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 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
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 matrices and vectors, 126, 259–261
for mutually exclusive events, 1021
of power series, 687
ADI (alternating direction implicit) method, 928–930
Adjacency matrix:
of a digraph, 973
Airy, Sir George Bidell, 556n.2, 918n.4
Airy equation, 556
Airy function:
Algebraic equations, 798
Algebraic multiplicity, 326, 878
Algorithms:
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
derivatives of, 664–668, 688–689, A95–A96
integration of:
indefinite, 647
Laurent series:
analytics at infinity, 718–719
maximum modulus theorem, 782–783
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
Area:
of a region, 428
of region bounded by ellipses, 436
Argand, Jean Robert, 611n.2
Argand diagram, 611n.2
Argument (complex numbers), 613
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
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):
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:
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
Beats (oscillation), 89
Bellman, Richard, 981n.3
Bellman equations, 981
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
Bernoulli's law of large numbers, 1051
Bessel, Friedrich Wilhelm, 187n.6
Bessel functions, 167, 187–191, 202
of order v, 191
orthogonality of, 506
of the second kind:
of the third kind, 200
Bessel's equation, 167, 187–196, 202
Bessel functions, 167, 187–191, 196–200
circular membrane, 587
Bessel's inequality:
for Fourier coefficients, 497
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
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
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
periodic, 501
two-dimensional wave equation, 577
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
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
BVP, see Boundary value problem
CAD (computer-aided design), 820
Canonical form, 344
Cantor, Georg, A72n.3
Cantor–Dedekind axiom, A72n.3, A95n.4
Capacity:
cut sets, 994
networks, 991
Cardano, Girolamo, 608n.1
linear element in, A75
writing, A74
Cartesian coordinate systems:
complex plane, 611
right-handed, 368–369, A83–A84
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
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
independence of path, 655
for multiply connected domains, 658–659
principle of deformation of path, 656
Cayley, Arthur, 748n.2
c-charts, 1092
Center:
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
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
CN (Crank–Nicolson) method, 938–941
Coefficients:
binomial:
Newton's forward difference formula, 816
constant:
higher-order homogeneous linear ODEs, 111–116
second-order homogeneous linear ODEs, 53–60
second-order nonhomogeneous linear ODEs, 81
Fourier, 476, 484, 538, 582–583
of kinetic friction, 19
of ODEs, 47
higher-order homogeneous linear ODEs, 105
second-order homogeneous linear ODEs, 53–60, 73
second-order nonhomogeneous linear ODEs, 81–85
of power series, 680
variable:
Laplace transforms ODEs with, 240–241
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
Column “sum” norm, 861
Column vectors, 126
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
flow augmenting paths, 992–993
paths, 992
Ford–Fulkerson algorithm for maximum flow, 998–1001
shortest path problems, 975–980
complexity of algorithms, 978–980
Moore's BFS algorithm, 977–980
shortest spanning trees:
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
Cauchy–Riemann equations, 623–629
circles and disks, 619
hyperbolic, 635
Cauchy's integral formula, 660–663, 670
Cauchy's integral theorem, 652–660, 669
derivatives of analytic
conjugate, 612
defined, 608
division of, 610
subtraction of, 610
complex plane, 611
geometry of analytic functions, 737–742
linear fractional transformations, 742–750
by trigonometric and hyperbolic analytic functions, 750–754
analytic or singular at infinity, 718–719
point at infinity, 718
Riemann sphere, 718
zeros of analytic functions, 717
convergence behavior of, 680–682
convergence tests, 674–676, A93–A94
Maclaurin series, 690
in powers of x, 168
radius of convergence, 682–684
formulas for residues, 721–722
several singularities inside contour, 723–725
Complex conjugate numbers, 612
Complex conjugate roots, 72–73
Complex Fourier integral, 523
hyperbolic, 635
Complex heat potential, 767
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
analytic or singular at infinity, 718–719
point at infinity, 718
Riemann sphere, 718
zeros of analytic functions, 717–718
basic properties of, 645
existence of, 646
indefinite integration and substitution of limits, 646–647
representation of a path, 647–650
convergence behavior of, 680–682
Maclaurin series, 690
radius of convergence of, 682–684
formulas for residues, 721–722
several singularities inside contour, 723–725
Complexity, of algorithms, 978–979
Complex line integrals, see Line integrals
Complex matrices and forms, 346–352
conjugate, 612
defined, 608
division of, 610
subtraction of, 610
Complex plane, 611
sets in, 620
Complex potential, 786
higher-order homogeneous linear ODEs:
multiple, 115
second-order homogeneous linear ODEs, 57–59
Complex trigonometric polynomials, 529
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
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
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
second-order homogeneous linear ODEs, 53–60
two distinct real roots, 54–55
second-order nonhomogeneous linear ODEs, 81
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
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
Control charts, 1088
for standard deviation, 1090
Controlled variables, in regression analysis, 1103
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
convergence tests, 674–676, A93–A94
radius of convergence of, 682–684, 706
radius of, 172
defined, 172
sequence of vectors, 378
speed of (numeric analysis), 804–805
superlinear, 806
uniform:
and absolute convergence, 704
Convergence interval, 171, 683
power series, 674–676, A93–A94
Convergent iteration processes, 800
Convergent sequence of functions, 507–508, 672
Convolution:
defined, 232
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
Cosine integral:
formula for, A69
table, A98
Cosine series, 781
Cotangent, formula for, A65
Coulomb, Charles Augustin de, 19n.6, 93n.7, 401n.6
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
asymptotically stable, 149
and conformal mapping, 738, 757
constant-coefficient systems of ODEs, 142–146
center, 144
improper node, 142
proper node, 143
saddle point, 143
isolated, 152
nonlinear systems, 152
stable and attractive, 140, 149
Critical region, 1079
Cross product, 368, 410. See also Vector product
Crout, Prescott Durand, 853n.2
Cubic spline, 821
Cumulative absolute frequencies (of values), 1012
Cumulative distribution functions, 1029
Cumulative relative frequencies (of values), 1012
Curl, A76
of vector fields, 406–409, 412
Curvature, of a curve, 389–390
Curves:
arc of, 383
Bezier, 827
deflection, 120
elastic, 120
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
twisted, 383
vector differential calculus, 381–392, 411
length of, 385
method of least squares, 872–874
by polynomials of degree m, 874–875
Curvilinear coordinates, 354, 412, A74
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
standard deviation, 1014
variation, 1014
and randomness, 1064
Decisions:
false, risks of making, 1080
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
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
Derived series, 687
Descartes, René, 356n.1, 391n.4
Cauchy, 113
defined, A81
general properties of, 295–298
of a matrix, 128
of order n, 293
second-order homogeneous linear ODEs, 76
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
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
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
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
ADI method, 929
Laplace equation, 593–596, 925–928, 934–935
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
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
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
Divergence theorem of Gauss, 405, 470
vector integral calculus, 453–457
Divergent sequence, 672
Division, of complex numbers, 610, 615–616
Domain (s), 393
bounded, 652
of f, 620
holes of, 653
multiply connected:
Cauchy's integral formula, 662–663
Cauchy's integral theorem, 658–659
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
Double integrals (vector integral calculus), 426–432, 470
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
incident, 971
Edge chromatic number, 1006
Edge condition, 991
Edge incidence list (graphs), 973
Efficient algorithms, 979
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
Eigenvalues, 129–130, 166, 353, 605, 877, 899. See also Matrix eigenvalue problems
circular membrane, 588
and critical points, 149
defined, 324
dominant, 883
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
convergent sequence of, 886
defined, 324
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
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
mixed boundary value problems, 931–933
Neumann problem, 931
Empirical Rule, 1014
Energies, 157
Entire function, 630, 642, 707, 718
Entries:
determinants, 294
Equal complex numbers, 609
Equality:
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–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
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
first-order ODEs, 10–11, 901–902
Euler trail, 980
Even periodic extension, 488–490
Events (probability theory), 1016–1017, 1060
complements of, 1016
defined, 1015
disjoint, 1016
equally likely, 1018
mutually exclusive, 1016, 1021
simple, 1015
Exact differential equation, 21
Exact differential form, 422, 470
defined, 21
Existence, problem of, 39
Existence theorems:
cubic splines, 822
homogeneous linear ODEs:
higher-order, 108
second-order, 74
linear systems, 138
power series solutions, 172
systems of ODEs, 137
Expectation, 1035, 1037–1038, 1057
in probability theory, 1015–1016
Experimental error, 794
Explicit formulas, 913
Explicit method:
wave equation, 943
Explicit solution, 21
Exponential function, 630–633, 642
formula for, A63
Taylor series, 694
Exponential growth, 5
Exponential integral, formula for, A69
Extended complex plane:
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
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
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
defined, 4
defined, 21
explicit form, 4
implicit form, 4
homogeneous, 28
Adams–Bashforth methods, 911–914
Adams–Moulton methods, 913–914
backward Euler method, 909–910
improved Euler's method, 902–904
Runge–Kutta–Fehlberg method, 906–908
orthogonal trajectories, 36–38
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
flow augmenting paths, 992–993
paths, 992
Fluid flow:
Laplace's equation, 593
Fluid state, 404
Flux (motion of a fluid), 404
Forced oscillations:
second-order nonhomogeneous linear ODEs, 85–92
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
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
reduced echelon, 279
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
cut sets, 994
initial flow, 998
of a path, 992
Four-color theorem, 1006
Fourier, Jean-Baptiste Joseph, 473n.1
approximation by trigonometric polynomials, 495–498
convergence and sum of, 480–481
derivation of Euler formulas, 479–480
even and odd functions, 486–488
half-range expansions, 488–490
complex form of Fourier integral, 522–523
spectrum representation, 525
orthogonal series (generalized Fourier series), 504–510
mean square convergence, 507–508
Sturm–Liouville Problems, 498–504
eigenvalues, eigenfunctions, 499–500
Fourier–Bessel series, 506–507, 589
Fourier coefficients, 476, 484, 538, 582–583
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
Fourier–Legendre series, 505–506, 596–598
Fourier matrix, 530
convergence and sum or, 480–481
derivation of Euler formulas, 479–480
even and odd functions, 486–488
half-range expansions, 488–490
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
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
Frenet, Jean-Frédéric, 392
Frenet formulas, 392
Frequency (in statistics):
cumulative absolute, 1012
cumulative relative, 1012
relative class, 1012
Frequency (of vibrating string), 547
Frequency distributions, mean and variance of:
moments, 1038
Fresnel, Augustin, 697n.4, A68n.1
Frobenius, Georg, 180n.4
Frobenius method, 167, 180–187, 201
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
second-order homogeneous linear ODEs, 48
Galilei, Galileo, 16n.4
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
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
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
Generalized Euler formula, 582
Generalized Fourier series, see Orthogonal series
Generalized solution (vibrating string), 550
Generalized triangle inequality, 615
General solution:
higher-order linear ODEs, 106, 110–111, 123
nonhomogeneous linear systems, 160
second-order linear ODEs:
homogeneous, 49–51, 77–78, 104
Generating functions, 179, 241
Geometric interpretation:
partial derivatives, A70
scalar triple product, 373, 374
Geometric multiplicity, 326, 878
Taylor series, 694
uniformly convergent, 698
Gerschgorin, Semyon Aranovich, 879n.6
Gerschgorin's theorem, 879–881, 899
Gibbs phenomenon, 515
Global error, 902
Gompertz model, 19
Gosset, William Sealy, 1086n.4
Goursat, Édouard, 654n.1
Goursat's proof, 654
Gradient, A75
fluid flow, 771
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
center of, 991
complete, 974
complete bipartite, 1005
computer representation of, 972–974
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
central, 991
double labeling of, 986
eccentricity of, 991
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
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
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:
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
under conformal mapping, 763
defined, 758
Laplace's equation, 593, 628–629
maximum modulus theorem, 783–784
potential theory, 781–784, 786
Heat equation, 459–460, 557–558
Laplace's equation, 564
numeric analysis, 936–941, 948
Crank–Nicolson method, 938–941
one-dimensional, 559
solution:
by Fourier transforms, 571–574
steady two-dimensional heat problems, 546–566
unifying power of methods, 566
Heat flow:
Laplace's equation, 593
Heat flow lines, 767
Heaviside, Oliver, 204n.1
Heaviside calculus, 204n.1
Heaviside expansions, 228
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
systems of, see Systems of ODEs
Higher order ODEs (numeric analysis), 915–922
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
with constant coefficients, 53–60
two distinct real-roots, 54–55
existence and uniqueness of solutions, 74–79
general solution, 49–51, 77–78
modeling free oscillations of mass–spring system, 62–70
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 functions, 635, 642
inverse, 640
Taylor series, 695
Hyperbolic PDEs:
defined, 923
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
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
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
Improper node, 142
Improved Euler's method:
Impulse, of a force, 225
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
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:
Infinity:
analytic of singular at, 718–719
point at, 718
Initial conditions:
higher-order linear ODEs:
homogeneous, 107
nonhomogeneous, 117
one-dimensional heat equation, 559
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
nonhomogeneous, 117
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
Input (driving force), 27, 86, 214
Instability, numeric vs. mathematical, 796
Integrals, see Line integrals
Integral equations:
defined, 236
Integral of a function, Laplace transforms of, 212–213
Integrating factors, 23–26, 45
defined, 24
Integration. See also Complex integration
constant of, 18
of Laplace transforms, 238–240
Gauss integration formulas, 836–838
rectangular rule, 828
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
Newton's backward difference formula, 818–819
Newton's divided difference, 812–815
Newton's forward difference formula, 815–818
Interpolation polynomial, 808, 842
Interquartile range, 1013
Intersection, of events, 1016, 1017
Intervals. See also Confidence intervals
class, 1012
closed, A72n.3
Interval estimates, 1065
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 of a matrix, 128, 301–309, 321
determinants of matrix products, 307–308
Gauss–Jordan method, 302–304, 856–857
Inverse sine, 640
Inverse tangent, 640
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
Iteration (iterative) methods:
fixed-point iteration, 798–801
Newton's (Newton–Raphson) method, 801–805
numeric linear algebra, 858–864, 898
Gauss–Seidel iteration, 858–862
IVP, see Initial value problem
Jacobi, Carl Gustav Jacob, 430n.3
Jordan, Wilhelm, 302n.3
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
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
Laguerre, Edmond, 504n.7
Laguerre polynomials, 241, 504
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
in cylindrical coordinates, 593–594
Fourier–Legendre series, 596–598
heat equation, 564
numeric analysis, 922–936, 948
Dirichlet problem, 925–928, 934–935
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
first shifting theorem (s-shifting), 208–209
general formulas, 248
initial value problems, 213–216
of integral of a function, 212–213
notation, 205
ODEs with variable coefficients, 240–241
partial differential equations, 600–603
second shifting theorem (t-shifting), 219–223
uniqueness, 210
unit step function (Heaviside function), 217–219
in cylindrical coordinates, 593–594
heat equation, 557
Laplace's equation, 593
in spherical coordinates, 594
of u in polar coordinates, 586
Laurent, Pierre Alphonse, 708n.1
analytic or singular at infinity, 718–719
point at infinity, 718
Riemann sphere, 718
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
Legendre, Adrien-Marie, 175n.1, 1103
Legendre function, 175
Legendre polynomials, 167, 177–179, 202
Legendre's equation, 167, 175–177, 201, 202
Leibniz, Gottfried Wilhelm, 15n.3
Leibniz test for real series, A73–A74
Length:
curves, 385
Leonardo of Pisa, 690n.2
Leontief, Wassily, 334n.1
Leontief input–output model, 334
Leslie model, 331
LFTs, see Linear fractional transformations
Libby, Willard Frank, 13n.2
Likelihood function, 1066
Limit (sequences), 672
Limit l, 378
Limit point, A93
Limit vector, 378
Linear algebra, 255. See also Numeric linear algebra
general properties of, 295–298
determinants of matrix products, 307–308
elementary row operations, 277
nonhomogeneous, 291
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
symmetric and skew-symmetric matrices, 267–268
triangular matrices, 268
matrix eigenvalue problems, 322–353
complex matrices and forms, 346–352
determining eigenvalues and
diagonalization of matrices, 341–342
orthogonal transformations, 336
symmetric and skew- symmetric matrices, 334–336
transformation to principal axes, 344
vector spaces:
linear transformations, 313–317
Linear combination:
homogeneous linear ODEs:
higher-order, 107
second-order, 48
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
scalar triple product, 373
Linear inequalities, 954
Linearity:
line integrals, 645
Linearity principle, see Superposition principle
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
homogeneous, 28
higher-order homogeneous, 105
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
degenerate feasible solution, 962–965
difficulties in starting, 965–968
Linear systems, 138–139, 165, 272–274, 320, 845
elementary row operations, 277
Gauss elimination, 274–280, 844–852
elementary row operations, 277
Gauss–Jordan elimination, 856–857
homogeneous, 138, 165, 272, 290–291
constant-coefficient systems, 140–151
condition number of a matrix, 868–870
vector norms, 866
Gauss–Seidel iteration, 858–882
of m equations in n unknowns, 272
nonhomogeneous, 138, 160–163, 272, 290, 291
Linear transformations, 320
motivation of multiplication by, 265–266
basic properties of, 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
Lines of constant revenue, 954
LINPACK, 789
Liouville, Joseph, 499n.4
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
Taylor series, 695
Logarithmic decrement, 70
Logarithmic integral, formula for, A69
Logarithm of base ten, formula for, A63
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 series, 690, 694–696
Main diagonal:
determinants, 294
Malthus, Thomas Robert, 5n.1
Maple, 789
Maple Computer Guide, 789
bijective, 737n.1
boundary value problems, 763–767, A96
defined, 738
geometry of analytic functions, 737–742
linear fractional transformations, 742–750
by trigonometric and hyperbolic analytic functions, 750–754
fixed points of, 745
of half-planes onto half-planes, 748
identity, 745
injective, 737n.1
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
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
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
comparison of means, 1084–1085
comparison of variances, 1086
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
for standard deviation, 1090
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
definitions and terms, 125–126, 128, 257
diagonal, 268
fundamental, 139
determinants of matrix products, 307–308
Gauss–Jordan method, 302–304, 856–857
matrix multiplication, 127, 263–266, 269–279
determinants of matrix
notation, 258
square, 126
symmetric and skew-symmetric, 267–268
triangular, 268
Matrix eigenvalue problems, 322–353, 876–896
choice of numeric method for, 879
complex matrices and forms, 346–352
determining eigenvalues and eigenvectors, 323–329
diagonalization of matrices, 341–342
orthogonal transformations, 336
symmetric and skew-symmetric
transformation to principal axes, 344
Matrix multiplication, 127, 263–266, 269–279
determinants of matrix products, 307–308
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
of normal distributions:
confidence intervals for, 1069–1073
hypothesis testing for, 1081–1084
probability distributions, 1035–1039
sample, 1064
Mean square convergence (orthogonal series), 507–508
Mean value (fluid flow), 774n.1
Mean value property:
harmonic functions, 782
Mean value theorem, 395
for double integrals, 427
for surface integrals, 448
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
satisfying boundary conditions, 546–548
two ODEs from wave equation, 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
Mixed type PDEs, 555
Mixing problems, 14
MLEs (maximum likelihood estimates), 1066–1067
Möbius, August Ferdinand, 447n.5
Möbius strip, 447
Möbius transformations, 743. See also Linear fractional transformations (LFTs)
Models, 2
defined, 2
initial value problem, 6
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
determinants of matrix products, 307–308
of power series, 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
Net flow, through cut set, 994–995
NETLIB, 789
Networks:
defined, 991
flow augmenting paths, 992–993
paths, 992
Neumann, Carl, 198n.7
Neumann, John von, 959n.1
Neumann boundary condition, 564
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
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:
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
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):
orthogonal functions, 500
Normal accelerations, 391
Normal acceleration vector, 387
Normal derivative, 437
defined, 437
Neumann problems, 931
solutions of Laplace's equation, 460
Normal distributions, 1045–1051, 1062
as approximation of binomial distribution, 1049–1050
confidence intervals:
distribution function, 1046–1047
means of:
confidence intervals for, 1069–1073
hypothesis testing for, 1081–1084
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 mode:
circular membrane, 588
Normal plane, 390
Normal random variables, 1045
Not rejecting a hypothesis, 1081
No trend hypothesis, 1101
n th order linear ODEs, 105, 123
n th partial sum, 170
Fourier series, 495
of series, 673
n th roots, 616
n th roots of unity, 617
Null hypothesis, 1078
Numbers:
acceptance, 1092
Bernoulli's law of large numbers, 1051
chromatic, 1006
conjugate, 612
defined, 608
division of, 610
subtraction of, 610
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
Newton's backward difference formula, 818–819
Newton's divided difference, 812–815
Newton's forward difference formula, 815–818
loss of significant digits, 793–794
numeric differentiation, 838–839
Gauss integration formulas, 836–838
rectangular rule, 828
solution of equations by iteration, 798–808
fixed-point iteration, 798–801
Newton's (Newton–Raphson) method, 801–805
Numeric differentiation, 838–839
Gauss integration formulas, 836–838
rectangular rule, 828
Numeric linear algebra, 844–899
linear systems, 845
Gauss–Jordan elimination, 856–857
ill-conditioning norms, 864–872
matrix eigenvalue problems, 876–896
Numeric methods:
defined, 791
n × n matrix, 125
Nyström, E. J., 919
OCs (operating characteristics), 1081
OC curve, see Operating characteristic curve
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 Leontief input–output model, 334
Open set, in complex plane, 620
Operating characteristic curve (OC curve), 1081, 1092, 1095
Operating characteristics (OCs), 1081
Operation count (Gauss elimination), 850
Optimal solutions (normal form of linear optimization problems), 957
Optimization:
assignment problems, 1001–1006
flow problems in networks, 991–997
Ford–Fulkerson algorithm for
shortest path problems, 975–980
constrained (linear), 951, 954–968
normal form of problems, 955–957
unconstrained:
method of steepest descent, 952–954
Optimization methods, 949
Optimization problems, 949, 954–958
normal form of problems, 955–957
objective, 951
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
orthogonal trajectories, 36–38
systems of, see Systems of ODEs
first shifting theorem (s-shifting), 208–209
general formulas, 248
initial value problems, 213–216
of integral of a function, 212–213
notation, 205
ODEs with variable coefficients, 240–241
partial differential equations, 600–603
second shifting theorem (t-shifting), 219–223
uniqueness, 210
unit step function (Heaviside function), 217–219
linear, 46
nonlinear, 46
second-order nonlinear, 46
series solutions of ODEs, 167–202
Bessel functions, 187–194, 196–200
conversion of n th-order ODEs to, 134–135
homogeneous, 138
linear, 124–130, 138–151, 160–163
as models of applications, 130–134
in phase plane, 124, 141–146, 157–159
qualitative methods for nonlinear systems, 152–160
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
mean square convergence, 507–508
Orthogonal trajectories:
defined, 36
Orthogonal transformations, 336, A85n.2
Orthogonal vectors, 312, 362, 410
Orthonormal functions, 500, 501, 508
Orthonormal system, 337
Oscillations:
second-order linear ODEs:
Outcomes:
probability theory, 1015
Outer normal derivative, 460, 931
Output (response to input), 27, 86, 214
Overdetermined linear systems, 277
Overflow (floating-point numbers), 792
Overrelaxation factor, 863
Pappus, theorem of, 452
Pappus of Alexandria, 452n.7
Parabolic PDEs:
defined, 923
Parallelogram law, 357
Parallel processing of products (on computer), 265
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
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
d'Alembert's solution, 553–556
defined, 540
double Fourier series solution, 577–585
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
unifying power of methods, 566
homogeneous, 541
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
satisfying boundary conditions, 546–548
two ODEs from wave equation, 545–546
nonhomogeneous, 541
nonlinear, 541
ODEs vs., 4
d'Alembert's solution, 553–556
solution by separating variables, 545–553
Partial fractions (Laplace transforms), 228–230
Partial pivoting, 276, 846–848, 898
Partial sums, of series, 477, 478, 495
Particular solution (s):
higher-order homogeneous linear ODEs, 106
nonhomogeneous linear systems, 160
second-order linear ODEs:
nonhomogeneous, 80
Partitioning, of a path, 645
Pascal, Blaise, 391n.4
Pascal, Étienne, 391n.4
Paths:
alternating, 1002
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
Stokes's Theorem applied to, 468
Pauli spin matrices, 351
PDEs, see Partial differential equations
Pearson, Egon Sharpe, 1077n.2
Period, 475
Periodic boundary conditions, 501
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
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
nonlinear systems, 152
Phase plane method, 124
linear systems:
nonlinear systems, 152
Lotka–Volterra population model, 155–156
transformation to first-order equation in, 157–159
Phase plane representations, 134
Phase portrait, 165
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 equation, 276, 846, 898, 960
Planar graphs, 1005
Plane:
complex, 611
finite, 718
sets in, 620
normal, 390
nonlinear systems, 152
rectifying, 390
vectors in, 309
Plane curves, 383
Planimeters, 436
Poincaré, Henri, 141n.1, 510n.8
Points:
branch, 755
asymptotically stable, 149
and conformal mapping, 738, 757
constant-coefficient systems of
isolated, 152
nonlinear systems, 152
stable and attractive, 140, 149
guidepoints, 827
at infinity, 718
initial (vectors), 355
limit, A93
regular, 181
regular singular, 180n.4
sample, 1015
analytic functions, 693
regular, 180n.4
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
mixed boundary value problem, 931–933
Poisson's integral formula:
series for potentials in disks, 779–780
Polar coordinates, 431
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
Chebyshev, 504
orthogonal, 179
trigonometric:
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
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
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
Poisson's integral formula, 777–781
Power function, of a test, 1081, 1113
Power method (matrix eigenvalue problems), 885–888, 899
convergence behavior of, 680–682
convergence tests, 674–676, A93–A94
Maclaurin series, 690
in powers of x, 168
radius of convergence, 682–684
and absolute convergence, 704
Power series method, 167–175, 201
extension of, see Frobenius method
idea and technique of, 168–170
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
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
independent events, 1023
Probability distributions, 1029, 1061
mean and variance of, 1035–1039
multinomial, 1045
of several random variables, 1051–1060
addition of variances, 1058–1059
continuous two-dimensional
distributions, 1053
discrete two-dimensional
function of random variables, 1056
independence of random
marginal distributions, 1053–1055
symmetric, 1036
two-dimensional, 1051
continuous, 1053
Probability function, 1030–1032, 1052, 1061
Probability theory, 1009, 1015–1062
binomial coefficients, 1027–1028
distributions (probability distributions), 1029
mean and variance of, 1035–1039
of several random variables, 1051–1060
factorial function, 1027
outcomes, 1015
probability:
independent events, 1023
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
inverting, 306
matrix multiplication, 263, 320
parallel processing of (on computer), 265
scalar multiplication, 260
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
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
Lotka–Volterra population model, 155–156
transformation to first-order equation in phase plane, 157–159
Quality control (statistics), 1087–1092, 1113
for standard deviation, 1090
Quantitative methods, 124
Quasilinear equations, 555, 923
Quotient:
complex numbers, 610
difference, 923
Radius:
of convergence, 172
defined, 172
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
function of, 1056
marginal distribution of, 1054, 1055
normal, 1045
occurrence of, 1063
probability distributions of, 1051–1060
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
Random variation, 1063
Range, 1013
defined, 1090
of f, 620
Rank:
of A, 279
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
Reactance (RLC circuits), 94
Real axis (complex plane), 611
Real different roots, 71
Real functions, complex analytic functions vs., 694
Real inner product space, 312
Real integrals, residue integration of, 725–733
of rational functions of cos θ sin θ, 725–729
Real part (complex numbers), 609
Real pre-Hilbert space, 312
Real roots:
different, 71
higher-order homogeneous linear ODEs:
second-order homogeneous linear ODEs:
Real sequence, 671
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
Representation, 315
by Fourier series, 476
by power series, 683
spectral, 525
at m th-order pole, 722
formulas for residues, 721–722
of rational functions of cos θ sin θ, 725–729
several singularities inside contour, 723–725
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
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
Roots:
complex:
higher-order homogeneous linear ODEs, 113–115
second-order homogeneous linear ODEs, 57–59
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
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
Row-equivalent matrices, 283–284
Row-equivalent systems, 277
Row operations (linear systems), 276, 277
Row scaling (Gauss elimination), 850
Row “sum” norm, 861
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
Runge–Kutta (RK) methods, 915, 947
error of, 908
Runge–Kutta–Nyström (RKN) methods, 919–921, 947
Rutherford–Geiger experiments, 1044, 1100
Rutishauser, Heinz, 892n.12
Samples:
for experiments, 1015
in mathematical statistics, 1063–1064
Sample covariance, 1105
Sampled function, 529
Sample distribution function, 1096
Sample points, 1015
Sample regression line, 1104
Sample space, 1015, 1016, 1060
Sample standard deviation, 1065
Sampling:
from a population, 1023
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
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
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
with constant coefficients, 53–60
existence and uniqueness of solutions, 74–79
general solution, 49–51, 77–78
modeling free oscillations of mass–spring system, 62–70
superposition principle, 47–48
defined, 47
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 (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 ODEs, 44
reduction of nonseparable ODEs to, 17–18
Separating variables, method of, 12–13
circular membrane, 587
partial differential equations, 545–553, 605
satisfying boundary conditions, 546–548
two ODEs from wave equation, 545–546
Separation constant, 546
Sequences (infinite sequences):
divergent, 672
limit point of, A93
real, 671
binomial, 696
conditionally convergent, 675
cosine, 781
derived, 687
double Fourier:
defined, 582
convergence and sum or, 480–481
derivation of Euler formulas, 479–480
even and odd functions, 486–488
half-range expansions, 488–490
Fourier–Legendre, 505–506, 596–598
one-dimensional heat equation, 561
vibrating string, 548
Taylor series, 694
uniformly convergent, 698
hypergeometric, 186
analytic or singular at infinity, 718–719
point at infinity, 718
Riemann sphere, 718
zeros of analytic functions, 717
mean square convergence, 507–508
convergence behavior of, 680–682
convergence tests, 674–676, A93–A94
Maclaurin series, 690
in powers of x, 168
radius of convergence, 682–684
Series solutions of ODEs, 167–202
Bessel functions, 187–188, 196–200
idea and technique of, 168–170
Sets:
complete orthonormal, 508
in the complex plane, 620
linearly independent, 128–129, 311
Shewhart, W. A., 1088
Shifted function, 219
Shortest path, 976
Shortest path problems (combinatorial optimization), 975–980, 1008
complexity of algorithms, 978–980
Moore's BFS algorithm, 977–980
Shortest spanning trees:
combinatorial optimization, 1008
defined, 984
Short impulses (Laplace transforms), 225–226
Sifting property, 226
Significance (in statistics), 1078
Significance level, 1078, 1080, 1113
Significance tests, 1078
Similarity transformation, 340
Similar matrices, 340–341, 878
Simple closed curves, 646
Simple closed path, 652
Simple curves, 383
Simple events, 1015
Simple general properties of the line integral, 415–416
Simple poles, 714
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
adaptive integration with, 835–836
Simultaneous corrections, 862
Sine function:
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
isolated, 715
isolated essential, 715
principal part of, 708
removable, 717
Singular matrices, 301
analytic functions, 693
regular, 180n.4
Singular solutions:
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
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
Slope field (direction field), 9–10
Smooth surfaces, 442
Sobolev, Sergei L'Vovich, 226n.2
Software:
for data representation in statistics, 1011
variable step size selection in, 902
Solenoid, 405
Solutions. See also specific methods defined, 4, 798
first-order ODEs:
explicit solutions, 21
family of solutions, 5
implicit solutions, 21
solution by calculus, 5
graphing in phase plane, 141–142
higher-order homogeneous linear ODEs, 106
general solution, 106, 110–111
particular solution, 106
singular solution, 110
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
second-order linear ODEs, 47
second-order nonhomogeneous linear ODEs:
particular solution, 80
Solution space, 290
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
Sparse graphs, 974
Sparse systems, 858
theory of, 175
Special vector spaces, 285–287
Specific circulation, of flow, 467
Spectral density, 525
Spectral mapping theorem, 878
Spectral representation, 525
Spectral shift, 896
Spectrum, 877
of matrix, 324
vibrating string, 547
angular (rotation), 372
Spherical coordinates, A74–A76
boundary value problem in, 594–596
defined, 594
Laplacian in, 594
Spring constant, 62
Square matrices, 126, 257, 258, 301–309, 320
Stability:
of critical points, 165
Stability chart, 149
Stable and attractive critical points, 140, 149
Stable critical points, 140, 149
Stable equilibrium solution, 33–34
Stable systems, 84
Stagnation points, 773
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 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
Stereographic projection, 718
Stirling, James, 1027n.2
Stochastic matrices, 270
Stochastic variables, 1029. See also Random variables
Stokes, Sir George Gabriel, 464n.9, 703n.5
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
Subgraphs, 972
Submarine cable equations, 599
Submatrices, 288
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 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
second-order homogeneous linear ODEs, 47–48, 104
undamped forced oscillations, 87
Surfaces, for surface integrals, 439–443
representation of surfaces, 439–441
tangent plane and surface normal, 441–442
Surface integrals, 470
defined, 443
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 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
critical points, 142–146, 148–151
graphing solutions in phase plane, 141–142
conversion of nth-order ODEs to, 134–135
homogeneous, 138
linear, 138–139. See also Linear systems
constant-coefficient systems, 140–151
definitions and terms, 125–126, 128–129
eigenvalues and eigenvectors, 129–130
systems of ODEs as vector equations, 127–128
as models of applications:
mixing problem involving two tanks, 130–132
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
graphing solutions in, 141–142
transformation to first-order equation in, 157–159
qualitative methods for nonlinear systems, 152–160
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
Target (networks), 991
Taylor, Brook, 690n.2
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 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
components of a vector, 356–357
scalar multiplication, 358–359
Three-sigma limits, 1047
Time (curves in mechanics), 386
TI-Nspire, 789
Todd, John, 855n.3
Tolerance (adaptive integration), 835
Torricelli, Evangelista, 16n.4
Torsion, curvature and, 389–390
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
Euler trail, 980
nonlinear systems, 152
Transcendental equations, 798
Transducers, 98
Transfer function, 214
Transformation (s), 313
orthogonal, 336
to principal axes, 344
Transient-state solution, 31
Translation (vectors), 355
Transposition (s):
of matrices or vectors, 128, 320
in samples, 1101
error bounds and estimate for, 829–831
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
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:
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
homogeneous linear systems, 290
linear systems, 273
Sturm–Liouville problem, 499
Truncating, 794
Tuning (vibrating string), 548
Twisted curves, 383
2-space (plane), vectors in, 354
components of a vector, 356–357
scalar multiplication, 358–359
2 × 2 matrix, 125
Two-dimensional heat equation, 564–566
Two-dimensional normal distribution, 1110
Two-dimensional probability distributions:
continuous, 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
UCL (upper control limit), 1088
Unacceptable lots, 1094
Unconstrained optimization, 969
method of steepest descent, 952–954
Uncorrelated related variables, 1109
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
Uniform convergence:
and absolute convergence, 704
properties of uniform convergence, 700–701
Uniform distributions, 1035–1036, 1053
Unifying power of mathematics, 97
Uniqueness:
of Laplace transforms, 210
of Laurent series, 712
of power series representation, 685–686
problem of, 39
Uniqueness theorems:
cubic splines, 822
higher-order homogeneous linear ODEs, 108
Laplace's equation, 462
linear systems, 138
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 impulse function, 226. See also Dirac delta function
Unit principal normal vector, 389
Unit step function (Heaviside function), 217–219
Unit tangent vector, 384
Universal gravitational constant, 63
Unknowns, 257
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
Variables:
basic, 960
control, 951
controlled, 1103
intermediate, 393
linearly, 1109
nonbasic, 960
continuous, 1029, 1032–1034, 1055
defined, 1030
dependent, 1055
function of, 1056
marginal distribution of, 1054, 1055
normal, 1045
occurrence of, 1063
probability distributions of, 1051–1060
skewness of, 1039
standardized, 1037
stochastic, 1029
uncorrelated related, 1109
Variable coefficients:
Laplace transforms ODEs with, 240–241
idea and technique of, 168–170
second-order homogeneous linear ODEs, 73
comparison of, 1086
equality of, 1084n.3
of normal distributions, confidence intervals for, 1073–1076
of probability distributions, 1035–1039
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
addition and scalar multiplication of, 259–261
definitions and terms, 126, 128–129, 257, 259, 309
linear independence and dependence of, 282–283
multiplying matrices by, 263–265
systems of ODEs as vector equations, 127–128
differential, see Vector differential calculus
integral, see Vector integral calculus
Vector differential calculus, 354–412
length of, 385
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
scalar functions, 376
that are gradients of scalar fields, 400–401
partial derivatives of, 380
vector product (cross product), 368–375
scalar triple product, 373–374
vectors in 2-space and 3-space:
components of a vector, 356–357
scalar multiplication, 358–359
Vector fields:
defined, 376
vector differential calculus, 377–378
that are gradients of scalar fields, 400–401
Vector functions:
differentiable, 379
divergence theorem of Gauss, 453–457
mean value theorem, 395
vector differential calculus, 375–376, 411
partial derivatives of, 380
Vectors in 2-space and 3-space:
components of a vector, 356–357
scalar multiplication, 358–359
Vector integral calculus, 413–471
divergence theorem of Gauss, 453–463
change of variables in, 429–431
evaluation of, by two successive integrations, 427–428
Green's theorem in the plane, 433–438
definition and evaluation of, 414–416
path dependence of line integrals, 418–426
defined, 418
and integration around closed curves, 421–425
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
Vector moment, 371
Vector norms, 866
Vector product (cross product):
in Cartesian coordinates, A83–A84
vector differential calculus, 368–375, 410
scalar triple product, 373–374
Vector spaces, 482
linear transformations, 313–317
Velocity potential, 771
Venn, John, 1017n.1
Venn diagrams, 1017
Verhulst, Pierre-François, 32n.8
Vertices (graphs), 971, 977, 1007
central, 991
double labeling of, 986
eccentricity of, 991
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
d'Alembert's solution, 553–556
numeric analysis, 942–944, 948
solution by separating variables, 545–553
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 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
z-score, 1014