

for range, 1090–1091

for standard deviation, 1090

for variance, 1089–1090

Quantitative methods, 124

Quasilinear equations, 555, 923

Quotient:

complex numbers, 610

difference, 923

Rayleigh, 885, 899

Radius:

of convergence, 172

defined, 172

power series, 682–684, 706

of a graph, 991

Random experiments, 1011, 1015–1016, 1060

Randomly selected samples, 1064

Randomness, 1015, 1064. See also Random variables

Random numbers, 1064

Random number generators, 1064

Random sampling (statistics), 1063–1065

Random selections, 1064

Random variables, 1011, 1029–1030, 1061

continuous, 1029, 1032–1034, 1055

defined, 1030

dependent, 1055

discrete, 1029–1032, 1054

function of, 1056

independence of, 1055–1056

marginal distribution of, 1054, 1055

normal, 1045

occurrence of, 1063

probability distributions of, 1051–1060

addition of means, 1057–1058

addition of variances, 1058–1059

continuous two-dimensional distributions, 1053

discrete two-dimensional distributions, 1052–1053

function of random variables, 1056

independence of random variables, 1055–1056

marginal distributions, 1053–1055

skewness of, 1039

standardized, 1037

two-dimensional, 1051, 1062

Random variation, 1063

Range, 1013

control chart for, 1090–1091

defined, 1090

of f, 620

Rank:

of A, 279

of a matrix, 279, 283, 321

in terms of column vectors, 284–285

in terms of determinants, 297

of R, 279

Raphson, Joseph, 801n.1

Rational functions, 624, 725–729

Ratio test (power series), 676–678

Rayleigh, Lord (John William Strutt), 160n.5, 885n.10

Rayleigh equation, 160

Rayleigh quotient, 885, 899

Reactance (RLC circuits), 94

Real axis (complex plane), 611

Real different roots, 71

Real double root, 55–56, 72

Real functions, complex analytic functions vs., 694

Real inner product space, 312

Real integrals, residue integration of, 725–733

Fourier integrals, 729–730

improper integrals, 730–732

of rational functions of cos θ sin θ, 725–729

Real part (complex numbers), 609

Real pre-Hilbert space, 312

Real roots:

different, 71

double, 55–56

higher-order homogeneous linear ODEs:

distinct, 112–113

multiple, 114–115

second-order homogeneous linear ODEs:

distinct, 54–55

double, 55–56

Real sequence, 671

Real series, A73–A74

Real vector spaces, 309–311, 359, 410

Recording, of sample values, 1011–1012

Rectangular cross-section, 120

Rectangular matrix, 258

Rectangular membrane R, 577–584

Rectangular rule (numeric integration), 828

Rectifiable (curves), 385

Rectification (acceptance sampling), 1094–1095

Rectifying plane, 390

Recurrence formula, 201

Recurrence relation, 176

Recursion formula, 176

Reduced echelon form, 279

Reduction of order (second-order homogeneous linear ODEs), 51–52

Regions, 426n.2

bounded, 426n.2

center of gravity of mass in, 429

closed, 426n.2

critical, 1079

feasibility, 954

fundamental (exponential function), 632

moments of inertia of, 429

polar moment of inertia of, 429

rejection, 1079

sets in complex plane, 620

total mass of, 429

volume of, 428

Regression analysis, 1063, 1103–1108, 1113

confidence intervals in, 1107–1108

defined, 1103

Regression coefficient, 1105, 1107–1108

Regression curve, 1103

Regression line, 1103, 1104, 1106

Regular point, 181

Regular singular point, 180n.4

Regular Sturm–Liouville problem, 501

Rejectable quality level (RQL), 1094

Rejection:

of a hypothesis, 1078

of products, 1092

Rejection region, 1079

Relative class frequency, 1012

Relative error, 794

Relative frequency (probability):

of an event, 1019

class, 1012

cumulative, 1012

Relaxation methods, 862

Remainder, 170

of a series, 673

of Taylor series, 691

Remarkable parallelogram, 375

Removable singularities, 717

Repeated factors, 220, 221

Representation, 315

by Fourier series, 476

by power series, 683

spectral, 525

Residual, 805, 862, 899

Residues, 708, 720, 735

at m th-order pole, 722

at simple poles, 721–722

Residue integration, 719–733

formulas for residues, 721–722

of real integrals, 725–733

Fourier integrals, 729–730

improper integrals, 730–732

of rational functions of cos θ sin θ, 725–729

several singularities inside contour, 723–725

Residue theorem, 723–724

Resistance, apparent, 95

Resonance:

practical, 90

undamped forced oscillations, 88–89

Resonance factor, 88

Response to input, see Output (response to input)

Resultant, of forces, 357

Riccati equation, 35

Riemann, Bernhard, 625n.4

Riemannian geometry, 625n.4

Riemann sphere, 718

Riemann surfaces (conformal mapping), 754–757

Right-hand derivatives (Fourier series), 480

Right-handed Cartesian coordinate system, 368–369, A83–A84

Right-handed triple, 369

Right-hand limit (Fourier series), 480

Right-sided tests, 1079, 1082

Risks of making false decisions, 1080

RKF method, see Runge–Kutta–Fehlberg method

RK methods, see Runge–Kutta methods

RKN methods, see Runge–Kutta–Nyström methods

Robin problem:

Laplace's equation, 593

two-dimensional heat equation, 564

Rodrigues, Olinde, 179n.2

Rodrigues's formula, 179, 241

Romberg integration, 840, 843

Roots:

complex:

higher-order homogeneous linear ODEs, 113–115

second-order homogeneous linear ODEs, 57–59

complex conjugate, 72–73

differing by an integer, 183

Frobenius method, 183

distinct (Frobenius method), 182

double (Frobenius method), 183

of equations, 798

multiple complex, 115

n th, 616

n th roots of unity, 617

simple complex, 113–114

Root test (power series), 678–679

Rotation (vorticity of flow), 774

Rounding, 792

Rounding unit, 793

Roundoff (numeric analysis), 792–793

Roundoff errors, 792, 794, 902

Roundoff rule, 793

Rows:

determinants, 294

matrix, 125, 257, 320

Row echelon form, 279–280

Row-equivalent matrices, 283–284

Row-equivalent systems, 277

Row operations (linear systems), 276, 277

Row scaling (Gauss elimination), 850

Row “sum” norm, 861

Row vectors, 126, 257, 320

RQL (rejectable quality level), 1094

Runge, Carl, 820n.3

Runge, Karl, 905n.1

RUNGE–KUTTA, ALGORITHM, 905

Runge–Kutta–Fehlberg (RKF) method, 947

error of, 908

first-order ODEs, 906–908

Runge–Kutta (RK) methods, 915, 947

error of, 908

first-order ODEs, 904–906

higher order ODEs, 917–919

Runge–Kutta–Nyström (RKN) methods, 919–921, 947

Rutherford, E., 1044, 1100

Rutherford–Geiger experiments, 1044, 1100

Rutishauser, Heinz, 892n.12

Saddle point, 143, 165

Samples:

for experiments, 1015

in mathematical statistics, 1063–1064

selection of, 1063–1064

Sample covariance, 1105

Sampled function, 529

Sample distribution function, 1096

Sample mean, 1064, 1113

Sample points, 1015

Sample regression line, 1104

Sample size, 1015, 1064

Sample space, 1015, 1016, 1060

Sample standard deviation, 1065

Sample variance, 1015, 1113

Sampling:

from a population, 1023

random, 1063–1065

with replacement, 1023

binomial distribution, 1042

hypergeometric distribution, 1043–1044

in statistics, 1063

without replacement, 1018, 1023

binomial distribution, 1042–1043

hypergeometric distribution, 1043–1044

Sampling plan, 1092–1093

Scalar (s), 260, 310, 354

Scalar fields, vector fields that are gradients of, 400–401

Scalar functions:

defined, 376

vector differential calculus, 376

Scalar matrices, 268

Scalar multiplication, 126–127, 310

of matrices and vectors, 259–261

vectors in 2-space and 3-space, 358–359

Scalar triple product, 373–374, 411

Scale (vectors), 886–887

Scanning labeled vertices, 998

Schrödinger, Erwin, 226n.2

Schur, Issai, 882n.7

Schur's inequality, 882

Schur's theorem, 882

Schwartz, Laurent, 226n.2

Secant, formula for, A65

Secant method (numeric analysis), 805–806, 842

Second boundary value problem, see Neumann problem

Second-order determinants, 291–292

Second-order differential operator, 60

Second-order linear ODEs, 46–104

homogeneous, 46–79

basis, 50–52

with constant coefficients, 53–60

differential operators, 60–62

Euler–Cauchy equations, 71–74

existence and uniqueness of solutions, 74–79

general solution, 49–51, 77–78

initial value problem, 49–50

modeling free oscillations of mass–spring system, 62–70

reduction of order, 51–52

superposition principle, 47–48

Wronskian, 75–78

nonhomogeneous, 79–102

defined, 47

general solution, 80–81

method of undetermined coefficients, 81–85

modeling electric circuits, 93–99

modeling forced oscillations, 85–92

solution by variation of parameters, 99–102

Second-order method, improved Euler method as, 904

Second-order nonlinear ODEs, 46

Second-order PDEs, 540–541

Second (second order) partial derivatives, A71

Second shifting theorem (t-shifting), 219–223

Second transmission line equation, 599

Seidel, Philipp Ludwig von, 858n.4

Self-starting methods, 911

Sense reversal (complex line integrals), 645

Separable equations, 12–13

Separable ODEs, 44

first-order, 12–20

extended method, 17–18

modeling, 13–17

reduction of nonseparable ODEs to, 17–18

Separating variables, method of, 12–13

circular membrane, 587

partial differential equations, 545–553, 605

Fourier series, 548–551

satisfying boundary conditions, 546–548

two ODEs from wave equation, 545–546

vibrating string, 545–546

Separation constant, 546

Sequences (infinite sequences):

bounded, A93–A95

convergent, 507–508, 672

divergent, 672

limit point of, A93

monotone real, A72–A73

power series, 671–673

real, 671

Series, A73–A74

binomial, 696

conditionally convergent, 675

convergent, 171, 673

cosine, 781

derived, 687

divergent, 171, 673

double Fourier:

defined, 582

rectangular membrane, 577–585

Fourier, 473–483, 538

convergence and sum or, 480–481

derivation of Euler formulas, 479–480

double, 577–585

even and odd functions, 486–488

half-range expansions, 488–490

heat equation, 558–563

from period 2 to 2L, 483–486

Fourier–Bessel, 506–507, 589

Fourier cosine, 484, 486, 538

Fourier–Legendre, 505–506, 596–598

Fourier sine, 477, 486, 538

one-dimensional heat equation, 561

vibrating string, 548

geometric, 168, 675

Taylor series, 694

uniformly convergent, 698

hypergeometric, 186

infinite, 673–674

Laurent, 708–719, 734

analytic or singular at infinity, 718–719

point at infinity, 718

Riemann sphere, 718

singularities, 715–717

zeros of analytic functions, 717

Maclaurin, 690, 694–696

orthogonal, 504–510

completeness, 508–509

mean square convergence, 507–508

power, 168, 671–707

convergence behavior of, 680–682

convergence tests, 674–676, A93–A94

functions given by, 685–690

Maclaurin series, 690

in powers of x, 168

radius of convergence, 682–684

ratio test, 676–678

root test, 678–679

sequences, 671–673

series, 673–674

Taylor series, 690–697

uniform convergence, 698–705

real, A73–A74

Taylor, 690–697, 707

trigonometric, 476, 484

value (sum) of, 171, 673

Series solutions of ODEs, 167–202

Bessel functions, 187–188

of the first kind, 189–194

of the second kind, 196–200

Bessel's equation, 187–196

Bessel functions, 187–188, 196–200

general solution, 194–200

Frobenius method, 180–187

indicial equation, 181–183

typical applications, 183–185

Legendre polynomials, 177–179

Legendre's equation, 175–179

power series method, 167–175

idea and technique of, 168–170

operations on, 173–174

theory of, 170–174

Sets:

complete orthonormal, 508

in the complex plane, 620

cut, 994–996, 1008

linearly dependent, 129, 311

linearly independent, 128–129, 311

Shewhart, W. A., 1088

Shifted function, 219

Shortest path, 976

Shortest path problems (combinatorial optimization), 975–980, 1008

Bellman's principle, 980–981

complexity of algorithms, 978–980

Dijkstra's algorithm, 981–983

Moore's BFS algorithm, 977–980

Shortest spanning trees:

combinatorial optimization, 1008

Greedy algorithm, 984–988

Prim's algorithm, 988–991

defined, 984

Short impulses (Laplace transforms), 225–226

Sifting property, 226

Significance (in statistics), 1078

Significance level, 1078, 1080, 1113

Significance tests, 1078

Significant digits, 791–792

Similarity transformation, 340

Similar matrices, 340–341, 878

Simple closed curves, 646

Simple closed path, 652

Simple complex roots, 113–114

Simple curves, 383

Simple events, 1015

Simple general properties of the line integral, 415–416

Simple poles, 714

Simplex method, 958–968

degenerate feasible solution, 962–965

difficulties in starting, 965–968

Simplex table, 960

Simplex tableau, 960

Simple zero, 717

Simply connected domains, 423, 646, 652, 653

SIMPSON, ALGORITHM, 832

Simpson, Thomas, 832n.4

Simpson's rule, 832, 843

adaptive integration with, 835–836

numeric integration, 831–835

Simultaneous corrections, 862

Sine function:

conformal mapping by, 750–751

formula for, A63–A65

Sine integral, 514, 697, A68–A69, A98

Single precision, floating-point standard for, 792

Singularities (singular, having a singularity), 693, 707, 715

analytic functions, 693

essential, 715–716

inside a contour, 723–725

isolated, 715

isolated essential, 715

Laurent series, 715–719

principal part of, 708

removable, 717

Singular matrices, 301

Singular point, 181, 201

analytic functions, 693

regular, 180n.4

Singular solutions:

first-order ODEs, 8, 35

higher-order homogeneous linear ODEs, 110

second-order homogeneous linear ODEs, 50, 78

Singular Sturm–Liouville problem, 501, 503

Sink (s):

motion of a fluid, 404, 458, 775, 776

networks, 991

Size:

of matrices, 258

sample, 1015, 1064

Skew-Hermitian form, 351

Skew-Hermitian matrices, 347, 348, 350, 353

Skewness, of a random variables, 1039

Skew-symmetric matrices, 268, 320, 334–336, 353

Slack variables, 956, 969

Slope field (direction field), 9–10

Smooth curves, 414, 644

Smooth surfaces, 442

Sobolev, Sergei L'Vovich, 226n.2

Software:

for data representation in statistics, 1011

numeric analysis, 788–789

variable step size selection in, 902

Solenoid, 405

Solutions. See also specific methods defined, 4, 798

first-order ODEs:

concept of, 4–6

equilibrium solutions, 33–34

explicit solutions, 21

family of solutions, 5

general solution, 6, 44

implicit solutions, 21

particular solution, 6, 44

singular solution, 8, 35

solution by calculus, 5

trivial solution, 28, 35

graphing in phase plane, 141–142

higher-order homogeneous linear ODEs, 106

general solution, 106, 110–111

particular solution, 106

singular solution, 110

linear systems, 273, 745

nonhomogeneous linear systems:

general solution, 160

particular solution, 160

PDEs, 541

second-order homogeneous linear ODEs:

general solution, 49–51, 77–78

linear dependence and independence of, 75

particular solution, 49–51

singular solution, 50, 78

second-order linear ODEs, 47

second-order nonhomogeneous linear ODEs:

general solution, 80–81

particular solution, 80

systems of ODEs, 137, 139

Solution curves, 4–6

Solution space, 290

Solution vector, 273, 745

SOR (successive overrelaxation), 863

SOR formula for Gauss–Seidel, 863

Sorting, of sample values, 1011–1012

Source (s):

motion of a fluid, 404, 458, 775

networks, 991

Source intensity, 458

Source line (flow modeling), 776

Span, of vectors, 286

Spanning trees, 984, 988

Sparse graphs, 974

Sparse matrices, 823, 925

Sparse systems, 858

Special functions, 167, 202

formulas for, A63–A69

theory of, 175

Special vector spaces, 285–287

Specific circulation, of flow, 467

Spectral density, 525

Spectral mapping theorem, 878

Spectral radius, 324, 861

Spectral representation, 525

Spectral shift, 896

Spectrum, 877

of matrix, 324

vibrating string, 547

Speed, 386, 391

angular (rotation), 372

of convergence, 804–805

Spherical coordinates, A74–A76

boundary value problem in, 594–596

defined, 594

Laplacian in, 594

Spiral point, 144–145, 165

Spline, 821, 843

Spline interpolation, 820–827

Spring constant, 62

Square error, 496–497, 539

Square matrices, 126, 257, 258, 301–309, 320

s-shifting, 208–209

Stability:

of critical points, 165

of solutions, 33–34, 124, 936

of systems, 84, 124

Stability chart, 149

Stable algorithms, 796, 842

Stable and attractive critical points, 140, 149

Stable critical points, 140, 149

Stable equilibrium solution, 33–34

Stable systems, 84

Stagnation points, 773

Standard basis, 314, 359, 365

Standard deviation, 1014, 1035, 1090

Standard form:

first-order ODEs, 27

higher-order homogeneous linear ODEs, 105

higher-order linear ODEs, 123

power series method, 172

second-order linear ODEs, 46, 103

Standardized normal distribution, 1046

Standardized random variables, 1037

Standard trick (confidence intervals), 1068

Stationary point (unconstrained optimization), 952

Statistics, 1015, 1063. See also Mathematical statistics

Statistical inference, 1059, 1063

Steady flow, 405, 458

Steady heat flow, 767

Steady-state case (heat problems), 591

Steady-state current, 98

Steady-state heat flow, 460

Steady-state solution, 31, 84, 89–91

Steady two-dimensional heat problems, 546–566, 605

Steepest descent, method of, 952–954

Steiner, Jacob, 451n.6

Stem-and-leaf plots, 1012

Stencil (pattern, molecule, star), 925

Step-by-step methods, 901

Step function, 828, 1031

Step size, 901, 902

Stereographic projection, 718

Stiff ODEs, 909–910

Stiff systems, 920–921

Stirling, James, 1027n.2

Stirling formula, 1027, A67

Stochastic matrices, 270

Stochastic variables, 1029. See also Random variables

Stokes, Sir George Gabriel, 464n.9, 703n.5

Stokes's Theorem, 463–470

Stream function, 771

Streamline, 771

Strength (flow modeling), 776

Strictly diagonally dominant matrices, 881

Sturm, Jacques Charles François, 499n.4

Sturm–Liouville equation, 499

Sturm–Liouville expansions, 474

Sturm–Liouville Problems, 498–504

eigenvalues, eigenfunctions, 499–500

orthogonal functions, 500–503

Subgraphs, 972

Submarine cable equations, 599

Submatrices, 288

Subsidiary equation, 203, 253

Subspace, of vector space, 286

Subtraction:

of complex numbers, 610

termwise, of power series, 687

Success corrections, 862

Successive overrelaxation (SOR), 863

Sufficient convergence condition, 861

Sum:

of matrices, 320

partial, of series, 477, 478, 495

of a series, 171, 673

of vectors, 357

Sum Rule (method of undetermined coefficients):

higher-order homogeneous linear ODEs, 115

second-order nonhomogeneous linear ODEs, 81, 83–84

Superlinear convergence, 806

Superposition (electrostatic fields), 761–762

Superposition (linearity) principle:

higher-order homogeneous linear ODEs, 106

higher-order linear ODEs, 123

homogeneous linear systems, 138

PDEs, 541–542

second-order homogeneous linear ODEs, 47–48, 104

undamped forced oscillations, 87

Surfaces, for surface integrals, 439–443

orientation of, 446–447

representation of surfaces, 439–441

tangent plane and surface normal, 441–442

Surface integrals, 470

defined, 443

surfaces for, 439–443

orientation of, 446–447

representation of surfaces, 439–441

tangent plane and surface normal, 441–442

vector integral calculus, 443–452

orientation of surfaces, 446–447

without regard to orientation, 448–450

Surface normal, 398–399, 442

Surface normal vector, 398–399

Surjective mapping, 737n.1

Sustainable yield, 36

Symbol O, 979

Symmetric coefficient matrix, 343

Symmetric distributions, 1036

Symmetric matrices, 267–268, 320, 334–336, 353

Systems of ODEs, 124–166

basic theory of, 137–139

constant-coefficient, 140–151

critical points, 142–146, 148–151

graphing solutions in phase plane, 141–142

conversion of nth-order ODEs to, 134–135

homogeneous, 138

Laplace transforms, 242–247

linear, 138–139. See also Linear systems

constant-coefficient systems, 140–151

matrices and vectors, 124–130

nonhomogeneous, 160–163

matrices and vectors, 124–130

calculations with, 125–127

definitions and terms, 125–126, 128–129

eigenvalues and eigenvectors, 129–130

systems of ODEs as vector equations, 127–128

as models of applications:

electrical network, 132–134

mixing problem involving two tanks, 130–132

nonhomogeneous, 138, 160–163

method of undetermined coefficients, 161

method of variation of parameters, 162–163

nonlinear systems:

qualitative methods for, 152–160

transformation to first-order equation in phase plane, 157–159

in phase plane, 124

critical points, 142–146

graphing solutions in, 141–142

transformation to first-order equation in, 157–159

qualitative methods for nonlinear systems, 152–160

linearization, 152–155

Lotka–Volterra population model, 155–156

Tangent:

to a curve, 384

formula for, A65

Tangent function, conformal mapping by, 752–753

Tangential accelerations, 391

Tangential acceleration vector, 387

Tangent plane, 398, 441–442

Tangent vector, 384, 411

Target (networks), 991

Taylor, Brook, 690n.2

Taylor series, 690–697, 707

Taylor's formula, 691

Taylor's theorem, 691

t-distribution, 1071–1073, 1078, A103

Telegraph equations, 599

Term (s):

of a sequence, 671

of a series, 673

Terminal point (vectors), 355

Termination criterion, 802–803

Termwise addition, 173, 687

Termwise differentiation, 173, 687–688, 703

Termwise integration, 687, 688, 701–703

Termwise multiplication, 173, 687

Termwise subtraction, 687

Tests, statistical, 1077, 1113

Theory of special functions, 175

Thermal diffusivity, 460

Third boundary value problem, see Robin problem

Third-order determinants, 292–293

Third (third order) partial derivatives, A71

3-space, vectors in, 309, 354

components of a vector, 356–357

scalar multiplication, 358–359

vector addition, 357–359

Three-sigma limits, 1047

Time (curves in mechanics), 386

TI-Nspire, 789

Todd, John, 855n.3

Tolerance (adaptive integration), 835

Torricelli, Evangelista, 16n.4

Torricelli's law, 16–17

Torsion, curvature and, 389–390

Total differential, 20, 45

Total energy, of physical system, 525

Total error, 902

Total mass, of a region, 429

Total orthonormal set, 508

Total pivoting, 846

Trace, 345

Trail (shortest path problems), 975

closed trails, 975–976

Euler trail, 980

Trajectories, 134, 165

linear systems, 141–142, 148

nonlinear systems, 152

Transcendental equations, 798

Transducers, 98

Transfer function, 214

Transformation (s), 313

orthogonal, 336

to principal axes, 344

Transient solution, 84, 89

Transient-state solution, 31

Translation (vectors), 355

Transposition (s):

of matrices or vectors, 128, 320

in samples, 1101

Trapezoidal rule, 828, 843

error bounds and estimate for, 829–831

numeric integration, 828–831

Trees (graphs), 984, 988. See also Shortest spanning trees

Trials (experiments), 1011, 1015

Triangle inequality, 363, 614–615

Triangular form (Gauss elimination), 846

Triangular matrices, 268

Tricomi, Francesco, 556n.2

Tricomi equation, 555, 556

Tridiagonalization (matrix eigenvalue problems), 888–892

Tridiagonal matrices, 823, 888, 928

Trigonometric analytic functions (conformal mapping), 750–754

Trigonometric function, 633–635, 642

inverse, 640

Taylor series, 695

Trigonometric polynomials:

approximation by, 495–498

complex, 529

of the same degree N, 495

Trigonometric series, 476, 484

Trigonometric system, 475, 479–480, 538

Trihedron, 390

Triple integrals, 470

defined, 452

mean value theorem for, 456–457

vector integral calculus, 452–458

Triply connected domains, 653, 658, 659

Trivial solution, 28, 35

homogeneous linear systems, 290

linear systems, 273

Sturm–Liouville problem, 499

Truncating, 794

t-shifting, 219–223

Tuning (vibrating string), 548

Twisted curves, 383

2-space (plane), vectors in, 354

components of a vector, 356–357

scalar multiplication, 358–359

vector addition, 357–359

2 × 2 matrix, 125

Two-dimensional heat equation, 564–566

Two-dimensional normal distribution, 1110

Two-dimensional probability distributions:

continuous, 1053

discrete, 1052–1053

Two-dimensional problems (potential theory), 759, 771

Two-dimensional random variables, 1051, 1062

Two-dimensional wave equation, 575–584, 586

Two-sided alternative (hypothesis testing), 1079–1080

Two-sided tests, 1079, 1082–1083

Type I errors, 1080, 1081

Type II errors, 1080–1081

UCL (upper control limit), 1088

Unacceptable lots, 1094

Unconstrained optimization, 969

basic concepts, 951–952

method of steepest descent, 952–954

Uncorrelated related variables, 1109

Underdamping, 65, 67

Underdetermined linear systems, 277

Underflow (floating-point numbers), 792

Undetermined coefficients, method of:

higher-order homogeneous linear ODEs, 115

higher-order linear ODEs, 123

nonhomogeneous linear systems of ODEs, 161

second-order linear ODEs:

homogeneous, 104

nonhomogeneous, 81–85

Uniform convergence:

and absolute convergence, 704

power series, 698–705

properties of uniform convergence, 700–701

termwise integration, 701–703

test for, 703–704

Uniform distributions, 1035–1036, 1053

Unifying power of mathematics, 97

Union, of events, 1016–1017

Uniqueness:

of Laplace transforms, 210

of Laurent series, 712

of power series representation, 685–686

problem of, 39

Uniqueness theorems:

cubic splines, 822

Dirichlet problem, 462, 784

first-order ODEs, 39–42

higher-order homogeneous linear ODEs, 108

Laplace's equation, 462

linear systems, 138

proof of, A77–A79

second-order homogeneous linear ODEs, 74

systems of ODEs, 137

Unitary matrices, 347–350, 353

Unitary systems, 349

Unitary transformation, 349

Unit binormal vector, 389

Unit circle, 617, 619

Unit impulse function, 226. See also Dirac delta function

Unit matrices, 128, 268

Unit normal vectors, 366, 441

Unit principal normal vector, 389

Unit step function (Heaviside function), 217–219

Unit tangent vector, 384

Unit vectors, 312, 355

Universal gravitational constant, 63

Unknowns, 257

Unrepeated factors, 220–221

Unstable algorithms, 796

Unstable critical points, 140, 149

Unstable equilibrium solution, 33–34

Unstable systems, 84

Upper bound, for flows, 995

Upper confidence limits, 1068

Upper control limit (UCL), 1088

Upper triangular matrices, 268

Value (sum) of series, 171, 673

Vandermonde, Alexandre Théophile, 113n.1

Vandermonde determinant, 113

Van der Pol, Balthasar, 158n.4

Van der Pol equation, 158–160

Variables:

artificial, 965–968

basic, 960

complex, 620–621

control, 951

controlled, 1103

dependent, 393, 1055, 1056

independent, 393, 1103

intermediate, 393

linearly, 1109

nonbasic, 960

random, 1011, 1029–1030, 1061

continuous, 1029, 1032–1034, 1055

defined, 1030

dependent, 1055

discrete, 1029–1032, 1054

function of, 1056

independence of, 1055–1056

marginal distribution of, 1054, 1055

normal, 1045

occurrence of, 1063

probability distributions of, 1051–1060

skewness of, 1039

standardized, 1037

two-dimensional, 1051, 1062

slack, 956, 969

stochastic, 1029

uncorrelated related, 1109

Variable coefficients:

Frobenius method, 180–187

indicial equation, 181–183

typical applications, 183–185

Laplace transforms ODEs with, 240–241

power series method, 167–175

idea and technique of, 168–170

operations on, 173–174

theory of, 170–174

second-order homogeneous linear ODEs, 73

Variance (s), 1014, 1061

comparison of, 1086

control chart for, 1089–1090

equality of, 1084n.3

of normal distributions, confidence intervals for, 1073–1076

of probability distributions, 1035–1039

addition of, 1058–1059

transformation of, 1036–1037

sample, 1015

Variation, random, 1063

Variation of parameters, method of:

higher-order linear ODEs, 123

high-order nonhomogeneous linear ODEs, 118–120

nonhomogeneous linear systems of ODEs, 162–163

second-order linear ODEs:

homogeneous, 104

nonhomogeneous, 99–102

Vectors, 256, 259

addition and scalar multiplication of, 259–261

calculations with, 126–127

definitions and terms, 126, 128–129, 257, 259, 309

eigenvalues, 129–130

eigenvectors, 129–130

linear independence and dependence of, 282–283

multiplying matrices by, 263–265

in the plane, 309, 355

systems of ODEs as vector equations, 127–128

in 3-space, 309

transposition of, 266–267

Vector addition, 309, 357–359

Vector calculus, 354, 378–380

differential, see Vector differential calculus

integral, see Vector integral calculus

Vector differential calculus, 354–412

curves, 381–392

arc length of, 385–386

length of, 385

in mechanics, 386–389

tangents to, 384–385

and torsion, 389–390

gradient of a scalar field, 395–402

directional derivatives, 396–397

maximum increase, 398

as surface normal vector, 398–399

vector fields that are, 400–401

inner product (dot product), 361–367

applications, 364–366

orthogonality, 361–363

scalar functions, 376

and vector calculus, 378–380

vector fields, 377–378

curl of, 406–409

divergence of, 402–406

that are gradients of scalar fields, 400–401

vector functions, 375–376

partial derivatives of, 380

of several variables, 392–395

vector product (cross product), 368–375

applications, 371–372

scalar triple product, 373–374

vectors in 2-space and 3-space:

components of a vector, 356–357

scalar multiplication, 358–359

vector addition, 357–359

Vector fields:

defined, 376

vector differential calculus, 377–378

curl of, 406–409, 412

divergence of, 402–406

that are gradients of scalar fields, 400–401

Vector functions:

continuous, 378–379

defined, 375–376

differentiable, 379

divergence theorem of Gauss, 453–457

of several variables, 392–395

chain rules, 392–394

mean value theorem, 395

vector differential calculus, 375–376, 411

partial derivatives of, 380

of several variables, 392–395

Vectors in 2-space and 3-space:

components of a vector, 356–357

scalar multiplication, 358–359

vector addition, 357–359

Vector integral calculus, 413–471

divergence theorem of Gauss, 453–463

double integrals, 426–432

applications of, 428–429

change of variables in, 429–431

evaluation of, by two successive integrations, 427–428

Green's theorem in the plane, 433–438

line integrals, 413–419

definition and evaluation of, 414–416

path dependence of, 418–426

work done by a force, 416–417

path dependence of line integrals, 418–426

defined, 418

and integration around closed curves, 421–425

Stokes's Theorem, 463–469

surface integrals, 443–452

orientation of surfaces, 446–447

without regard to orientation, 448–450

surfaces for surface integrals, 439–443

representation of surfaces, 439–441

tangent plane and surface normal, 441–442

triple integrals, 452–458

Vector moment, 371

Vector norms, 866

Vector product (cross product):

in Cartesian coordinates, A83–A84

vector differential calculus, 368–375, 410

applications, 371–372

scalar triple product, 373–374

Vector spaces, 482

complex, 309–310, 349

inner product spaces, 311–313

linear transformations, 313–317

real, 309–311

special, 285–287

Velocity, 391, 411, 771

Velocity potential, 771

Velocity vector, 386, 771

Venn, John, 1017n.1

Venn diagrams, 1017

Verhulst, Pierre-François, 32n.8

Verhulst equation, 32–33

Vertices (graphs), 971, 977, 1007

adjacent, 971, 977

central, 991

coloring, 1005–1006

double labeling of, 986

eccentricity of, 991

exposed, 1001, 1003

four-color theorem, 1006

scanning, 998

Vertex condition, 991

Vertex incidence list (graphs), 973

Volta, Alessandro, 93n.7

Voltage drop, 29

Volterra, Vito, 155n.3, 198n.7, 236n.3

Volterra integral equations, of the second kind, 236–237

Volume, of a region, 428

Vortex (fluid flow), 777

Vorticity, 774

Walk (shortest path problems), 975

Wave equation, 544–545, 942

d'Alembert's solution, 553–556

numeric analysis, 942–944, 948

one-dimensional, 544–545

solution by separating variables, 545–553

two-dimensional, 575–584

Weber's equation, 510

Weber's functions, 198n.7

Weierstrass, Karl, 625n.4, 703n.5

Weierstrass approximation theorem, 809

Weierstrass M-test for uniform convergence, 703–704

Weighted graphs, 976

Weight function, 500

Well-conditioned problems, 864

Well-conditioning (linear systems), 865

Wessel, Caspar, 611n.2

Work done by a force, 416–417

Work integral, 415

Wronski, Josef Maria Höne, 76n.5

Wronskian (Wronski determinant):

second-order homogeneous linear ODEs, 75–78

systems of ODEs, 139

Zeros, of analytic functions, 717

Zero matrix, 260

Zero surfaces, 598

Zero vector, 129, 260, 357

z-score, 1014