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 |  موضوع: كتاب Advanced Engineering Mathematics 10th Edition الجمعة 18 أغسطس 2017, 10:03 pm | |
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أخواني في الله
أحضرت لكم كتاب
Advanced Engineering Mathematics 10ed
Erwin Kreyszig
Professor of Mathematics
Ohio State University
Columbus, Ohio
In Collaboration With
Herbert Kreyszig
New York, New York
Edward J. Norminton
Associate Professor of Mathematics
Carleton University
Ottawa, Ontario
و المحتوى كما يلي :
C O N T E N T S
P A R T A Ordinary Differential Equations (ODEs)
CHAPTER First-Order ODEs
. Basic Concepts. Modeling
. Geometric Meaning of y ƒ(x, y). Direction Fields, Euler’s Method
. Separable ODEs. Modeling
. Exact ODEs. Integrating Factors
. Linear ODEs. Bernoulli Equation. Population Dynamics
. Orthogonal Trajectories. Optional
. Existence and Uniqueness of Solutions for Initial Value Problems
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Second-Order Linear ODEs
. Homogeneous Linear ODEs of Second Order
. Homogeneous Linear ODEs with Constant Coefficients
. Differential Operators. Optional
. Modeling of Free Oscillations of a Mass–Spring System
. Euler–Cauchy Equations
. Existence and Uniqueness of Solutions. Wronskian
. Nonhomogeneous ODEs
. Modeling: Forced Oscillations. Resonance
. Modeling: Electric Circuits
. Solution by Variation of Parameters
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Higher Order Linear ODEs
. Homogeneous Linear ODEs
. Homogeneous Linear ODEs with Constant Coefficients
. Nonhomogeneous Linear ODEs
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Systems of ODEs. Phase Plane. Qualitative Methods
. For Reference: Basics of Matrices and Vectors
. Systems of ODEs as Models in Engineering Applications
. Basic Theory of Systems of ODEs. Wronskian
. Constant-Coefficient Systems. Phase Plane Method
. Criteria for Critical Points. Stability
. Qualitative Methods for Nonlinear Systems
. Nonhomogeneous Linear Systems of ODEs
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Series Solutions of ODEs. Special Functions
. Power Series Method
. Legendre’s Equation. Legendre Polynomials Pn(x) . Extended Power Series Method: Frobenius Method
. Bessel’s Equation. Bessel Functions J(x)
. Bessel Functions of the Y
(x). General Solution
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Laplace Transforms
. Laplace Transform. Linearity. First Shifting Theorem (s-Shifting)
. Transforms of Derivatives and Integrals. ODEs
. Unit Step Function (Heaviside Function).
Second Shifting Theorem (t-Shifting)
. Short Impulses. Dirac’s Delta Function. Partial Fractions
. Convolution. Integral Equations
. Differentiation and Integration of Transforms.
ODEs with Variable Coefficients
. Systems of ODEs
. Laplace Transform: General Formulas
. Table of Laplace Transforms
Chapter Review Questions and Problems
Summary of Chapter
P A R T B Linear Algebra. Vector Calculus
CHAPTER Linear Algebra: Matrices, Vectors, Determinants.
Linear Systems
. Matrices, Vectors: Addition and Scalar Multiplication
. Matrix Multiplication
. Linear Systems of Equations. Gauss Elimination
. Linear Independence. Rank of a Matrix. Vector Space
. Solutions of Linear Systems: Existence, Uniqueness
. For Reference: Second- and Third-Order Determinants
. Determinants. Cramer’s Rule
. Inverse of a Matrix. Gauss–Jordan Elimination
. Vector Spaces, Inner Product Spaces. Linear Transformations. Optional
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Linear Algebra: Matrix Eigenvalue Problems
. The Matrix Eigenvalue Problem.
Determining Eigenvalues and Eigenvectors
. Some Applications of Eigenvalue Problems
. Symmetric, Skew-Symmetric, and Orthogonal Matrices
. Eigenbases. Diagonalization. Quadratic Forms
. Complex Matrices and Forms. Optional
Chapter Review Questions and Problems
Summary of Chapter
xvi ContentsCHAPTER Vector Differential Calculus. Grad, Div, Curl
. Vectors in -Space and -Space
. Inner Product (Dot Product)
. Vector Product (Cross Product)
. Vector and Scalar Functions and Their Fields. Vector Calculus: Derivatives
. Curves. Arc Length. Curvature. Torsion
. Calculus Review: Functions of Several Variables. Optional
. Gradient of a Scalar Field. Directional Derivative
. Divergence of a Vector Field
. Curl of a Vector Field
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Vector Integral Calculus. Integral Theorems
. Line Integrals
. Path Independence of Line Integrals
. Calculus Review: Double Integrals. Optional
. Green’s Theorem in the Plane
. Surfaces for Surface Integrals
. Surface Integrals
. Triple Integrals. Divergence Theorem of Gauss
. Further Applications of the Divergence Theorem
. Stokes’s Theorem
Chapter Review Questions and Problems
Summary of Chapter
P A R T C Fourier Analysis. Partial Differential Equations (PDEs)
CHAPTER Fourier Analysis
. Fourier Series
. Arbitrary Period. Even and Odd Functions. Half-Range Expansions
. Forced Oscillations
. Approximation by Trigonometric Polynomials
. Sturm–Liouville Problems. Orthogonal Functions
. Orthogonal Series. Generalized Fourier Series
. Fourier Integral
. Fourier Cosine and Sine Transforms
. Fourier Transform. Discrete and Fast Fourier Transforms
. Tables of Transforms
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Partial Differential Equations (PDEs)
. Basic Concepts of PDEs
. Modeling: Vibrating String, Wave Equation
. Solution by Separating Variables. Use of Fourier Series
. D’Alembert’s Solution of the Wave Equation. Characteristics
. Modeling: Heat Flow from a Body in Space. Heat Equation
Contents xvii . Heat Equation: Solution by Fourier Series.
Steady Two-Dimensional Heat Problems. Dirichlet Problem
. Heat Equation: Modeling Very Long Bars.
Solution by Fourier Integrals and Transforms
. Modeling: Membrane, Two-Dimensional Wave Equation
. Rectangular Membrane. Double Fourier Series
. Laplacian in Polar Coordinates. Circular Membrane. Fourier–Bessel Series
. Laplace’s Equation in Cylindrical and Spherical Coordinates. Potential
. Solution of PDEs by Laplace Transforms
Chapter Review Questions and Problems
Summary of Chapter
P A R T D Complex Analysis
CHAPTER Complex Numbers and Functions.
Complex Differentiation
. Complex Numbers and Their Geometric Representation
. Polar Form of Complex Numbers. Powers and Roots
. Derivative. Analytic Function
. Cauchy–Riemann Equations. Laplace’s Equation
. Exponential Function
. Trigonometric and Hyperbolic Functions. Euler’s Formula
. Logarithm. General Power. Principal Value
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Complex Integration
. Line Integral in the Complex Plane
. Cauchy’s Integral Theorem
. Cauchy’s Integral Formula
. Derivatives of Analytic Functions
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Power Series, Taylor Series
. Sequences, Series, Convergence Tests
. Power Series
. Functions Given by Power Series
. Taylor and Maclaurin Series
. Uniform Convergence. Optional
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Laurent Series. Residue Integration
. Laurent Series
. Singularities and Zeros. Infinity
. Residue Integration Method
. Residue Integration of Real Integrals
Chapter Review Questions and Problems
Summary of Chapter
xviii ContentsCHAPTER Conformal Mapping
. Geometry of Analytic Functions: Conformal Mapping
. Linear Fractional Transformations (Möbius Transformations)
. Special Linear Fractional Transformations
. Conformal Mapping by Other Functions
. Riemann Surfaces. Optional
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Complex Analysis and Potential Theory
. Electrostatic Fields
. Use of Conformal Mapping. Modeling
. Heat Problems
. Fluid Flow
. Poisson’s Integral Formula for Potentials
. General Properties of Harmonic Functions.
Uniqueness Theorem for the Dirichlet Problem
Chapter Review Questions and Problems
Summary of Chapter
P A R T E Numeric Analysis
Software
CHAPTER Numerics in General
. Introduction
. Solution of Equations by Iteration
. Interpolation
. Spline Interpolation
. Numeric Integration and Differentiation
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Numeric Linear Algebra
. Linear Systems: Gauss Elimination
. Linear Systems: LU-Factorization, Matrix Inversion
. Linear Systems: Solution by Iteration
. Linear Systems: Ill-Conditioning, Norms
. Least Squares Method
. Matrix Eigenvalue Problems: Introduction
. Inclusion of Matrix Eigenvalues
. Power Method for Eigenvalues
. Tridiagonalization and QR-Factorization
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Numerics for ODEs and PDEs
. Methods for First-Order ODEs
. Multistep Methods
. Methods for Systems and Higher Order ODEs
Contents xix . Methods for Elliptic PDEs
. Neumann and Mixed Problems. Irregular Boundary
. Methods for Parabolic PDEs
. Method for Hyperbolic PDEs
Chapter Review Questions and Problems
Summary of Chapter
P A R T F Optimization, Graphs
CHAPTER Unconstrained Optimization. Linear Programming
. Basic Concepts. Unconstrained Optimization: Method of Steepest Descent
. Linear Programming
. Simplex Method
. Simplex Method: Difficulties
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Graphs. Combinatorial Optimization
. Graphs and Digraphs
. Shortest Path Problems. Complexity
. Bellman’s Principle. Dijkstra’s Algorithm
. Shortest Spanning Trees: Greedy Algorithm
. Shortest Spanning Trees: Prim’s Algorithm
. Flows in Networks
. Maximum Flow: Ford–Fulkerson Algorithm
. Bipartite Graphs. Assignment Problems
Chapter Review Questions and Problems
Summary of Chapter
P A R T G Probability, Statistics
Software
CHAPTER Data Analysis. Probability Theory
. Data Representation. Average. Spread
. Experiments, Outcomes, Events
. Probability
. Permutations and Combinations
. Random Variables. Probability Distributions
. Mean and Variance of a Distribution
. Binomial, Poisson, and Hypergeometric Distributions
. Normal Distribution
. Distributions of Several Random Variables
Chapter Review Questions and Problems
Summary of Chapter
CHAPTER Mathematical Statistics
. Introduction. Random Sampling
. Point Estimation of Parameters
. Confidence Intervals
xx Contents . Testing Hypotheses. Decisions
. Quality Control
. Acceptance Sampling
. Goodness of Fit. -Test
. Nonparametric Tests
. Regression. Fitting Straight Lines. Correlation
Chapter Review Questions and Problems
Summary of Chapter
APPENDIX References A
APPENDIX Answers to Odd-Numbered Problems A
APPENDIX Auxiliary Material A
A . Formulas for Special Functions A
A . Partial Derivatives A
A . Sequences and Series A
A . Grad, Div, Curl, in Curvilinear Coordinates A
APPENDIX Additional Proofs A
APPENDIX Tables A
INDEX I
I N D E X
Abel, Niels Henrik, n.
Abel’s formula,
Absolute convergence (series):
defined,
and uniform convergence,
Absolute frequency (probability):
of an event,
cumulative,
of a value,
Absolutely integrable nonperiodic
function, –
Absolute value (complex numbers),
Acceleration, –
Acceleration of gravity,
Acceleration vector,
Acceptable lots,
Acceptable quality level (AQL),
Acceptance:
of a hypothesis,
of products,
Acceptance number,
Acceptance sampling, – ,
errors in, –
rectification, –
Adams, John Couch, n.
Adams–Bashforth methods, – ,
Adams–Moulton methods, – ,
Adaptive integration, – ,
Addition:
for arbitrary events, –
of complex numbers, ,
of matrices and vectors, ,
–
of means, –
for mutually exclusive events,
of power series,
termwise, ,
of variances, –
vector, , –
ADI (alternating direction implicit)
method, –
Adjacency matrix:
of a digraph,
of a graph, –
Adjacent vertices, ,
Airy, Sir George Bidell, n. ,
n.
Airy equation,
RK method, –
RKN method, –
Airy function:
RK method, –
RKN method, –
Algebraic equations,
Algebraic multiplicity, ,
Algorithms:
complexity of, –
defined,
numeric analysis,
numeric methods as,
numeric stability of, ,
ALGORITHMS:
BISECT, A
DIJKSTRA,
EULER,
FORD–FULKERSON,
GAUSS,
GAUSS–SEIDEL,
INTERPOL,
KRUSKAL,
MATCHING,
MOORE,
NEWTON,
PRIM,
RUNGE–KUTTA,
SIMPSON,
Aliasing,
Alternating direction implicit (ADI)
method, –
Alternating path,
Alternative hypothesis,
Ampère, André Marie, n.
Amplification,
Amplitude,
Amplitude spectrum,
Analytic functions, , ,
complex analysis, –
conformal mapping, –
derivatives of, – , – ,
A –A
integration of:
indefinite,
by use of path, –
Analytic functions (Cont.)
Laurent series:
analytics at infinity, –
zeros of, –
maximum modulus theorem,
–
mean value property, –
power series representation of,
–
real functions vs.,
Analyticity,
Angle of intersection:
conformal mapping,
between two curves,
Angular speed (rotation),
Angular velocity (fluid flow),
AOQ (average outgoing quality),
AOQL (average outgoing quality
limit),
Apparent resistance (RLC circuits),
Approximation(s):
errors involved in,
polynomial,
by trigonometric polynomials,
–
Approximation theory,
A priori estimates,
AQL (acceptable quality level),
Arbitrary positive,
Arc, of a curve,
Archimedes, n.
Arc length (curves), –
Area:
of a region,
of region bounded by ellipses,
of a surface, –
Argand, Jean Robert, n.
Argand diagram, n.
Argument (complex numbers),
Artificial variables, –
Assignment problems (combinatorial
optimization), –
Associative law,
Asymptotically equal, , ,
Asymptotically normal, I Index
Asymptotically stable critical points,
Augmented matrices, , , ,
, ,
Augmenting path, – . See
also Flow augmenting paths
Autonomous ODEs, ,
Autonomous systems, ,
Auxiliary equation, . See also
Characteristic equation
Average flow,
Average outgoing quality (AOQ),
Average outgoing quality limit
(AOQL),
Axioms of probability,
Back substitution (linear systems),
– ,
Backward edges:
cut sets,
initial flow,
of a path,
Backward Euler formula,
Backward Euler method (BEM):
first-order ODEs, –
stiff systems, –
Backward Euler scheme,
Balance law,
Band matrices,
Bashforth, Francis, n.
Basic feasible solution:
normal form of linear optimization
problems,
simplex method,
Basic Rule (method of undetermined
coefficients):
higher-order homogeneous linear
ODEs,
second-order nonhomogeneous
linear ODEs, ,
Basic variables,
Basis:
eigenvectors, –
of solutions:
higher-order linear ODEs, ,
,
homogeneous linear systems,
homogeneous ODEs, – ,
, , ,
second-order homogeneous
linear ODEs, – , ,
systems of ODEs,
standard,
vector spaces, , ,
Beats (oscillation),
Bellman, Richard, n.
Bellman equations,
Bellman’s principle, –
Bell-shaped curve, ,
BEM, see Backward Euler method
Benoulli, Niklaus, n.
Bernoulli, Daniel, n.
Bernoulli, Jakob, n.
Bernoulli, Johann, n.
Bernoulli distribution, . See also
Binomial distributions
Bernoulli equation,
defined,
linear ODEs, –
Bernoulli’s law of large numbers,
Bessel, Friedrich Wilhelm, n.
Bessel functions, , – ,
of the first kind, –
with half-integer v, –
of order ,
of order v,
orthogonality of,
of the second kind:
general solution, –
of order v, –
table, A –A
of the third kind,
Bessel’s equation, , – ,
Bessel functions, , – ,
–
circular membrane,
general solution, –
Bessel’s inequality:
for Fourier coefficients,
orthogonal series, –
Beta function, formula for, A
Bezier curve,
BFS algorithms, see Breadth First
search algorithms
Bijective mapping, n.
Binomial coefficients:
Newton’s forward difference
formula,
probability theory, –
Binomial distributions, – ,
normal approximation of,
–
sampling with replacement for,
table, A
Binomial series,
Binomial theorem,
Bipartite graphs, – ,
BISECT, ALGORITHM, A
Bisection method, –
Bolzano, Bernard, A n.
Bolzano–Weierstrass theorem,
A –A
Bonnet, Ossian, n.
Bonnet’s recursion,
Borda, J. C., n.
Boundaries:
ODEs,
of regions, n.
sets in complex plane,
Boundary conditions:
one-dimensional heat equation,
PDEs, ,
periodic,
two-dimensional wave equation,
vibrating string, –
Boundary points, n.
Boundary value problem (BVP),
conformal mapping for, – ,
A
first, see Dirichlet problem
mixed, see Mixed boundary value
problem
second, see Neumann problem
third, see Mixed boundary value
problem
two-dimensional heat equation,
Bounded domains,
Bounded regions, n.
Bounded sequence, A –A
Boxplots,
Boyle, Robert, n.
Boyle–Mariotte’s law for idea gases,
Bragg, Sir William Henry, n.
Bragg, Sir William Lawrence, n.
Branch, of logarithm,
Branch cut, of logarithm,
Branch point (Riemann surfaces),
Breadth First search (BFS)
algorithms,
defined, ,
Moore’s, –
BVP, see Boundary value problem
CAD (computer-aided design),
Cancellation laws, –
Canonical form,
Cantor, Georg, A n.
Cantor–Dedekind axiom, A n. ,
A n.
Capacity:
cut sets,
networks,
Cardano, Girolamo, n.
Cardioid, , Index I
Cartesian coordinates:
linear element in, A
transformation law, A –A
vector product in, A –A
writing, A
Cartesian coordinate systems:
complex plane,
left-handed, , , A
right-handed, – , A –A
in space, ,
transformation law for vector
components, A –A
Cartesius, Renatus, n.
Cauchy, Augustin-Louis, n. ,
n. , n.
Cauchy determinant,
Cauchy–Goursat theorem, see
Cauchy’s integral theorem
Cauchy–Hadamard formula,
Cauchy principal value, ,
Cauchy–Riemann equations, ,
complex analysis, –
proof of, A –A
Cauchy–Schwarz inequality, ,
–
Cauchy’s convergence principle,
– , A –A
Cauchy’s inequality,
Cauchy’s integral formula, – ,
Cauchy’s integral theorem, – ,
existence of indefinite integral,
–
Goursat’s proof of, A –A
independence of path,
for multiply connected domains,
–
principle of deformation of path,
Cayley, Arthur, n.
c-charts,
Center:
as critical point, ,
of a graph,
of power series,
Center control line (CL),
Center of gravity, of mass in a
region,
Central difference notation,
Central limit theorem,
Central vertex,
Centrifugal force,
Centripetal acceleration, –
Chain rules, –
Characteristics,
Characteristics, method of,
Characteristic determinant, of a
matrix, , , , ,
Characteristic equation:
matrices, , , , ,
PDEs,
second-order homogeneous linear
ODEs,
Characteristic matrix,
Characteristic polynomial, , ,
Characteristic values, , , .
See also Eigenvalues
Characteristic vectors, , . See
also eigenvectors
Chebyshev, Pafnuti, n.
Chebyshev equation,
Chebyshev polynomials,
Checkerboard pattern (determinants),
Chi-square ( ) distribution,
– , A
Chi-square ( ) test, – ,
Choice of numeric method, for matrix
eigenvalue problems,
Cholesky, André-Louis, n.
Cholesky’s method, – ,
Chopping, error caused by,
Chromatic number,
Circle,
Circle of convergence (power series),
Circulation, of flow, ,
CL (center control line),
Clairaut equation,
Clamped condition (spline
interpolation),
Class intervals,
Class marks,
Closed annulus,
Closed circular disk,
Closed integration formulas, ,
Closed intervals, A n.
Closed Newton–Cotes formulas,
Closed paths, , , –
Closed regions, n.
Closed sets,
Closed trails, –
Closed walks, –
CN (Crank–Nicolson) method,
–
Coefficients:
binomial:
Newton’s forward difference
formula,
probability theory, –
constant:
higher-order homogeneous
linear ODEs, –
second-order homogeneous
linear ODEs, –
Coefficients: (Cont.)
second-order nonhomogeneous
linear ODEs,
systems of ODEs, –
correlation, – ,
Fourier, , , , –
of kinetic friction,
of linear systems, ,
of ODEs,
higher-order homogeneous
linear ODEs,
second-order homogeneous
linear ODEs, – ,
second-order nonhomogeneous
linear ODEs, –
series of ODEs, ,
variable, , –
of power series,
regression, , –
variable:
Frobenius method, –
Laplace transforms ODEs
with, –
of ODEs, , –
power series method, –
second-order homogeneous
linear ODEs,
Coefficient matrices, ,
Hermitian or skew-Hermitian
forms,
linear systems,
quadratic form,
Cofactor (determinants),
Collatz, Lothar, n.
Collatz inclusion theorem, –
Columns:
determinants,
matrix, , ,
Column “sum” norm,
Column vectors,
matrices, , – ,
rank in terms of, –
Combinations (probability theory),
, –
of n things taken k at a time
without repetitions,
of n things taken k at a time with
repetitions,
Combinatorial optimization, ,
–
assignment problems, –
flow problems in networks,
–
cut sets, –
flow augmenting paths,
–
paths,
Ford–Fulkerson algorithm for
maximum flow, – I Index
Combinatorial optimization (Cont.)
shortest path problems, –
Bellman’s principle, –
complexity of algorithms,
–
Dijkstra’s algorithm, –
Moore’s BFS algorithm,
–
shortest spanning trees:
Greedy algorithm, –
Prim’s algorithm, –
Commutation (matrices),
Complements:
of events,
of sets in complex plane,
Complementation rule,
–
Complete bipartite graphs,
Complete graphs,
Complete matching,
Completeness (orthogonal series),
–
Complete orthonormal set,
Complex analysis,
analytic functions, –
Cauchy–Riemann equations,
–
circles and disks,
complex functions, –
exponential, –
general powers, –
hyperbolic,
logarithm, –
trigonometric, –
complex integration, –
Cauchy’s integral formula,
– ,
Cauchy’s integral theorem,
– ,
derivatives of analytic
functions, –
Laurent series, –
line integrals, – ,
power series, –
residue integration, –
complex numbers, –
addition of, ,
conjugate,
defined,
division of,
multiplication of, ,
polar form of, –
subtraction of,
complex plane,
conformal mapping, –
geometry of analytic functions,
–
linear fractional
transformations,
–
Complex analysis (Cont.)
Riemann surfaces, –
by trigonometric and
hyperbolic analytic
functions, –
half-planes, –
harmonic functions, –
Laplace’s equation, –
Laurent series, – ,
analytic or singular at infinity,
–
point at infinity,
Riemann sphere,
singularities, –
zeros of analytic functions,
power series, , –
convergence behavior of,
–
convergence tests, – ,
A –A
functions given by, –
Maclaurin series,
in powers of x,
radius of convergence,
–
ratio test, –
root test, –
sequences, –
series, –
Taylor series, –
uniform convergence,
–
residue integration, –
formulas for residues, –
of real integrals, –
several singularities inside
contour, –
Taylor series, – ,
Complex conjugate numbers,
Complex conjugate roots, –
Complex Fourier integral,
Complex functions, –
exponential, –
general powers, –
hyperbolic,
logarithm, –
trigonometric, –
Complex heat potential,
Complex integration, –
Cauchy’s integral formula,
– ,
Cauchy’s integral theorem,
– ,
existence of indefinite integral,
–
independence of path,
for multiply connected
domains, –
principle of deformation of
path,
Complex integration (Cont.)
derivatives of analytic functions,
–
Laurent series, –
analytic or singular at infinity,
–
point at infinity,
Riemann sphere,
singularities, –
zeros of analytic functions,
–
line integrals, – ,
basic properties of,
bounds for, –
definition of, –
existence of,
indefinite integration and
substitution of limits,
–
representation of a path,
–
power series, –
convergence behavior of,
–
convergence tests, –
functions given by, –
Maclaurin series,
radius of convergence of,
–
ratio test, –
root test, –
sequences, –
series, –
Taylor series, –
uniform convergence,
–
residue integration, –
formulas for residues, –
of real integrals, –
several singularities inside
contour, –
Complexity, of algorithms, –
Complex line integrals, see Line
integrals
Complex matrices and forms,
–
Complex numbers, – ,
addition of, ,
conjugate,
defined,
division of,
multiplication of, ,
polar form of, –
subtraction of,
Complex plane,
extended, , –
sets in,
Complex potential,
electrostatic fields, –
of fluid flow, , – Index I
Complex roots:
higher-order homogeneous linear
ODEs:
multiple,
simple, –
second-order homogeneous linear
ODEs, –
Complex trigonometric polynomials,
Complex variables, –
Complex vector space, , ,
Components (vectors), , ,
Composition, of linear
transformations, –
Computer-aided design (CAD),
Condition:
of incompressibility,
spline interpolation,
Conditionally convergent series,
Conditional probability, – ,
Condition number, – ,
Confidence intervals, ,
– ,
interval estimates,
for mean of normal distribution:
with known variance,
–
with unknown variance,
–
for parameters of distributions
other than normal,
in regression analysis, –
for variance of a normal
distribution, –
Confidence level,
Conformality,
Conformal mapping, –
boundary value problems,
– , A
defined,
geometry of analytic functions,
–
linear fractional transformations,
–
extended complex plane,
–
mapping standard domains,
–
Riemann surfaces, –
by trigonometric and hyperbolic
analytic functions,
–
Connected graphs, , ,
Connected set, in complex plane,
Conservative physical systems,
Conservative vector fields, ,
Consistent linear systems,
Constant coefficients:
higher-order homogeneous linear
ODEs, –
distinct real roots, –
multiple real roots, –
simple complex roots, –
second-order homogeneous linear
ODEs, –
complex roots, –
real double root, –
two distinct real roots, –
second-order nonhomogeneous
linear ODEs,
systems of ODEs, –
critical points, – ,
–
graphing solutions in phase
plane, –
Constant of gravity, at the Earth’s
surface,
Constant of integration,
Constant revenue, lines of,
Constrained (linear) optimization,
, – ,
normal form of problems, –
simplex method, –
degenerate feasible solution,
–
difficulties in starting, –
Constraints,
Consumers,
Consumer’s risk,
Consumption matrix,
Continuity equation (compressible
fluid flow),
Continuous complex functions,
Continuous distributions, ,
–
marginal distribution of,
two-dimensional,
Continuous random variables, ,
– ,
Continuous vector functions, –
Contour integral,
Contour lines, ,
Control charts,
for mean, –
for range, –
for standard deviation,
for variance, –
Controlled variables, in regression
analysis,
Control limits, ,
Control variables,
Convergence:
absolute:
defined,
and uniform convergence,
of approximate and exact
solutions,
Convergence: (Cont.)
circle of,
defined,
Gauss–Seidel iteration, –
mean square (orthogonal series),
–
in the norm,
power series, –
convergence tests, – ,
A –A
radius of convergence of,
– ,
uniform convergence, –
radius of,
defined,
power series, – ,
sequence of vectors,
speed of (numeric analysis),
–
superlinear,
uniform:
and absolute convergence,
power series, –
Convergence interval, ,
Convergence tests, –
power series, – , A –A
uniform convergence, –
Convergent iteration processes,
Convergent sequence of functions,
– ,
Convergent series, ,
Convolution:
defined,
Fourier transforms, –
Laplace transforms, –
Convolution theorem, –
Coriolis, Gustave Gaspard, n.
Coriolis acceleration, –
Corrector (improved Euler method),
Correlation analysis, ,
– ,
defined,
test for correlation coefficient,
–
Correlation coefficient, – ,
Cosecant, formula for, A
Cosine function:
conformal mapping by,
formula for, A –A
Cosine integral:
formula for, A
table, A
Cosine series,
Cotangent, formula for, A
Coulomb, Charles Augustin de,
n. , n. , n.
Coulomb’s law, , I Index
Covariance:
in correlation analysis,
defined,
Cramer, Gabriel, n. , n.
Cramer’s rule, , – ,
for three equations,
for two equations,
Cramer’s Theorem,
Crank, John, n.
Crank–Nicolson (CN) method,
–
Critical damping, ,
Critical points, ,
asymptotically stable,
and conformal mapping, ,
constant-coefficient systems of
ODEs, –
center,
criteria for, –
degenerate node, –
improper node,
proper node,
saddle point,
spiral point, –
stability of, –
isolated,
nonlinear systems,
stable, ,
stable and attractive, ,
unstable, ,
Critical region,
Cross product, , . See also
Vector product
Crout, Prescott Durand, n.
Crout’s method, ,
Cubic spline,
Cumulative absolute frequencies (of
values),
Cumulative distribution functions,
Cumulative relative frequencies (of
values),
Curl, A
invariance of, A –A
of vector fields, – ,
Curvature, of a curve, –
Curves:
arc of,
bell-shaped, ,
Bezier,
deflection,
elastic,
equipotential, , ,
one-parameter family of, –
operating characteristic, ,
,
oriented,
orthogonal coordinate, A
parameter,
plane,
Curves: (Cont.)
regression,
simple,
simple closed,
smooth, ,
solution, –
twisted,
vector differential calculus,
– ,
arc length of, –
length of,
in mechanics, –
tangents to, –
and torsion, –
Curve fitting, –
method of least squares, –
by polynomials of degree m,
–
Curvilinear coordinates, , , A
Cut sets, – ,
Cycle (paths), ,
Cylindrical coordinates, – ,
A –A
D’Alembert, Jean le Rond, n.
D’Alembert’s solution, –
Damped oscillations,
Damping constant,
Dantzig, George Bernard,
Data processing:
frequency distributions,
–
and randomness,
Data representation:
frequency distributions,
–
Empirical Rule,
graphic,
mean, –
standard deviation,
variation,
and randomness,
Decisions:
false, risks of making,
statistics for, –
Dedekind, Richard, A n.
Defect (eigenvalue),
Defectives,
Definite integrals, complex, see Line
integrals
Deflection curve,
Deformation of path, principle of,
Degenerate feasible solution (simplex
method), –
Degenerate node, –
Degrees of freedom (d.f.), number of,
,
Degree of incidence,
Degree of precision (DP),
Deleted neighborhood,
Demand vector,
De Moivre, Abraham, n.
De Moivre–Laplace limit theorem,
De Moivre’s formula,
De Morgan’s laws,
Density,
continuous two-dimensional
distributions,
of a distribution,
Dependent random variables, ,
Dependent variables, , ,
Depth First Search (DFS) algorithms,
Derivatives:
of analytic functions, – ,
– , A –A
of complex functions, ,
Laplace transforms of, –
of matrices or vectors,
of vector functions, –
Derived series,
Descartes, René, n. , n.
Determinants, – ,
Cauchy,
Cramer’s rule, –
defined, A
general properties of, –
of a matrix,
of matrix products, –
of order n,
proof of, A –A
second-order, –
second-order homogeneous linear
ODEs,
third-order, –
Vandermonde,
Wronski:
second-order homogeneous
linear ODEs, –
systems of ODEs,
Developed, in a power series,
D.f. (degrees of freedom), number of,
,
DFS (Depth First Search) algorithms,
DFTs (discrete Fourier transforms),
–
Diagonalization of matrices, –
Diagonally dominant matrices,
Diagonal matrices,
inverse of, –
scalar,
Diameter (graphs),
Difference:
complex numbers,
scalar multiplication, Index I
Difference equations (elliptic PDEs),
–
Difference quotients,
Difference table,
Differentiable complex functions,
–
Differentiable vector functions,
Differential (total differential),
,
Differential equations:
applications of,
defined,
Differential form,
exact, ,
first fundamental form, of S,
floating-point, of numbers,
–
path independence and exactness
of, ,
Differential geometry,
Differential operators:
second-order,
for second-order homogeneous
linear ODEs, –
Differentiation:
of Laplace transforms, –
matrices or vectors,
numeric, –
of power series, – ,
termwise, , – ,
Diffusion equation, – , .
See also Heat equation
Digraphs (directed graphs), – ,
computer representation of,
–
defined,
incidence matrix of,
subgraphs,
Dijkstra, Edsger Wybe, n.
Dijkstra’s algorithm, – ,
DIJKSTRA, ALGORITHM,
Dimension of vector spaces, ,
,
Diocles, n.
Dirac, Paul, n.
Dirac delta function, – ,
Directed graphs, see Digraphs
(directed graphs)
Directed path,
Directional derivatives (scalar
functions), – ,
Direction field (slope field), – ,
Direct methods (linear system
solutions), , . See also
iteration
Dirichlet, Peter Gustav LeJeune,
n.
Dirichlet boundary condition,
Dirichlet problem, ,
ADI method,
heat equation, –
Laplace equation, – ,
– , –
Poisson equation, –
two-dimensional heat equation,
–
uniqueness theorem for, ,
Dirichlet’s discontinuous factor,
Discharge (flow modeling),
Discrete distributions, –
marginal distributions of,
–
two-dimensional, –
Discrete Fourier transforms (DFTs),
–
Discrete random variables, ,
– ,
defined,
marginal distributions of,
Discrete spectrum,
Disjoint events,
Disks:
circular, open and closed,
mapping, –
Poisson’s integral formula, –
Dissipative physical systems,
Distance:
graphs,
vector norms,
Distinct real roots:
higher-order homogeneous linear
ODEs, –
second-order homogeneous linear
ODEs, –
Distinct roots (Frobenius method),
Distributions, n. . See also
Frequency distributions;
Probability distributions
Distribution-free tests,
Distribution function, –
cumulative,
normal distributions, –
of random variables, , A
sample,
two-dimensional probability
distributions, –
Distributive laws,
Distributivity,
Divergence, A
fluid flow,
of vector fields, –
of vector functions, ,
Divergence theorem of Gauss, ,
applications, –
vector integral calculus, –
Divergent sequence,
Divergent series, ,
Division, of complex numbers, ,
–
Domain(s),
bounded,
doubly connected, ,
of f,
holes of,
mapping, , –
multiply connected:
Cauchy’s integral formula,
–
Cauchy’s integral theorem,
–
p-fold connected, –
sets in complex plane,
simply connected, , , ,
triply connected, , ,
Dominant eigenvalue,
Doolittle, Myrick H., n.
Doolittle’s method, – ,
Dot product, , . See also Inner
product
Double Fourier series:
defined,
rectangular membrane, –
Double integrals (vector integral
calculus), – ,
applications of, –
change of variables in, –
evaluation of, by two successive
integrations, –
Double precision, floating-point
standard for,
Double root (Frobenius method),
Double subscript notation,
Doubly connected domains, ,
DP (degree of precision),
Driving force, see Input (driving
force)
Duffing equation,
Duhamel, Jean-Marie Constant,
n.
Duhamel’s formula,
Eccentricity, of vertices,
Edges:
backward:
cut sets,
initial flow,
of a path,
forward:
cut sets,
initial flow,
of a path,
graphs, ,
incident,
Edge chromatic number, I Index
Edge condition,
Edge incidence list (graphs),
Efficient algorithms,
Eigenbases, –
Eigenfunctions,
circular membrane,
one-dimensional heat equation,
Sturm–Liouville Problems,
–
two-dimensional heat equation,
two-dimensional wave equation,
,
vibrating string,
Eigenfunction expansion,
Eigenspaces, ,
Eigenvalues, – , , , ,
, . See also Matrix
eigenvalue problems
circular membrane,
complex matrices, –
and critical points,
defined,
determining, –
dominant,
finding, –
one-dimensional heat equation,
Sturm–Liouville Problems,
– , A
two-dimensional wave equation,
vibrating string,
Eigenvalues of A,
Eigenvalue problem,
Eigenvectors, – , , ,
,
basis of, –
convergent sequence of,
defined,
determining, –
finding, –
Eigenvectors of A,
EISPACK,
Elastic curve,
Electric circuits:
analogy of electrical and
mechanical quantities,
–
second-order nonhomogeneous
linear ODEs, –
Electrostatic fields (potential theory),
–
complex potential, –
superposition, –
Electrostatic potential,
Electrostatics (Laplace’s equation),
Elementary matrix,
Elementary row operations (linear
systems),
Ellipses, area of region bounded by,
Elliptic PDEs:
defined,
numeric analysis, –
ADI method, –
difference equations, –
Dirichlet problem, –
irregular boundary, –
mixed boundary value
problems, –
Neumann problem,
Empirical Rule,
Energies,
Entire function, , , ,
Entries:
determinants,
matrix, ,
Equal complex numbers,
Equality:
of matrices, ,
of vectors,
Equally likely events,
Equal spacing (interpolation):
Newton’s backward difference
formula, –
Newton’s forward difference
formula, –
Equilibrium harvest,
Equilibrium solutions (equilibrium
points), –
Equipotential curves, , ,
Equipotential lines,
electrostatic fields, ,
fluid flow,
Equipotential surfaces,
Equivalent vector norms,
Error(s):
in acceptance sampling,
–
of approximations,
in numeric analysis,
basic error principle,
error propagation,
errors of numeric results,
–
roundoff,
in statistical tests, –
and step size control, –
trapezoidal rule,
vector norms,
Error bounds,
Error estimate,
Error function, , A –A , A
Essential singularity, –
Estimation of parameters,
EULER, ALGORITHM,
Euler, Leonhard, n. , n.
Euler–Cauchy equations, – ,
higher-order nonhomogeneous
linear ODEs, –
Laplace’s equation,
third-order, IVP for,
Euler–Cauchy method,
Euler constant,
Euler formulas,
complex Fourier integral,
derivation of, –
exponential function,
Fourier coefficients given by, ,
generalized,
Taylor series,
trigonometric function,
Euler graph,
Euler’s method:
defined,
error of, – , ,
first-order ODEs, – , –
backward method, –
improved method, –
higher order ODEs, –
Euler trail,
Even functions, –
Even periodic extension, –
Events (probability theory),
– ,
addition rule for, –
arbitrary, –
complements of,
defined,
disjoint,
equally likely,
independent, –
intersection, ,
mutually exclusive, ,
simple,
union, –
Exact differential equation,
Exact differential form, ,
Exact ODEs, – ,
defined,
integrating factors, –
Existence, problem of,
Existence theorems:
cubic splines,
first-order ODEs, –
homogeneous linear ODEs:
higher-order,
second-order,
of the inverse, –
Laplace transforms, –
linear systems,
power series solutions,
systems of ODEs,
Expectation, , – ,
Index I
Experiments:
defined, ,
in probability theory, –
random, , – ,
Experimental error,
Explicit formulas,
Explicit method:
heat equation, , –
wave equation,
Explicit solution,
Exponential decay, ,
Exponential function, – ,
formula for, A
Taylor series,
Exponential growth,
Exponential integral, formula for, A
Exposed vertices, ,
Extended complex plane:
conformal mapping, –
defined,
Extended method (separable ODEs),
–
Extended problems,
Extrapolation,
Extrema (unconstrained
optimization),
Factorial function, , A , A .
See also Gamma functions
Failing to reject a hypothesis,
Fair die, ,
False decisions, risks of making,
False position, method of, –
Family of curves, one-parameter,
–
Family of solutions,
Faraday, Michael, n.
Fast Fourier transforms (FFTs),
–
F-distribution, , A –A
Feasibility region,
Feasible solutions, –
basic, ,
degenerate, –
normal form of linear optimization
problems,
Fehlberg, E.,
Fehlberg’s fifth-order RK method,
–
Fehlberg’s fourth-order RK method,
–
FFTs (fast Fourier transforms),
–
Fibonacci (Leonardo of Pisa), n.
Fibonacci numbers,
Fibonacci’s rabbit problem,
Finite complex plane, . See also
Complex plane
Finite jumps,
First boundary value problem, see
Dirichlet problem
First fundamental form, of S,
First-order method, Euler method as,
First-order ODEs, – ,
defined,
direction fields, –
Euler’s method, –
exact, – ,
defined,
integrating factors, –
explicit form,
geometric meanings of, –
implicit form,
initial value problem, –
linear, –
Bernoulli equation, –
homogeneous,
nonhomogeneous, –
population dynamics, –
modeling, –
numeric analysis, –
Adams–Bashforth methods,
–
Adams–Moulton methods,
–
backward Euler method,
–
Euler’s method, –
improved Euler’s method,
–
multistep methods, –
Runge–Kutta–Fehlberg
method, –
Runge–Kutta methods,
–
orthogonal trajectories, –
separable, – ,
extended method, –
modeling, –
systems of,
transformation of systems to,
–
First (first order) partial derivatives,
A
First shifting theorem (s-shifting),
–
First transmission line equation,
Fisher, Sir Ronald Aylmer,
Fixed points:
defined,
of a mapping,
Fixed-point iteration (numeric
analysis), – ,
Fixed-point systems, numbers in,
Floating,
Floating-point form of numbers,
–
Flow augmenting paths, – ,
,
Flow problems in networks
(combinatorial optimization),
–
cut sets, –
flow augmenting paths, –
paths,
Fluid flow:
Laplace’s equation,
potential theory, –
Fluid state,
Flux (motion of a fluid),
Flux integral, ,
Forced motions, ,
Forced oscillations:
Fourier analysis, –
second-order nonhomogeneous
linear ODEs, –
damped, –
resonance, –
undamped, –
Forcing function,
Ford, Lester Randolph, Jr., n.
FORD–FULKERSON,
ALGORITHM,
Ford–Fulkerson algorithm for
maximum flow, – ,
Forest (graph),
Form(s):
canonical,
complex,
differential,
exact, ,
path independence and
exactness of,
Hesse’s normal,
Lagrange’s,
normal (linear optimization
problems), – , ,
Pfaffian,
polar, of complex numbers,
– ,
quadratic, – ,
reduced echelon,
row echelon, –
skew-Hermitian and Hermitian,
standard:
first-order ODEs,
higher-order homogeneous
linear ODEs,
higher-order linear ODEs,
power series method,
second-order linear ODEs, ,
triangular (Gauss elimination),
I Index
Forward edge:
cut sets,
initial flow,
of a path,
Four-color theorem,
Fourier, Jean-Baptiste Joseph, n.
Fourier analysis, –
approximation by trigonometric
polynomials, –
forced oscillations, –
Fourier integral, –
applications, –
complex form of, –
sine and cosine, –
Fourier series, –
convergence and sum of,
–
derivation of Euler formulas,
–
even and odd functions,
–
half-range expansions, –
from period to L,
–
Fourier transforms, –
complex form of Fourier
integral, –
convolution, –
cosine, – ,
discrete, –
fast, –
and its inverse, –
linearity, –
sine, – ,
spectrum representation,
orthogonal series (generalized
Fourier series), –
completeness, –
mean square convergence,
–
Sturm–Liouville Problems,
–
eigenvalues, eigenfunctions,
–
orthogonal functions, –
Fourier–Bessel series, – ,
Fourier coefficients, , , ,
–
Fourier constants, –
Fourier cosine integral, –
Fourier cosine series, , ,
Fourier cosine transforms, – ,
Fourier cosine transform method,
Fourier integrals, – ,
applications, –
complex form of, –
heat equation, –
residue integration, –
sine and cosine, –
p
Fourier–Legendre series, – ,
–
Fourier matrix,
Fourier series, – ,
convergence and sum or, –
derivation of Euler formulas,
–
double, –
even and odd functions, –
half-range expansions, –
heat equation, –
from period to L, –
Fourier sine integral, –
Fourier sine series, , ,
one-dimensional heat equation,
vibrating string,
Fourier sine transforms, – ,
Fourier transforms, – ,
complex form of Fourier integral,
–
convolution, –
cosine, – , ,
defined, ,
discrete, –
fast, –
heat equation, –
and its inverse, –
linearity of, –
sine, – , ,
spectrum representation,
Fourier transform method,
Four-point formulas,
Fraction defective chars, –
Francis, J. G. F.,
Fredholm, Erik Ivar, n. , n.
Free condition (spline interpolation),
Free oscillations of mass–spring
system (second-order ODEs),
–
critical damping, ,
damped system, –
overdamping, –
undamped system, –
underdamping, ,
Frenet, Jean-Frédéric,
Frenet formulas,
Frequency (in statistics):
absolute, ,
cumulative absolute,
cumulative relative,
relative class,
Frequency (of vibrating string),
Frequency distributions, mean and
variance of:
expectation, –
moments,
transformation of, –
p
Fresnel, Augustin, n. , A n.
Fresnel integrals, , A
Frobenius, Georg, n.
Frobenius method, , – ,
indicial equation, –
proof of, A –A
typical applications, –
Frobenius norm,
Fulkerson, Delbert Ray, n.
Function, of complex variable,
–
Function spaces,
Fundamental matrix,
Fundamental period,
Fundamental region (exponential
function),
Fundamental system, , . See
also Basis, of solutions
Fundamental Theorem:
higher-order homogeneous linear
ODEs,
for linear systems,
PDEs, –
second-order homogeneous linear
ODEs,
Galilei, Galileo, n.
Gamma functions, – ,
formula for, A –A
incomplete, A
table, A
GAMS (Guide to Available
Mathematical Software),
GAUSS, ALGORITHM,
Gauss, Carl Friedrich, n. ,
n. ,
Gauss distribution, . See also
Normal distributions
Gauss “Double Ring,”
Gauss elimination, ,
linear systems, – ,
– ,
back substitution, – ,
elementary row operations,
if infinitely many solutions
exist,
if no solution exists, –
operation count, –
row echelon form, –
operation count, –
Gauss integration formulas, ,
– ,
Gauss–Jordan elimination, – ,
–
GAUSS–SEIDEL, ALGORITHM,
Index I
Gauss–Seidel iteration, – ,
Gauss’s hypergeometric ODE, ,
Geiger, H., ,
Generalized Euler formula,
Generalized Fourier series, see
Orthogonal series
Generalized solution (vibrating
string),
Generalized triangle inequality,
General powers, – ,
General solution:
Bessel’s equation, –
first-order ODEs, ,
higher-order linear ODEs, ,
– ,
nonhomogeneous linear systems,
second-order linear ODEs:
homogeneous, – , – ,
nonhomogeneous, –
systems of ODEs, – ,
Generating functions, ,
Geometric interpretation:
partial derivatives, A
scalar triple product, ,
Geometric multiplicity, ,
Geometric series, ,
Taylor series,
uniformly convergent,
Gerschgorin, Semyon Aranovich,
n.
Gerschgorin’s theorem, – ,
Gibbs phenomenon,
Global error,
Golden Rule, ,
Gompertz model,
Goodness of fit, –
Gosset, William Sealy, n.
Goursat, Édouard, n.
Goursat’s proof,
Gradient, A
fluid flow,
of a scalar field, –
directional derivatives,
–
maximum increase,
as surface normal vector,
–
vector fields that are, –
of a scalar function, ,
unconstrained optimization,
Gradient method, . See also
Method of steepest descent
Graphs, – ,
bipartite, – ,
center of,
complete,
Graphs (Cont.)
complete bipartite,
computer representation of,
–
connected, , ,
diameter of,
digraphs (directed graphs),
– ,
computer representation of,
–
defined,
incidence matrix of,
subgraphs,
Euler,
forest,
incidence matrix of,
planar,
radius of,
sparse,
subgraphs,
trees,
vertices, , ,
adjacent, ,
central,
coloring, –
double labeling of,
eccentricity of,
exposed, ,
four-color theorem,
scanning,
weighted,
Graphic data representation,
Gravitation (Laplace’s equation),
Gravity, acceleration of,
Gravity constant, at the Earth’s
surface,
Greedy algorithm, –
Green, George, n.
Green’s first formula, ,
Green’s second formula, ,
Green’s theorem:
first and second forms of,
in the plane, – ,
Gregory, James, n.
Gregory–Newton’s (Newton’s)
backward difference
interpolation formula,
–
Gregory–Newton’s (Newton’s)
forward difference
interpolation formula,
–
Growth restriction,
Guidepoints,
Guide to Available Mathematical
Software (GAMS),
Guldin, Habakuk, n.
Guldin’s theorem, n.
Hadamard, Jacques, n.
Half-planes:
complex analysis, –
mapping, –
Half-range expansions (Fourier
series), – ,
Hamilton, William Rowan, n.
Hamiltonian cycle,
Hankel, Hermann, n.
Hankel functions,
Harmonic conjugate function
(Laplace’s equation),
Harmonic functions, , ,
complex analysis, –
under conformal mapping,
defined,
Laplace’s equation, , –
maximum modulus theorem,
–
potential theory, – ,
Harmonic oscillation, –
Heat equation, – , –
Dirichlet problem, –
Laplace’s equation,
numeric analysis, – ,
Crank–Nicolson method,
–
explicit method, , –
one-dimensional,
solution:
by Fourier integrals, –
by Fourier series, –
by Fourier transforms,
–
steady two-dimensional heat
problems, –
two-dimensional, –
unifying power of methods,
Heat flow:
Laplace’s equation,
potential theory, –
Heat flow lines,
Heaviside, Oliver, n.
Heaviside calculus, n.
Heaviside expansions,
Heaviside function, –
Helix,
Henry, Joseph, n.
Hermite, Charles, n.
Hermite interpolation,
Hermitian form,
Hermitian matrices, , , ,
Hertz, Heinrich, n.
Hesse, Ludwig Otto, n.
Hesse’s normal form,
Heun, Karl, n.
Heun’s method, . See also
Improved Euler’s method
Higher functions, . See also
Special functionsI Index
Higher-order linear ODEs, –
homogeneous, – ,
nonhomogeneous, –
systems of, see Systems
of ODEs
Higher order ODEs (numeric
analysis), –
Euler method, –
Runge–Kutta methods, –
Runge–Kutta–Nyström methods,
–
Higher transcendental functions,
High-frequency line equations,
Hilbert, David, n. , n.
Hilbert spaces,
Histograms,
Holes, of domains,
Homogeneous first-order linear
ODEs,
Homogeneous higher-order linear
ODEs, –
Homogeneous linear systems, ,
, , – ,
constant-coefficient systems,
–
matrices and vectors, – ,
trivial solution,
Homogeneous PDEs,
Homogeneous second-order linear
ODEs, –
basis, –
with constant coefficients, –
complex roots, –
real double root, –
two distinct real-roots, –
differential operators, –
Euler–Cauchy equations, –
existence and uniqueness of
solutions, –
general solution, – , –
initial value problem, –
modeling free oscillations of
mass–spring system, –
particular solution, –
reduction of order, –
Wronskian, –
Hooke, Robert,
Hooke’s law,
Householder, Alston Scott, n.
Householder’s tridiagonalization
method, –
Hyperbolic analytic functions
(conformal mapping), –
Hyperbolic cosine, ,
Hyperbolic functions, ,
formula for, A –A
inverse,
Taylor series,
Hyperbolic PDEs:
defined,
numeric analysis, –
Hyperbolic sine, ,
Hypergeometric distributions,
– ,
Hypergeometric equations, ,
–
Hypergeometric functions, ,
Hypergeometric series,
Hypothesis,
Hypothesis testing (in statistics),
, –
comparison of means, –
comparison of variances,
errors in tests, –
for mean of normal distribution
with known variance,
–
for mean of normal distribution
with unknown variance,
–
one- and two-sided alternatives,
–
Idempotent matrices,
Identity mapping,
Identity matrices,
Identity operator (second-order
homogeneous linear ODEs),
Ill-conditioned equations,
Ill-conditioned problems,
Ill-conditioned systems, , ,
Ill-conditioning (linear systems),
–
condition number of a matrix,
–
matrix norms, –
vector norms,
Image:
conformal mapping,
linear transformations,
Imaginary axis (complex plane),
Imaginary part (complex numbers),
Imaginary unit,
Impedance (RLC circuits),
Implicit formulas,
Implicit method:
backward Euler scheme as,
for hyperbolic PDEs,
Implicit solution,
Improper integrals:
defined,
residue integration, –
Improper node,
Improved Euler’s method:
error of, , ,
first-order ODEs, –
Impulse, of a force,
short impulses, –
unit impulse function,
Incidence matrices (graphs and
digraphs),
Incident edges,
Inclusion theorems:
defined,
matrix eigenvalue problems,
–
Incomplete gamma functions,
formula for, A
Inconsistent linear systems,
Indefinite (quadratic form),
Indefinite integrals:
defined,
existence of, –
Indefinite integration (complex line
integral), –
Independence:
of path,
of path in domain (integrals), ,
of random variables, –
Independent events, – ,
Independent sample values,
Independent variables:
in calculus,
in regression analysis,
Indicial equation, – , ,
Indirect methods (solving linear
systems), ,
Inference, statistical, ,
Infinite dimensional vector space,
Infinite populations,
Infinite sequences:
bounded, A –A
monotone real, A –A
power series, –
Infinite series, –
Infinity:
analytic of singular at, –
point at,
Initial conditions:
first-order ODEs, , ,
heat equation, , ,
higher-order linear ODEs:
homogeneous,
nonhomogeneous,
one-dimensional heat equation,
PDEs, ,
second-order homogeneous linear
ODEs, – ,
systems of ODEs,
two-dimensional wave equation,
vibrating string,
Initial point (vectors),
Initial value problem (IVP):
defined,
first-order ODEs, , , , Index I
Initial value problem (IVP): (Cont.)
bell-shaped curve,
existence and uniqueness of
solutions for, –
higher-order linear ODEs,
homogeneous, –
nonhomogeneous,
Laplace transforms, –
for RLC circuit,
second-order homogeneous linear
ODEs, , – ,
systems of ODEs,
Injective mapping, n.
Inner product (dot product),
for complex vectors,
invariance of,
vector differential calculus,
– ,
applications, –
orthogonality, –
Inner product spaces, –
Input (driving force), , ,
Instability, numeric vs. mathematical,
Integrals, see Line integrals
Integral equations:
defined,
Laplace transforms, –
Integral of a function, Laplace
transforms of, –
Integral transforms, ,
Integrand, ,
Integrating factors, – ,
defined,
finding, –
Integration. See also Complex
integration
constant of,
of Laplace transforms, –
numeric, –
adaptive, –
Gauss integration formulas,
–
rectangular rule,
Simpson’s rule, –
trapezoidal rule, –
termwise, of power series, ,
Intermediate value theorem, –
Intermediate variables,
Intermittent harvesting,
INTERPOL, ALGORITHM,
Interpolation,
defined,
numeric analysis, – ,
equal spacing, –
Lagrange, –
Newton’s backward difference
formula, –
Newton’s divided difference,
–
Interpolation (Cont.)
Newton’s forward difference
formula, –
spline, –
Interpolation polynomial, ,
Interquartile range,
Intersection, of events, ,
Intervals. See also Confidence
intervals
class,
closed, A n.
convergence, ,
open, , A n.
Interval estimates,
Invariance, of curl, A –A
Invariant rank,
Invariant subspace,
Inverse cosine,
Inverse cotangent,
Inverse Fourier cosine transform,
Inverse Fourier sine transform,
Inverse Fourier sine transform
method,
Inverse Fourier transform,
Inverse hyperbolic function,
Inverse hyperbolic sine,
Inverse mapping, ,
Inverse of a matrix, , – ,
cancellation laws, –
determinants of matrix
products, –
formulas for, –
Gauss–Jordan method,
– , –
Inverse sine,
Inverse tangent,
Inverse transform, ,
Inverse transformation,
Inverse trigonometric function,
Irreducible,
Irregular boundary (elliptic PDEs),
–
Irrotational flow,
Isocline,
Isolated critical point,
Isolated essential singularity,
Isolated singularity,
Isotherms, , , ,
Iteration (iterative) methods:
numeric analysis, –
fixed-point iteration, –
Newton’s (Newton–Raphson)
method, –
secant method, –
speed of convergence, –
numeric linear algebra, – ,
Gauss–Seidel iteration, –
Jacobi iteration, –
IVP, see Initial value problem
Jacobi, Carl Gustav Jacob, n.
Jacobians, ,
Jacobi iteration, –
Jordan, Wilhelm, n.
Joukowski airfoil, –
Kantorovich, Leonid Vitaliyevich,
n.
KCL (Kirchhoff’s Current Law),
n. ,
Kernel,
Kinetic friction, coefficient of,
Kirchhoff, Gustav Robert, n.
Kirchhoff’s Current Law (KCL),
n. ,
Kirchhoff’s law,
Kirchhoff’s Voltage Law (KVL), ,
,
Koopmans, Tjalling Charles, n.
Kreyszig, Erwin, n.
Kronecker, Leopold, n.
Kronecker delta, A
Kronecker symbol,
Kruskal, Joseph Bernard, n.
KRUSKAL, ALGORITHM,
Kruskal’s Greedy algorithm,
– ,
kth backward difference,
kth central moment,
kth divided difference,
kth forward difference, –
kth moment, ,
Kublanovskaya, V. N.,
Kutta, Wilhelm, n.
Kutta’s third-order method,
KVL, see Kirchhoff’s Voltage Law
Lagrange, Joseph Louis, n.
Lagrange interpolation, –
Lagrange’s form, ,
Laguerre, Edmond, n.
Laguerre polynomials, ,
Laguerre’s equation, –
LAPACK,
Laplace, Pierre Simon Marquis de,
n.
Laplace equation, , , – ,
,
boundary value problem in
spherical coordinates,
–
complex analysis, –
in cylindrical coordinates,
–
Fourier–Legendre series, –
heat equation,
numeric analysis, – ,
ADI method, –
difference equations, – I Index
Laplace equation (Cont.)
Dirichlet problem, – ,
–
Liebmann’s method, –
in spherical coordinates,
theory of solutions of, , .
See also Potential theory
two-dimensional heat equation,
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