**Advanced Engineering Mathematic 6edDennis G. ZillLoyola Marymount UniversityContentsPreface xiiiIntroduction to Differential Equations. Definitions and Terminology. Initial-Value Problems. Differential Equations as Mathematical ModelsChapter in ReviewFirst-Order Differential Equations. Solution Curves Without a Solution. Direction Fields. Autonomous First-Order DEs. Separable Equations. Linear Equations. Exact Equations. Solutions by Substitutions. A Numerical Method. Linear Models. Nonlinear Models. Modeling with Systems of First-Order DEsChapter in ReviewPART Ordinary Differential Equationsiv ContentsHigher-Order Differential Equations. Theory of Linear Equations. Initial-Value and Boundary-Value Problems. Homogeneous Equations. Nonhomogeneous Equations. Reduction of Order. Homogeneous Linear Equations with Constant Coefficients. Undetermined Coefficients. Variation of Parameters. Cauchy–Euler Equations. Nonlinear Equations. Linear Models: Initial-Value Problems. Spring/Mass Systems: Free Undamped Motion. Spring/Mass Systems: Free Damped Motion. Spring/Mass Systems: Driven Motion. Series Circuit Analogue. Linear Models: Boundary-Value Problems. Green’s Functions. . Initial-Value Problems. . Boundary-Value Problems. Nonlinear Models. Solving Systems of Linear EquationsChapter in ReviewThe Laplace Transform. Definition of the Laplace Transform. The Inverse Transform and Transforms of Derivatives. Inverse Transforms. Transforms of Derivatives. Translation Theorems. Translation on the s-axis. Translation on the t-axis. Additional Operational Properties. Derivatives of Transforms. Transforms of Integrals. Transform of a Periodic Function. The Dirac Delta Function. Systems of Linear Differential EquationsChapter in Review**

© azharjggt/ShutterStock, Inc. © Tim Jenner/ShutterStock, Inc.Contents v

Series Solutions of Linear Differential Equations

. Solutions about Ordinary Points

. Review of Power Series

. Power Series Solutions

. Solutions about Singular Points

. Special Functions

. Bessel Functions

. Legendre Functions

Chapter in Review

Numerical Solutions of Ordinary Differential Equations

. Euler Methods and Error Analysis

. Runge–Kutta Methods

. Multistep Methods

. Higher-Order Equations and Systems

. Second-Order Boundary-Value Problems

Chapter in Review

Vectors

. Vectors in -Space

. Vectors in -Space

. Dot Product

. Cross Product

. Lines and Planes in -Space

. Vector Spaces

. Gram–Schmidt Orthogonalization Process

Chapter in Review

PART Vectors, Matrices, and Vector Calculus

© Vaclav Volrab/Shutterstock, Inc. © sixninepixels/Shutterstock, Inc. © Cecilia Lim H M/ShutterStock, Inc.vi Contents

Matrices

. Matrix Algebra

. Systems of Linear Algebraic Equations

. Rank of a Matrix

. Determinants

. Properties of Determinants

. Inverse of a Matrix

. Finding the Inverse

. Using the Inverse to Solve Systems

. Cramer’s Rule

. The Eigenvalue Problem

. Powers of Matrices

. Orthogonal Matrices

. Approximation of Eigenvalues

. Diagonalization

. LU-Factorization

. Cryptography

. An Error-Correcting Code

. Method of Least Squares

. Discrete Compartmental Models

Chapter in Review

Vector Calculus

. Vector Functions

. Motion on a Curve

. Curvature and Components of Acceleration

. Partial Derivatives

. Directional Derivative

. Tangent Planes and Normal Lines

. Curl and Divergence

. Line Integrals

. Independence of the Path

. Double Integrals

**. Double Integrals in Polar Coordinates. Green’s Theorem. Surface Integrals. Stokes’ Theorem. Triple Integrals. Divergence Theorem. Change of Variables in Multiple IntegralsChapter in ReviewSystems of Linear Differential Equations. Theory of Linear Systems. Homogeneous Linear Systems. Distinct Real Eigenvalues. Repeated Eigenvalues. Complex Eigenvalues. Solution by Diagonalization. Nonhomogeneous Linear Systems. Undetermined Coefficients. Variation of Parameters. Diagonalization. Matrix ExponentialChapter in ReviewSystems of Nonlinear Differential Equations. Autonomous Systems. Stability of Linear Systems. Linearization and Local Stability. Autonomous Systems as Mathematical Models. Periodic Solutions, Limit Cycles, and Global StabilityChapter in ReviewPART Systems of Differential Equations**

Orthogonal Functions and Fourier Series

. Orthogonal Functions

. Fourier Series

. Fourier Cosine and Sine Series

. Complex Fourier Series

. Sturm–Liouville Problem

. Bessel and Legendre Series

. Fourier–Bessel Series

. Fourier–Legendre Series

Chapter in Review

Boundary-Value Problems in

Rectangular Coordinates

. Separable Partial Differential Equations

. Classical PDEs and Boundary-Value Problems

. Heat Equation

. Wave Equation

. Laplace’s Equation

. Nonhomogeneous Boundary-Value Problems

. Orthogonal Series Expansions

. Fourier Series in Two Variables

Chapter in Review

Boundary-Value Problems in

Other Coordinate Systems

. Polar Coordinates

. Cylindrical Coordinates

. Spherical Coordinates

Chapter in Review

PART Partial Differential Equations

**Integral Transform Method. Error Function. Applications of the Laplace Transform. Fourier Integral. Fourier Transforms. Fast Fourier TransformChapter in ReviewNumerical Solutions of Partial Differential Equations. Laplace’s Equation. Heat Equation. Wave EquationChapter in ReviewFunctions of a Complex Variable. Complex Numbers. Powers and Roots. Sets in the Complex Plane. Functions of a Complex Variable. Cauchy–Riemann Equations. Exponential and Logarithmic Functions. Trigonometric and Hyperbolic Functions. Inverse Trigonometric and Hyperbolic FunctionsChapter in Review**

PART Complex Analysis

**Integration in the Complex Plane. Contour Integrals. Cauchy–Goursat Theorem. Independence of the Path. Cauchy’s Integral FormulasChapter in ReviewSeries and Residues. Sequences and Series. Taylor Series. Laurent Series. Zeros and Poles. Residues and Residue Theorem. Evaluation of Real IntegralsChapter in ReviewConformal Mappings. Complex Functions as Mappings. Conformal Mappings. Linear Fractional Transformations. Schwarz–Christoffel Transformations. Poisson Integral Formulas. ApplicationsChapter in ReviewAppendicesI Derivative and Integral Formulas APP-II Gamma Function APP-III Table of Laplace Transforms APP-IV Conformal Mappings APP-Answers to Selected Odd-Numbered Problems ANS-IndexIndexI-AAbsolute convergence:of a complex series,definition of,of a power series, ,Absolute error,Absolute value of a complex number,Absolutely integrable, ,Acceleration:centripetal,due to gravity, ,normal component of,tangential component of,as a vector function,Adams–Bashforth–Moulton method,Adams–Bashforth predictor,Adams–Moulton corrector,Adaptive numerical method,Addition:of matrices,of power series,of vectors, , ,Adjoint matrix:definition of,use in finding an inverse,Age of a fossil,Aging spring, ,Agnew, Ralph Palmer,Air resistance:nonlinear, , ,projectile motion with, ,projectile motion with no, ,proportional to square of velocity, , ,proportional to velocity, , , –Airy, George Biddell,Airy’s differential equation:definition of, ,solution as power series,solution in terms of Bessel functions,various forms of,Algebraic equations,Aliasing,Allee, Warder Clyde,Allee effect,Alternative form of second translationtheorem,Ambient temperature, ,Amperes (A),Amplitude:damped,of free vibrations,time varying, ,Analytic function:criterion for,definition of,derivatives of,Analytic part of a Laurent series,Analyticity, vector fields and,Analyticity and path independence,Analyticity at a point:criterion for,definition of, ,Angle between two vectors,Angle preserving mappings,Anharmonic overtones,Annular domain,Annulus in the complex plane,Anticommute,Antiderivative:of a complex function,definition of,existence of,Applications of differential equations:aging springs, ,air exchange,air resistance, , , , – , , –Allee effect,bacterial growth,ballistic pendulum,bending of a circular plate, –buckling of a tapered column,buckling of a thin vertical column, ,cantilever beam,carbon-dating, I- IndexINDEXApplications of differentialequations:—(Cont.)caught pendulum,chemical reactions, – ,column bending under it’s ownweight,competing species of animals,– ,continuous compound interest, ,cooling fin, –cooling/warming, ,coupled spring/mass system, –cycloid,damped motion,deflection of a beam, – ,double pendulum, –double spring systems,draining a tank, ,electrical networks, , , ,electrical series circuits, , ,,emigration,evaporating raindrop, ,evaporation,falling bodies with air resistance, ,, , –falling bodies with no air resistance,– ,falling chain,floating barrel,fluctuating population,forgetfulness,growth of microorganisms in achemostat,hard spring,harvesting,heart pacemaker, ,hitting bottom,hole drilled through the Earth,immigration, ,infusion of a drug,leaking tanks,linear spring,lifting a heavy rope,logistic population growth, –marine toads, invasion of,memorization,mixtures, ,networks,nonlinear springs, –nonlinear pendulum, , ,–orthogonal families of curves,oscillating chain,Ötzi (the iceman),Paris guns, –pendulum of varying length, –population dynamics, ,potassium-argon dating, ,potassium- decay,predator-prey, – , –projectile motion, , – ,–pursuit curves, –radioactive decay, , – , – ,reflecting surface, –restocking,rocket motion, , – ,rope pulled upward by a constantforce, , –rotating fluid, shape of a,rotating pendulum,rotating rod, sliding bead on a,rotating shaft,rotating string, –sawing wood,series circuit, , –Shroud of Turin,sinking in water,skydiving,sliding bead, ,sliding box on an inclined plane,snowplow problem,soft spring,solar collector,spread of a disease,spring coupled pendulums,spring/mass systems, , – ,– , –spring pendulum, –streamlines, ,suspended cables and telephone wires,– , –temperature in an annular cooling fin,–temperature in an annular plate,,temperature between concentriccylinders, ,temperature between concentricspheres,temperature in a circular plate,temperature in a cylinder,– ,temperature in a quarter-circularplate,temperature in a ring,temperature in a semiannular plate,temperature in a semicircular plate,temperature in a sphere, ,– ,temperature in a wedge-shapedplate,terminal velocity, , – ,time of death,tractrix,tsunami,variable mass, – , –vibrating beam, ,vibrating string, –water clock,Aquatic food chain,Arc,Arc length as a parameter,Archimedes’ principle,Area as a double integral, ,Area:of a parallelogram,of a surface,of a triangle,Argument of a complex number:definition of,principal,properties of,Arithmetic modulo ,Arithmetic of power series,Associated homogeneous equation,Associated homogeneous system,Associated Legendre differentialequation,Associated Legendre functions,Associative laws:of complex numbers,of matrix addition,of matrix multiplication,Asymptotically stable critical point, ,,Attractor, , ,Augmented matrix:definition of,elementary row operations on,–in reduced row-echelon form,in row-echelon form,row equivalent,Autonomous differential equation:critical points for,definition of, ,direction field for,first-order,second-order, ,translation property for,Autonomous system of differentialequations,Auxiliary equation:for a Cauchy–Euler equation,for a linear equation with constantcoefficients,rational roots of,Axis of symmetry of a beam,BBack substitution,Backward difference,Bacterial growth, ,Balancing chemical equations, –Ballistic pendulum,Banded matrix,Band-limited signals,Basis of a vector space:definition of,standard, –BC,Beams:axis of symmetry,cantilever,clamped,deflection curve of,elastic curve of, Index I-INDEXCauchy’s integral formula forderivatives,Cauchy’s residue theorem,Cauchy’s theorem,Caught pendulum,Cavalieri, Bonaventura,Cayley, Arthur,Cayley–Hamilton theorem,Center:as a critical point,of curvature,of mass, ,Central difference:approximation for derivatives,definition of,Central force,Centripetal acceleration,Centroid,Chain Rule, , APP-Chain Rule of partial derivatives,–Chain rule for vector functions,Change of scale theorem,Change of variables:in a definite integral, –in a double integral, , ,in a triple integral,Characteristic equation of a matrix,Characteristic values of a matrix,Characteristic vectors of a matrix,Chebyshev, Pafnuty,Chebyshev polynomials,Chebyshev’s differential equation,,Chemical equations, balancing of,–Chemical reactions:first-order, –second-order, , –Chemostat,Cholesky, Andre-Louis,Cholesky’s method,Circle:in complex plane,of convergence,of curvature,Circle-preserving property, –Circuits, differential equations of, , ,– , ,Circular helix,Circulation,Circulation of a vector field,Clamped end conditions of a beam,Classification of ordinary differentialequations:by linearity, ,by order, ,by type,Classification of linear partial differentialequations by type,Classifying critical points, ,– ,Clepsydra,Boundary points, , ,Boundary-value problem (BVP):deflection of a beam, –eigenfunctions for, ,eigenvalues for, , ,the Euler load,homogeneous,nonhomogeneous, ,nontrivial solutions of,numerical methods for ODEs,– ,numerical methods for PDEs, ,,for an ordinary differential equation,, – ,for a partial differential equation, ,, , , , , ,periodic, ,regular,rotating string, –second-order, ,singular,Bounding theorem for contourintegrals,Boxcar function,Branch cut,Branch of the complex logarithm,Branch point,Branch point of an electricalnetwork,Buckling modes,Buckling of a tapered column,Buckling of a thin vertical column,,Buoyant force,BVP,CCalculation of order hn,Cambridge half-life of C- ,Cantilever beam,Capacitance,Carbon dating, –Carrying capacity,Cartesian coordinates,Cartesian equation of a plane,Catenary,Cauchy, Augustin-Louis,Cauchy–Euler differential equation:auxiliary equation for,definition of,general solutions of, –method of solution,reduction to constant coefficients,Cauchy–Goursat theorem,Cauchy–Goursat theorem for multiplyconnected domains,Cauchy principal value of anintegral,Cauchy–Riemann equations,Cauchy–Schwarz inequality,Cauchy’s inequality,Cauchy’s integral formula,embedded,free,simply supported,static deflection of a homogeneousbeam, –use of the Laplace transform, –Beats,Bell curve,Bending of a thin column, ,Bendixson negative criterion,Bernoulli, Jacob,Bernoulli’s differential equation:definition of,solution of,Bessel, Friedrich Wilhelm,Bessel function(s):aging spring and,differential equations solvable in termsof, –differential recurrence relations for,–of the first kind,graphs of, , , ,of half-integral order,modified of the first kind,modified of the second kind,numerical values of,of order n,of order , ,of order ,orthogonal set of,properties of,recurrence relation for,of the second kind,spherical,zeros of,Bessel series,Bessel’s differential equation:general solution of, ,modified of order n,of order n,of order n ,parametric form of,parametric form of modifiedequation,series solution of, –Biharmonic function,Binary string of length n,Binormal,Bits,Boundary conditions (BC):homogeneous, ,mixed,nonhomogeneous,for an ordinary differential equation,, ,periodic,for a partial differential equation,–separated,time dependent,time independent,Boundary of a set, I- IndexINDEXConstants of a linear system,Constructing an orthogonal basis:for R ,for R ,for Rn,Continuing numerical method,Continuity equation, –Continuity of a complex function,Continuity of a vector function,Continuous compound interest,Contour:definition of,indented,Contour integral:bounding theorem for,definition of,evaluation of, , ,fundamental theorem for,independent of the path, – ,for the inverse Laplace transform,properties of,Contourplot,Convergence:of a complex sequence,of a complex series, –of an improper integral,of a Fourier integral,of a Fourier series,of a Fourier-Bessel series,of a Fourier-Legendre series,of an improper integral,of a complex geometric series,of a power series, ,Convolution integral, ,Convolution theorem:for the Fourier transform,inverse form,for the Laplace transform,Cooling of a cake,Cooling fin, temperature in a, –Cooling and warming, Newton’s law of,,Coordinate planes,Coordinates of a midpoint,Coordinates of a vector relative to abasis,Coordinates of a vector relative to anorthonormal basis,Coplanar vectors,Coplanar vectors, criterion for,Cosine series,Cosine series in two variables,Coulomb (C),Coulomb’s law,Counterclockwise direction,Coupled pendulums,Coupled spring/mass system,–Coupled systems,Cover-up method,Cramer’s rule, –Crank–Nicholson method, –Critical loads,Complex number(s):absolute value of,addition of,argument of,associative laws for,commutative laws for,complex powers of,conjugate of a,definition of,distributive law for,division of, ,equality of,geometric interpretation of,imaginary part of,imaginary unit,integer powers of,logarithm of, –modulus of,multiplication of, ,polar form of, –principal argument of,principal nth root of,pure imaginary,real part of,roots of a, –subtraction of,triangle inequality for,vector interpretation,Complex plane:definition of,imaginary axis of,real axis of,sets in, –Complex potential,Complex powers:of a complex number,principal value of,Complex sequence,Complex series, –Complex vector space,Complex velocity potential,Components of a vector, , ,Component of a vector on anothervector,Conformal mapping,Conformal mapping and the Dirichletproblem, –Conformal mappings, table of, APP-Conjugate complex roots, , – ,– , , –Conjugate of a complex number,,Connected region, ,Conservation of energy,Conservative force field,Conservative vector field:definition of, ,potential function for, , ,,test for, , ,Consistent system of linearequations,Constant Rules, , APP-Clockwise direction,Closed curve, ,Closed region in the complex plane,Closure axioms of a vector space,Cn[a, b] vector space,Code,Code word,Coefficient matrix,Coefficients of variables in a linearsystem,Cofactor,Cofactor expansion of a determinant,–Column bending under its ownweight,Column vector, ,Commutative laws of complexnumbers,Compartmental analysis,Compartmental models,Compatibility condition,Competition models, – ,Competitive interaction,Complementary error function, ,Complementary function:for a linear differential equation,for a system of linear differentialequations, ,Complete set of functions,Completing the square,Complex eigenvalues of amatrix,Complex form of Fourier series,–Complex function:analytic, , , ,continuous, ,definition of,derivative of,differentiable,domain of,entire, , ,exponential,hyperbolic,inverse hyperbolic,inverse trigonometric,limit of,logarithmic,as a mapping,periodic, ,polynomial,power,range of,rational,as a source of harmonic functions,as a transformation,trigonometric,as a two-dimensional fluid flow,Complex impedance,Complex line integrals:definition of,evaluation of, – , ,properties of, Index I-INDEXexpansion by cofactors,of a matrix product,minor of,of order n,of a transpose,of a triangular matrix,properties of,Diagonal matrix, ,Diagonalizability:criterion for, ,sufficient condition for, ,Diagonalizable matrix:definition of,orthogonally,Diagonalization, solution of a linearsystem of DEs by, –Difference equation replacement:for heat equation, ,for Laplace’s equation,for a second-order ODE,for wave equation, –Difference quotients,Differentiable at a point,Differential:of arc length, ,of a function of several variables,nth order operator,operator,recurrence relations, –of surface area,Differential equation (ordinary):Airy’s, , , ,associated Legendre’s,autonomous, , ,Bernoulli’s,Bessel’s, ,Cauchy–Euler,Chebyshev’s,with constant coefficients,definitions and terminology,differential form of,Duffing’s,exact,explicit solution of,families of solutions of,first-order, ,first-order with homogeneouscoefficients,general form of,general solution of, , , , ,– , –Gompertz,Hermite’s, ,higher-order, ,homogeneous, , , , ,implicit solution of,Laguerre’s, ,Legendre’s,linear, ,as a mathematical model, –modified Bessel’s,nonhomogeneous, , ,nonlinear, , , ,Curvilinear motion in the plane,Cycle of a plane autonomous system,Cycloid,Cylindrical coordinates:conversion to rectangularcoordinates,definition of,Laplacian in,triple integrals in,Cylindrical functions,Cylindrical wedge,DD’Alembert’s solution,Da Vinci, Leonardo,Damped amplitude,Damped motion, , , –Damping constant,Damping factor,Daughter isotope,DE,Decay, radioactive,Decay constant,Decoding a message, –Definite integral, definition of,Deflation, method of,Deflection curve of a beam,Deflection of a beam, – , ,Deformation of contours,Degenerate nodes:stable, –unstable, –Del operator, –DeMoivre’s formula,Density-dependent hypothesis,Dependent variables,Derivative of a complex function:of complex exponential function,of complex hyperbolic functions,of complex inverse hyperbolicfunctions,of complex inverse trigonometricfunctions,of the complex logarithmfunction,of complex trigonometricfunctions,definition of,of integer powers of z,rules for,Derivative of a definite integral, ,Derivative and integral formulas,APP- , APP-Derivative of a Laplace transform,Derivative of real function, notation for,Derivative of vector function, definitionof,Determinant(s):of a matrix,of a matrix,cofactors of,definition of,evaluating by row reduction,Critical points of an autonomous firstorder differential equation:asymptotically stable,attractor,definition of,isolated,repeller,semi-stable,unstable,Critical points for autonomous linearsystems:attractor,center,classifying,definition of,degenerate stable node, –degenerate unstable node, –locally stable,repeller,saddle point,stable node,stable spiral point,stability criteria for,unstable,unstable node,unstable spiral point,Critical points for plane autonomoussystems:asymptotically stable,classifying,stability criteria for,stable,unstable,Critical speeds,Critically damped electrical circuit,Critically damped spring/masssystem,Cross product:component for of,as a determinant,magnitude of,properties of, –test for parallel vectors,Cross ratio,Crout, Preston D.,Crout’s method,Cryptography,Curl of a vector field:definition of,as a matrix product,physical interpretation of, ,Curvature, ,Curve integral,Curves:closed,defined by an explicit function,of intersection,parallel,parametric,piecewise smooth,positive direction on,simple closed,smooth, I- IndexINDEXchange of variables,definition of,evaluation of,as an iterated integral,in polar coordinates,properties of,reversing the order of integrationin,as volume,Double pendulum,Double sine series,Double spring systems,Doubly connected domain,Downward orientation of a surface,Drag,Drag coefficient,Drag force,Draining a tank, ,Driven motion:with damping, –without damping, –Driving function, ,Drosophila,Drug dissemination, model for,Drug infusion,Duffing’s differential equation,Dulac negative criterion,Dynamical system, ,EEcosystem, states of,Effective spring constant,Effective weight,Eigenfunctions:of a boundary-value problem, ,of a Sturm-Liouville problem,–Eigenvalues of a boundary-valueproblem, – , –Eigenvalues of a matrix:approximation of,complex, ,definition of, ,of a diagonal matrix,distinct-real,dominant, –of an inverse matrix,of multiplicity m,of multiplicity three,of multiplicity two,repeated,of a singular matrix,of a symmetric matrix, ,of a triangular matrix,Eigenvector(s) of a matrix:complex,definition of, ,dominant,of an inverse matrix,orthogonal,Elastic curve,Electrical circuits, , , , – ,–Directional derivative:computing,definition of,for functions of three variables,for functions of two variables,–maximum values of, –Dirichlet condition,Dirichlet problem:for a circular plate,for a cylinder, – ,definition of, – ,exterior,harmonic functions and,for a planar region,for a rectangular region,for a semicircular plate,solving using conformalmapping,for a sphere,superposition principle for,Disconnected region,Discontinuous coefficients, –Discrete compartmental models,–Discrete Fourier transform,Discrete Fourier transform pair,Discrete signal,Discretization error,Distance formula,Distance from a point to a line,Distributions, theory of,Distributive law:for complex numbers,for matrices,Divergence of a vector field:definition of,physical interpretation of, ,Divergence theorem,Division of two complex numbers,,Domain:in the complex plane,of a complex function,of a function,of a function of two variables,of a solution of an ODE,Dominant eigenvalue,Dominant eigenvector,Doolittle, Myrick H.,Doolittle’s method, –Dot notation for differentiation,Dot product:component form of,definition of, ,properties of,in terms of matrices,as work,Double cosine series,Double eigenvalues,Double integral:as area of a region, ,as area of a surface,Differential equation (ordinary):—(Cont.)with nonpolynomial coefficients,normal form of,notation for,order of,ordinary,ordinary points of, –parametric Bessel,parametric modified Bessel,particular solution of, , ,piecewise linear,with polynomial coefficients, ,Ricatti’s,second-order, , , , – ,self-adjoint form of, –separable, –singular points of,singular solution of,solution of, –standard form of a linear, , ,substitutions in,superposition principles for linear,,system of, , , ,Van der Pol’s, ,with variable coefficients, , ,,Differential equation (partial):classification of linear second-order,**

**definition of,diffusion, ,heat, – , – ,homogeneous linear second-order,Laplace’s, , ,linear second-order,nonhomogeneous linear second-order,**

**order of,Poisson’s,separable, –solution of,superposition principle forhomogeneous linear,time dependent,time independent,wave, , – , – ,Differential form, ,Differential operator, nth order,Differential recurrence relation,–Differentiation of vector functions, rulesof,Diffusion equation, ,Dimension of a vector space,Dirac delta function:definition of,Laplace transform of,Direction angles,Direction cosines,Direction field,Direction numbers of a line,Direction vector of a line, Index I-INDEXFast Fourier transform, computingwith,Fibonacci, Leonardo,Fibonacci sequence,Fick’s law,Filtered signals,Finite difference approximations,– , , ,Finite difference equation,Finite difference method:explicit, ,implicit,Finite differences,Finite dimensional vector space,First buckling mode,First harmonic,First moments,First normal mode,First octant,First shifting theorem,First standing wave,First translation theorem:form of,inverse form of,First-order chemical reaction,First-order differential equations:applications of, , ,solution of, , , , –First-order initial-value problem,First-order Runge–Kutta method,First-order system,Five-point approximation for Laplace’sequation,Flexural rigidity,Flow:around a corner,around a cylinder,of heat,steady-state fluid,Fluctuating population,Flux and Cauchy’s integral formula,Flux through a surface,Folia of Descartes, ,Forced electrical vibrations,Forced motion:with damping,without damping,Forcing function,Forgetfulness,Formula error,Forward difference,Fossil, age of,Fourier coefficients,Fourier cosine transform:definition of,operational properties of,Fourier integral:complex form, –conditions for convergence,cosine form,definition of, –sine form,Fourier integrals,Euler’s constant,Euler’s formula,Euler’s method:error analysis of, , –for first-order differential equations,,for second-order differentialequations,for systems of differentialequations,Evaluation of real integrals by residues,–Evaporating raindrop, ,Evaporation,Even function:definition of,properties of,Exact differential:definition of,test for,Exact differential equation:definition of,solution of,Existence and uniqueness of a solution,, ,Existence of Fourier transforms,Existence of Laplace transform,Expansion of a function:in a complex Fourier series,in a cosine series,in a Fourier series, –in a Fourier–Bessel series,in a Fourier–Legendre series,half-range,in a Laurent series, –in a power series, –in a sine series,in terms of orthogonal functions,–Explicit finite difference method,Explicit solution,Exponential form of a Fourierseries,Exponential function:definition of,derivative of,fundamental region of,period of,properties of,Exponential order,Exponents of a singularity,Exterior Dirichlet problem,External force,Extreme displacement,FFalling bodies, mathematical models of,– ,Falling chain,Falling raindrops, ,Family of solutions,Farads (f),Fast Fourier transform, ,Electrical networks, , ,Electrical vibrations:critically damped,forced, –free,overdamped,simple harmonic,underdamped,Elementary functions,Elementary matrix,Elementary operations for solving linearsystems,Elementary row operations on a matrix:definition of,notation for,Elimination method(s):for a system of algebraic equations,– ,for a system of ordinary differentialequations,Elliptic partial differentialequation,Elliptical helix,Embedded end conditions of a beam,,Empirical laws of heat conduction,Encoding a message,Encoding a message in the Hamming( , ) code,Entire function,Entries in a matrix,Epidemics, , ,Equality of complex numbers,Equality of matrices,Equality of vectors, , ,Equation of continuity, –Equation of motion,Equidimensional equation,Equilibrium point,Equilibrium position of a spring/masssystem,Equilibrium solution, ,Equipotential curves,Error(s):absolute,discretization,formula,global truncation,local truncation,percentage relative,relative,round-off,sum of square,Error function, ,Error-correcting code, –Error-detecting code, ,Escape velocity,Essential singularity,Euclidean inner product,Euler, Leonhard,Euler equation,Euler load,Euler–Cauchy equation, I- IndexINDEXof a homogeneous system of lineardifferential equations,of a linear first-order equation,of linear higher-order equations,–of modified Bessel’s equation,of a nonhomogeneous lineardifferential equation,of a nonhomogeneous system of lineardifferential equations,of parametric form of Bessel’sequation,of parametric form of modifiedBessel’s equation,of a second-order Cauchy–Eulerequation, –Generalized factorial function, APP-Generalized Fourier series,Generalized functions,Generalized length,Geometric series,Geometric vectors,George Washington monument,Gibbs phenomenon,Global truncation error,Globally stable critical point,Gompertz differential equation,Goursat, Edouard,Gradient:of a function of three variables,–of a function of two variables, –geometric interpretation of, –vector field, ,Gram–Schmidt orthogonalizationprocess, – ,Graphs:of a function of two variables,of level curves,of level surfaces,of a plane,Great circles,Green, George,Green’s function:for an initial-value problem,for a boundary-value problem,relationship to Laplace transform,–for a second-order differentialequation,for a second-order differentialoperator,Green’s identities,Green’s theorem in the plane,Green’s theorem in -space,Growth and decay, , –Growth constant,Growth rate, relative,HHalf-life:of carbon- ,definition of,generalized,gradient of,graph of,harmonic, , – ,inner product of,integral defined, –input, ,odd,orthogonal,output, ,partial derivative of, –periodic,polynomial,potential, ,power, ,of a real variable,range of,rational,sine integral, ,stream,of three variables,as a two-dimensional flow,of two variables,vs. solution,vector,weight,Fundamental angular frequency,Fundamental critical speed,Fundamental frequency,Fundamental matrix:definition of,matrix exponential as a,Fundamental mode of vibration,Fundamental period, ,Fundamental region of the complexexponential function,Fundamental set of solutions:definition of, ,existence of, ,Fundamental theorem:of algebra, –of calculus, ,for contour integrals,for line integrals,Gg, ,Galileo Galilei, ,Gamma function, , , APP-Gauss’ law, ,Gauss’ theorem,Gaussian elimination,Gauss–Jordan elimination,Gauss–Seidel iteration, ,General form of an ordinary differentialequation,General solution:of Bessel’s equation, ,definition of, , , ,of a homogeneous linear differentialequation,of a homogeneous second-order lineardifferential equation, –Fourier series:complex, –conditions for convergence,cosine,definition of,generalized,sine,in two variables,Fourier sine transform:definition of,operational properties of,Fourier transform pairs,Fourier transforms:definitions of,existence of,operational properties of, –Fourier–Bessel series:conditions for convergence,definition of, –Fourier–Legendre series:conditions for convergence,definition of, – , ,Fourth-order partial differential equation,,Fourth-order Runge–Kutta methods:for first-order differential equations,,for second-order differentialequations,for systems of differentialequations,Free electrical vibrations,Free motion of a spring/mass system:damped, –undamped,Free vectors,Free-end conditions of a beam, ,Frequency of free vibrations,Frequency filtering,Frequency response curve,Frequency spectrum,Fresnel sine integral function, ,Frobenius, Georg Ferdinand,Frobenius, method of,Frobenius’ theorem,Fubini, Guido,Fubini’s theorem,Fulcrum supported ends of a beam,Full-wave rectification of sine,Function(s):complementary,complementary error, ,of a complex variable,continuous,defined by an integral, ,differentiable,directional derivative of, –domain of,driving, , ,error, ,even,forcing, , ,Fresnel sine integral, Index I-INDEXIndependent variables,Indicial equation,Indicial roots,Inductance, , –Infinite-dimensional vector space,Infinite linearly independent set,Infinite series of complex numbers:absolute convergence,definition of,convergence of,geometric,necessary condition forconvergence,nth term test for divergence,sum of,Initial conditions (IC), , ,Initial-value problem (IVP):definition of, ,first-order, ,nth-order, ,second-order,for systems of linear differentialequations,Inner partition,Inner product:of two column matrices,definition of, , ,properties of, ,space,of two functions,of two vectors, ,Inner product space,Input function, ,Insulated boundary,Integers:modulo ,modulo ,Integrable function:of three variables,of two variables,Integral-defined function, –Integral equation,Integral transform:definition of,Fourier,Fourier cosine,Fourier sine,inverse, ,kernel of,Laplace, ,pair,Integral of a vector function,Integrating factor, , –Integration along a curve,Integration by parts,Integrodifferential equation,Interest, compoundedcontinuously,Interior point,Interior mesh points,Interior point of a set in the complexplane,Interpolating function,Homogeneous systems of linearalgebraic equations:definition of, ,matrix form of,nontrivial solutions of,properties of,trivial solution of,Homogeneous systems of lineardifferential equations:complex eigenvalues, –definition of,distinct-real eigenvalues,fundamental set of solutions for,general solution of,matrix form of,repeated eigenvalues, –superposition principle for,Hoëné-Wronski, Jósef Maria,Hooke’s law, ,Horizontal component of a vector,Hurricane Hugo, –Huygens, Christiaan,Hydrogen atoms, distance between,–Hyperbolic functions, complex:definitions of,derivatives of,zeros of,Hyperbolic partial differentialequation,IIC,i, j vectors,i, j, k vectors,Iceman (Ötzi),Identity matrix,Identity property of power series,Ill-conditioned system of equations,Image of a point under atransformation,Images of curves,Imaginary axis,Imaginary part of a complex number,Imaginary unit,Immigration model, ,Impedance,Implicit finite difference method,Implicit solution,Improper integral:convergent, ,divergent, ,Improved Euler method,Impulse response,Incompressible flow, ,Incompressible fluid,Inconsistent system of linearequations,Indefinite integral, ,Indented contours,Independence of path:definition of, ,test for, , , ,of a drug,of plutonium- ,of radium- ,of uranium- ,Half-plane,Half-range expansions,Half-wave rectification of sine,Hamilton, William Rowan,Hamming ( , ) code,Hamming ( , ) code,Hamming, Richard W.,Hard spring,Harmonic conjugate functions,Harmonic function, , – ,Harmonic function, transformationtheorem for,Harmonic functions and the Dirichletproblem,Harvesting,Heart pacemaker, model for, ,Heat equation:derivation of one-dimensionalequation,difference equation replacement for,,and discrete Fourier series,and discrete Fourier transform,–one-dimensional, –in polar coordinates,solution of,two-dimensional,Heaviside, Oliver,Heaviside function,Helmholtz’s partial differentialequation,Helix:circular,elliptical,pitch of,Henrys (h),Hermite, Charles,Hermite polynomials,Hermite’s differential equation,,Higher-order ordinary differentialequations, , ,Hinged end of a beam,Hitting bottom,Hole through the Earth,Homogeneous boundary conditions,, ,Homogeneous boundary-value problem,,Homogeneous first-order differentialequation:definition of,solution of,Homogeneous function,Homogeneous linear differentialequation:ordinary, ,partial, I- IndexINDEXLeaking tank, –Leaning Tower of Pisa,Learning theory,Least squares, method of, –Least squares line, ,Least squares parabola, –Least squares solution,Legendre, Adrien-Marie,Legendre associated functions,Legendre functions, – , ,–Legendre polynomials:first six,graphs of,properties of,recurrence relation for,Rodriques’ formula for,Legendre’s differential equation:associated,of order n,series solution of, –Leibniz notation,Leibniz’s rule,Length of a space curve,Length of a vector:in -space,in n-space,Leonardo da Vinci,Level curves, ,Level of resolution of a mathematicalmodel,Level surfaces,L’Hôpital’s rule, – , ,,Liber Abbaci,Libby half-life,Libby, Willard F.,Liebman’s method,Limit cycle, ,Limit of a function of a complexvariable:definition of,properties of,Limit of a vector function,Line of best fit,Line integrals:around closed paths, , ,–as circulation,complex,in the complex plane,definition of, –evaluation of, , ,fundamental theorem for,independent of the path,in the plane,in space,as work,Line segment,Lineal element,Linear algebraic equations, systems of,–Linear combination of vectors,Kirchhoff’s point and loop rules,Kirchhoff’s second law, ,LLagrange’s identity,Laguerre polynomials,Laguerre’s differential equation,,Laplace, Pierre-Simon Marquis de,Laplace transform:behavior as s S q,of Bessel function of order n ,conditions for existence,convolution theorem for, –definition of, ,derivatives of a,of derivatives,of differential equations,differentiation of,of Dirac delta function,existence of,inverse of, ,of an integral,as a linear transform,and the matrix exponential,of a partial derivative,of a periodic function,of systems of ordinary differentialequations,tables of, , APP-translation theorems for, ,of unit step function,Laplace’s equation, , , ,– , , ,Laplace’s partial differential equation:in cylindrical coordinates,difference equation replacementfor,in polar coordinates,maximum principle for,solution of,in three dimensions,in two dimensions, – ,Laplacian:in cylindrical coordinates,definition of, ,in polar coordinates,in rectangular coordinates,in spherical coordinates,in three dimensions,in two dimensions,Lascaux cave paintings, dating of,Latitude,Lattice points,Laurent series,Laurent’s theorem, –Law of conservation of mechanicalenergy,Law of mass action,Law of universal gravitation,Laws of exponents for complexnumbers,Laws of heat conduction,Interval:of convergence,of definition of a solution,of existence and uniqueness,of validity of solution,Invariant region:definition of,Types I and II,Invasion of the marine toads,Inverse cosine function:derivative of,as a logarithm,Inverse hyperbolic functions:definition of,derivatives of,as logarithms,Inverse integral transform:Fourier,Fourier cosine,Fourier sine,Laplace, ,Inverse of a matrix:definition of,by the adjoint method, –by elementary row operations,properties of,using to solve a system, –Inverse power method,Inverse sine function:definition of,derivative of,as a logarithm,Inverse tangent function:derivative of,as a logarithm,Inverse transform, , ,Inverse transformation,Inverse trigonometric functions:definitions of,derivatives of,Invertible matrix,Irregular singular point,Irrotational flow, ,Isocline, ,Isolated critical point,Isolated singularity:classification of, –definition of,Iterated integral,IVP,JJacobian determinant,Jacobian matrix,Joukowski airfoil,Joukowski transformation,KKepler’s first law of planetarymotion,Kernel of an integral transform,Kinetic friction,Kirchhoff’s first law, Index I-INDEXLR-series circuit, differential equationof,LU-decomposition of a matrix,LU-factorization of a matrix, –MMaclaurin series, , ,Maclaurin series representation:for the cosine function,for the exponential function,for the sine function,Magnification in the z-plane,Magnitude of a complex number,Magnitude of the cross product, ,Magnitude of a vector, ,Main diagonal entries of a matrix,Malthus, Thomas,Malthusian model,Mapping, ,Mapping, conformal,Marine toad invasion model,Mass:center of,as a double integral,of a surface,Mass action, law of,Mathematical model, – , ,,Matrix (matrices):addition of,adjoint,associative law, ,augmented,banded,characteristic equation of,coefficient,column vector,commutative law,definition of,determinant of, –diagonal, ,diagonalizable,difference of,distributive law,dominant eigenvalue of, –eigenvalues of, , , ,eigenvectors of, , , ,elementary,elementary row operations on,–entries (or elements) of,equality of,exponential, –fundamental,identity,inverse of, , – ,invertible,Jacobian,lower triangular, ,LU-factorization of, –main diagonal entries of,multiplication,multiplicative identity,nonhomogeneous,solution of,superposition principle for,Linear spring,Linear system:of algebraic equations,definition of,of differential equations, ,rank and,Linear transform, ,Linearity:of a differential operator,of the inverse Laplace transform,of the Laplace transform,Linearity property,Linearization:of a function f(x) at a number, ,of a function f(x, y) at a point,of a nonlinear differential equation,of a nonlinear system of differentialequations, –Linearly dependent set of functions,Linearly independent set offunctions,Lines of force,Lines in space:direction numbers for,direction vector for,normal,parametric equations of,symmetric equations of,vector equation for,Liouville’s theorem,Lissajous curve, ,Local linear approximation, , ,Local truncation error,Locally stable critical point,Logarithm of a complex number:branch cut for,branch of,definition of,derivative of,principal branch,principal value of,properties of,Logistic curve,Logistic equation:definition of, , –modifications of,solution of,Logistic function,Logistic growth, –Longitude,Loop rule, Kirchhoff’s,Losing a solution,Lotka–Volterra competition model,Lotka–Volterra predator-prey model,Lower bound for the radius ofconvergence,Lower triangular matrix,LRC-series circuit:differential equation of, , –integrodifferential equation of,Linear dependence:of a set of functions, –of a set of vectors, ,of solution vectors,Linear donor-controlled hypothesis,Linear equation in n variables,Linear first-order differential equation:definition of, ,general solution of,homogeneous,integrating factor for,method of solution,nonhomogeneous,singular points of,standard form of,variation of parameters for,Linear fractional transformation,Linear independence:of a set of functions, –of a set of vectors,of solution vectors,of solutions of linear DEs, –Linear momentum,Linear operator, ,Linear ordinary differential equations:applications of, , – , ,associated homogeneous,auxiliary equation for, ,boundary-value problems for, ,,with constant coefficients,complementary function for,definition, ,first-order, ,general solution of, , , ,– , –higher-order, , , ,homogeneous, , ,indicial equation for,initial-value problems for, , , ,, ,infinite series solutions for, ,nonhomogeneous, , ,nth-order initial-value problem for,ordinary points of, –particular solution for, , ,,piecewise,reduction of order, –second-order, , , – ,singular points of, , – ,,standard forms of, , , ,,superposition principles for, ,with variable coefficients, , ,, , – ,Linear partial differential equation,Linear regression,Linear second-order partial differentialequations:classification of,homogeneous, I- IndexINDEXNegative of a vector,Neighborhood,Net flux,Networks, ,Neumann condition,Neumann problem:for a circular plate,for a rectangle,Newton, Isaac,Newton’s dot notation,Newton’s law of air resistance,Newton’s law of cooling/warming,,Newton’s law of universal gravitation,Newton’s laws of motion:first,second, , , –Nilpotent matrix, , ,Nodal line,Nodes:of a plane autonomous system,– ,of a standing wave,Nonconservative force,Nonelementary integral, ,Nonhomogeneous boundarycondition,Nonhomogeneous boundary-valueproblem, , , ,Nonhomogeneous linear differentialequation:definition of,general solution of,initial-value problem for,ordinary, ,partial,particular solution of,Nonhomogeneous systems of algebraicequations,Nonhomogeneous systems of lineardifferential equations:complementary function of,definition of,general solution of,initial-value problem for,matrix form of,normal form of,particular solution of,solution vector of,Nonisolated singular point,Nonlinear mathematical models,,Nonlinear ordinary differentialequation, ,Nonlinear oscillations,Nonlinear pendulum, , –Nonlinear spring,Nonlinear systems of differentialequations, ,Nonoriented surface,Nonpolynomial coefficients,Nonsingular matrix, , ,Nontrivial solution,for nonhomogeneous systems of linearDEs,Method of Frobenius, –Method of isoclines,Method of least squares, –Method of separation of variables:for ordinary differential equations,–for partial differential equations,Method of undetermined coefficients:for nonhomogeneous linear DEs,for nonhomogeneous systems of linearDEs, –Method of variation of parameters:for nonhomogeneous linear DEs, ,–for nonhomogeneous systems of linearDEs, –Midpoint of a line segment in space,Minor determinant,Mises, Richard von,Mixed boundary conditions,Mixed partial derivatives:definition of,equality of,Mixtures, , ,ML-inequality,Mm,n vector space,Möbius strip,Modeling process, steps in,Modifications of the logistic equation,–Modified Bessel equation:of order n,parametric form of,Modified Bessel function:of the first kind,of the second kind,Modulus of a complex number,Moments of inertia,Moments of inertia, polar,Motion:on a curve,in a force field,Moving trihedral,Multiplication:of complex numbers, ,of matrices,of power series,by scalars, , ,Multiplication rule for undeterminedcoefficients,Multiplicative inverse of a matrix,Multiplicity of eigenvalues, ,–Multiply connected domain,Multiply connected region,Multistep numerical method,Nn-dimensional vector, –Negative criteria, ,Negative direction on a curve,Matrix (matrices):—(Cont.)multiplicative inverse,nilpotent, , ,null-space of,nonsingular, ,order n,orthogonal, , –orthogonally diagonalizable,partitioned,powers of,product of,rank of, –reduced row-echelon form,rotation,row-echelon form,row equivalent,row reduction of,row space of,row vector,scalar,scalar multiple of,similar,singular,size,skew-symmetric,sparse,square,stochastic,sum of,symmetric, ,of a system,trace of a,transpose of,triangular, ,tridiagonal,upper triangular,zero,Matrix addition, properties of,Matrix exponential:computation of, ,definition of,derivative of,as a fundamental matrix,as an inverse Laplace transform,Matrix form of a system of linearalgebraic equations,Matrix form of a system of lineardifferential equations,Maximum principle,Maxwell, James Clerk,Maxwell’s equations,Meander function,Memorization, mathematical model for,Meridian,Mesh:points, ,size,Message,Methane molecule, –Method of deflation,Method of diagonalization:for homogeneous systems of linearDEs, Index I-INDEXOrthogonal diagonalizability:criterion for,definition of,Orthogonal eigenvectors, –Orthogonal family of curves, ,Orthogonal functions,Orthogonal matrix:constructing an,definition of,Orthogonal projection of a vector onto asubspace,Orthogonal series expansion, –Orthogonal set of functions,Orthogonal with respect to a weightfunction,Orthogonal surfaces at a point,Orthogonal trajectories,Orthogonal vectors,Orthogonally diagonalizable matrix,Orthonormal basis:definition of,for Rn,for a vector space,Orthonormal set of functions,Orthonormal set of vectors,Oscillating chain,Osculating plane,Ötzi (the iceman),Output function, ,Overdamped electrical circuit,Overdamped spring/mass system,Overdamped system,Overdetermined system of linearalgebraic equations,Overtones,PPacemaker, heart, ,Parabolic partial differentialequation,Parallel vectors:definition of,criterion for,Parallels,Parametric curve:closed,definition of,piecewise smooth,positive direction on,simple closed,smooth, ,in space,Parametric equations for a line inspace,Parametric form of Bessel equation:of order n,of order n,in self-adjoint form,Parametric form of modified Besselequation of order n,Parent isotope,Paris Guns, –Parity,predictor-corrector methods, ,Runge–Kutta methods, , ,,shooting method,single-step method,stability of,stable,starting method,unstable,using the tangent line,Numerical solution curve,Numerical solver,Numerical values of Besselfunctions,OOctants,Odd function:definition of,properties of,ODE,Ohms (),Ohm’s law,One-dimensional heat equation:definition of, –derivation of, –One-dimensional phase portrait,One-dimensional wave equation:definition of, –derivation of,One-parameter family of solutions,One-to-one transformation,Open annulus,Open disk,Open region,Open set,Operational properties of the Laplacetransform, , , , , ,, , , ,Operator, differential, ,Order of a differential equation, ,Order, exponential,Order of a Runge–Kutta method,–Order of integration, , –Ordered n-tuple, – ,Ordered pair, , , ,Ordered triple, , ,Ordinary differential equation,Ordinary point of an ordinary differentialequation:definition of, ,solution about, –Orientable surface:definition of,of a closed,Orientation of a surface:downward,inward,outward,upward,Orthogonal basis for a vector space, ,, ,Norm:of a column vector (matrix),of a function,of a partition, ,square,of a vector, , ,Normal component of acceleration,Normal form:of an ordinary differential equation,of a system of linear first-orderequations,Normal line to a surface,Normal modes,Normal plane,Normal vector to a plane,Normalization of a vector, ,Normalized eigenvector,Normalized set of orthogonalfunctions,Notation for derivatives,n-parameter family of solutions,n-space (Rn):coordinates relative to an orthonormalbasis,dot (or inner product) in,length (or norm) in,orthogonal vectors in,orthonormal basis for,standard basis for,unit vector in,vector in,zero vector in,nth root of a nonzero complex number,–nth roots of unity,nth term test for divergence,nth-order differential equation expressedas a system,nth-order differential operator,nth-order initial-value problem, ,nth-order ordinary differential equation,– , ,Nullcline,Null-space of a matrix,Number of parameters in a solution of alinear system of equations,Numerical methods:absolute error in,Adams–Bashforth–Moulton,adaptive methods,continuing method,Crank–Nicholson method,deflation method,errors in, ,Euler’s method, , – , ,,finite-difference methods, , ,,Gauss–Seidel iteration,improved Euler’s method,inverse power method,multistep method,power method, I- IndexINDEXPoisson integral formula:for unit disk,for upper half-plane, –Poisson’s partial differentialequation,Polar coordinates, – ,Polar form of a complex number,–Polar moment of inertia,Polar rectangle,Pole:definition of, –of order n, – ,residue of, –simple, –Polynomial function,Population, mathematical models for, ,, , , – , – , ,Position vector, ,Positive criteria,Positive direction on a curve, ,–Potassium-argon dating, ,Potassium- decay,Potential:complex,complex velocity,energy,function, ,Power function,Power method,Power rule of differentiation, , APP-Power series:absolute convergence of,arithmetic of,center of,circle of convergence,convergence of,defines a function,definition of,differentiation of,identity property of,integration of,interval of convergence,Maclaurin,radius of convergence,ratio test for,represents a continuous function,represents an analytic function,review of, –shift of summation index,solutions of differential equations,–Taylor,Powers, complex,Powers, integer,Powers of a matrix, –Predator-prey model, – , –Predictor-corrector methods, ,Prime meridian,Prime notation,Principal argument of a complexnumber,of the complex hyperbolic sine andcosine,Period of free vibrations,Periodic boundary conditions, ,Periodic boundary-value problem,Periodic driving force, ,Periodic extension,Periodic functions:definition of, , ,Laplace transform of,Periodic solution of a plane autonomoussystem,Phase angle,Phase line,Phase plane, , ,Phase portrait:for first-order differentialequations,for systems of two linear first-orderdifferential equations, ,for systems of two nonlinear firstorder differential equations,–Phase-plane method,Physical pendulum,Piecewise-continuous function:definition of,Laplace transform of,Piecewise-defined solution of an ordinarydifferential equation, ,Piecewise-linear differential equation,,Piecewise-smooth curve,Pin supported end of a beam,Pitch, –Pitch of a helix,Planar transformation,Plane(s):Cartesian equation of,curvilinear motion in,graphs of, –line of intersection of two,normal vector to,perpendicular to a vector, –phase, ,point-normal form of,trace of,vector equation of,Plane autonomous system:changed to polar coordinates,types of solutions of a,in two variables,Plane autonomous system, solutions of:arc,constant,periodic (cycle),Plucked string, , – ,Plutonium- , half-life of,Poincare–Bendixson theorems, ,–Point-normal form of an equation of aplane,Point rule, Kirchhoff’s,Parity check bits,Parity check code,Parity check equations,Parity check matrix,Parity error,Partial derivatives:Chain Rule for, –definition of,generalizations of,higher-order,mixed,with respect to x,with respect to y,second-order,symbols for,third-order,tree diagrams for,Partial differential equation, linearsecond order:definition of,elliptic, ,homogeneous,hyperbolic, , ,linear,nonhomogeneous,parabolic, , ,separable,solution of,Partial fractions, use of, , –Particular solution:definition of, ,of Legendre’s equation, –of a nonhomogeneous system of linearDEs, , , ,by undetermined coefficients, –by variation of parameters, –Partitioned matrix,Path independence:definition of,tests for, ,Path of integration,Pauli spin matrices,PDE,Pendulum:ballistic,double, –free damped,linear,nonlinear,oscillating,physical,rotating,simple, –spring, –spring-coupled,of varying length, –whirling,Percentage relative error,Perihelion,Period:of the complex exponentialfunction,of the complex sine and cosine, Index I-INDEXzero-input,zero-state,Rest point,Rest solution,Restocking a fishery,Reversing the order of integration,Review of power series, –Riccati’s differential equation,Riemann mapping theorem,Riemann sum,Right-hand rule,RK method, ,RK method for systems, ,RKF method,Robin condition,Robins, Benjamin, ,Rocket motion, ,Rodrigues’ formula,Roll,Root mean square,Root test,Roots of a complex number,–Rope pulled upward by a constantforce,Rotational flow,Rotating fluid, shape of,Rotating pendulum,Rotating rod with a sliding bead,Rotating shaft,Rotating string, –Rotation and translation,Rotation in the z-plane,Round-off error,Row-echelon form,Row equivalent matrices,Row operations, use in finding theinverse of a nonsingularmatrix,Row reduction,Row space,Row vector, ,Row vector form of an autonomoussystem,R ,R ,Rules of differentiation, , APP-Runge–Kutta methods:first-order,fourth-order, ,second-order,Runge–Kutta–Fehlberg method,Rutherford, Ernest,SSaddle point,Sample point, , , ,Sampling Theorem,Sawing wood,Sawtooth function,Scalar, ,Scalar acceleration,Scalar matrix,Reactance,Reactions, chemical, – ,Real axis,Real integrals, evaluation by residues,–Real part of a complex number,Real power function,Real vector space,Real-valued function, periodic,Reciprocal lattice,Rectangular coordinates, –Rectangular pulse,Rectified sine wave,Rectifying plane,Recurrence relation:three term,two term,Reduced row-echelon form of amatrix,Reduction of order, –Reflecting surface, –Region:closed,in the complex plane,connected,disconnected,with holes,image of,of integration,invariant,multiply connected,open,simply connected,type I (II), ,Regression line,Regular singular point of an ordinarydifferential equation:definition of,solution about,Regular Sturm–Liouville problem:definition of,properties of,Relative error:definition of,percentage,Relative growth rate,Removable singularity, ,Repeller, , ,Residue(s):definition of,evaluation of integrals by, ,at a pole of order n,at a simple pole,Residue theorem,Resistance,Resonance, pure, –Resonance curve,Resonance frequency,Response:definition of,impulse,of a series circuit,of a system, , ,Principal axes of a conic,Principal branch of the logarithm,Principal logarithmic function,Principal normal vector,Principal nth root of a complexnumber,Principal part of a Laurent series,,Principal value:of a complex power,of an integral,of logarithmic function,Product Rule, , APP-Projectile motion, , – ,–Projection of a vector onto another,p-series,Pure imaginary number,Pure resonance, –Pursuit curve, –Pythagorean theorem,QQuadratic form:definition of,as a matrix product,Qualitative analysis:of first-order differential equations,,of second-order differential equations,, – , –of systems of differential equations,– ,Quasi frequency,Quasi period,Quotient Rule, , APP-RRadial symmetry,Radial vibrations,Radioactive decay, , –Radioactive decay series,Radiogenic isotope,Radiometric dating methods,Radius of convergence, ,Radius of curvature,Radius of gyration,Raindrop, velocity of evaporating,Raleigh differential equation,Range of a complex function,Range of a projectile:with air resistance,with no air resistance,Rank of a matrix:definition of,by row reduction, –Ratio test, ,Rational function,Rational roots of a polynomialequation,Rayleigh quotient,RC-series circuit, differential equationof, I- IndexINDEXSoft spring,Solar collector,Solenoidal vector field,Solution curve:of an autonomous differentialequation,definition of,Solution of a linear second-order partialdifferential equation:definition of,particular, –Solution of a linear system of DEs,methods of, ,Solution of an ordinary differentialequation:about an ordinary point,about a regular singular point,definition of,domain of,existence and uniqueness of,– ,explicit,general, , , ,implicit,integral-defined, – , ,interval of definition,losing a,nontrivial,n-parameter family of,particular, ,piecewise-defined,singular,trivial,verification of a, ,Solution of a system of differentialequations, , ,Solution of a system of linear algebraicequations:definition of, ,number of parameters in a, –Solution space,Solution vector,Source,Space curve:definition of,length of,Span,Spanning set,Sparse matrix,Specific growth rate,Speed,Spherical Bessel function:of the first kind,of the second kind,Spherical coordinates:conversion to cylindrical coordinates,–conversion to rectangular coordinates,–definition of,Laplacian in,triple integrals in,Spherical wedge,radius of convergence, ,solutions of ordinary differentialequations,sine,Taylor, , –tests for convergence, ,trigonometric,Series circuits, , , –Sets in the complex plane, –Shaft through the Earth,Shifting summation index,Shooting method,Shroud of Turin, ,Sifting property,Signal processing,Similar matrices,Simple closed curve,Simple harmonic electricalvibrations,Simple harmonic motion,Simple pendulum,Simple pole,Simply connected domain, ,Simply connected region,Simply supported end of a beam,Simply supported end conditions of abeam,Sine integral function, , ,Sine series,Sine series in two variables,Single-step numerical method,Singular boundary-value problem,Singular matrix, ,Singular point of a complex function:definition of,essential,isolated,nonisolated,pole,removable,Singular point of a linear ordinarydifferential equation:definition of, , ,at infinity,irregular,regular,solution about, ,Singular solution,Singular Sturm–Liouville problem:definition of,properties of,Sink,Sinking in water,SIR model,Skew-symmetric matrix,Skydiving, , ,Sliding bead,Sliding box on an inclined plane,Slope field,Smooth curve,Smooth function,Smooth surface,Snowplow problem,Scalar multiple:of a matrix,of vectors, , ,Scalar triple product,Scaling,Schwartz, Laurent,Schwarz–Christoffel transformations,–Second derivative of a complex function,–Second moments,Second-order boundary-value problem,, , ,Second-order chemical reaction, ,–Second-order DE as a system, ,Second-order difference equation,Second-order differential operator,Second-order initial-value problem, ,,Second-order Runge–Kuttamethod,Second shifting theorem,Second translation theorem:alternative form of,form of,inverse form of,Self-adjoint form of a linear secondorder DE,Semi-stable critical point,Separable first-order differentialequation:definition of,solution of,Separable partial differentialequations,Separated boundary conditions,Separation constant,Sequence:convergent,criterion for convergence,definition of,Sequence of partial sums,Series (infinite):absolutely convergent, ,circle of convergence,complex Fourier,of complex numbers,convergent, ,cosine,definition of,Fourier,Fourier–Bessel,Fourier–Legendre,geometric,interval of convergence,Laurent, –Maclaurin, , ,necessary condition forconvergence,nth term test for,orthogonal, –power, , Index I-INDEXoverdetermined,solution of,superposition principle for,underdetermined,Systems of first-order differentialequations:autonomous, ,definition of, , , ,linear form of,matrix form of, ,solution of, ,Systems of linear algebraic equations,methods for solving:using augmented matrices, –using Cramer’s rule, –using elementary operations,using elementary row operations,–using the inverse of a matrix,using LU-factorization,Systems of linear first-order differentialequations, methods for solving:using diagonalization, ,using the Laplace transform, ,using matrices, –using a matrix exponential, –using systematic elimination, –using undetermined coefficients,using variation of parameters, –TTables:of conformal mappings, APP-of derivatives and integrals, APP-of Laplace transforms, , APP-of trial particular solutions,Tangent line,Tangent plane to a surface:definition of,equation of,vector equation of,Tangent vectors, , –Tangential component ofacceleration,Taylor series, ,Taylor’s theorem,Telegraph equation,Telephone wires, shape of, – ,–Temperature:in an annular cooling fin, –in an annular plate, ,in a circular plate, ,in a circular cylinder, – ,between concentric cylinders,between concentric spheres, ,in a cooling/warming body, , ,–in an infinite cylinder,in an infinite plate,in a one-eighth annular plate,in a quarter-annular plate,in a quarter-circular plate,Subscript notation,Subspace:criteria for,definition of,Substitutions:in differential equations,in integrals, ,Subtraction of vectors, ,Successive mappings,Sum of square errors,Sum Rule, , APP-Summation index, shifting of,Superposition principle:for BVPs involving the waveequation,for Dirichlet’s problem for arectangular plate, –for homogeneous linear ODEs,for homogeneous linear PDEs,for homogeneous systems of linearalgebraic equations,for homogeneous systems of linearDEs,for nonhomogeneous linear ODEs,Surface, orientable, –Surface area:differential of,as a double integral,Surface integral:applications of, ,definition of,evaluation of,over a piecewise defined surface,Suspended cable, mathematical model of,– , , –Suspension bridge, , ,Sylvester, James Joseph,Symmetric equations for a line,Symmetric matrix:definition of,eigenvalues for,eigenvectors of,orthogonality of eigenvectors,–Syndrome,Systematic elimination,Systems of DEs:higher-order DEs reduced to, ,numerical solution of, ,reduced to first-order systems,Systems of linear algebraic equations:as an augmented matrix,coefficients of,consistent,elementary operations on a,homogeneous, ,ill-conditioned,inconsistent,Gaussian elimination,Gauss–Jordan elimination,general form of,matrix form of,nonhomogeneous,Spiral points:stable,unstable,Spread of a disease,Spring constant,Spring coupled pendulums,Spring/mass systems, –Spring pendulum, –Square errors, sum of,Square matrix,Square norm of a function,Square wave,Stability criteria:for first-order autonomousequations,for linear systems,for plane autonomous systems,Stability for explicit finite differencemethod, ,Stability of linear systems,Stable node, ,Stable numerical method,Stable critical point,Stable spiral point, ,Staircase function,Standard basis:for Pn,for R , ,for R , ,for Rn,Standard Euclidean inner product,Standard form for a linear differentialequation, , , , ,Standard inner product in Rn,Standing waves, ,Starting methods,State of a system, , , ,State variables,Stationary point,Steady-state current, ,Steady-state fluid flow,Steady-state solution, , , ,Steady-state temperature, , , ,, ,Steady-state term, ,Stefan’s law of radiation,Step size,Stochastic matrix,Stokes, George G.,Stokes’ law of air resistance,Stokes’ theorem,Stream function,Streamlines, ,Streamlining,String falling under its own weight, –String of length n,Sturm–Liouville problem:definition of,orthogonality of solutions,properties of,regular,singular,Submatrix, I- IndexINDEXUnstable numerical method, ,Unstable spiral point, ,Unsymmetrical vibrations,Upper triangular matrix,Upward orientation of a surface,USS Missouri,VVan der Pol’s differential equation,,Van der Waal’s equation,Variable mass, , –Variable spring constant,Variables, separable, , ,Variation of parameters:for linear DEs, , –for systems of linear DEs, –Vector(s):acceleration,addition of, – ,angle between,binormal,component on another vector,components of a, ,in a coordinate plane,coplanar,cross product of, –difference of, , ,differential operator,direction,direction angles of,direction cosines of,dot product of, ,equality of, , ,equation for a line,equation for a plane,fields,free,function,geometric,horizontal component of,initial point of,inner product, ,length of, , ,linear combination of,linearly dependent,linearly independent,magnitude of, , , ,multiplication by scalars, , ,, ,negative of,norm of, ,normal to a plane, –normalization of, ,in n-space,orthogonal,orthogonal projection onto asubspace,orthonormal basis,parallel, ,position, ,principal normal,projection on another vector,Triple integral:applications of,in cylindrical coordinates,definition of, –evaluation of, –in spherical coordinates,as volume,Triple scalar product,Triple vector product,Triply connected domain,Trivial solution:defined,for a homogeneous system of linearequations,Trivial vector space,Truncation error:for Euler’s method,global,for improved Euler’s method,local,for RK method,Tsunami, mathematical model of,Twisted cubic,Twisted shaft,Two-dimensional definite integral,Two-dimensional fluid flow, ,,Two-dimensional heat equation,Two-dimensional Laplace’sequation,Two-dimensional Laplacian,Two-dimensional vector field, –Two-dimensional wave equation,Two-point boundary-value problem,,-space (R ),Two-term recurrence relation,Type I (II) invariant region,Type I (II) region,UUncoupled linear system,Undamped forced motion,Undamped spring/mass system, ,Underdamped electrical circuit,Underdamped spring/mass system,Underdamped system,Underdetermined system of linearalgebraic equations,Undetermined coefficients:for linear differential equations,–for linear systems,Uniqueness theorems, ,Unit impulse,Unit step function:definition of,graph of,Laplace transform of,Unit tangent,Unit vector,Unstable critical point, , ,Unstable node, ,Temperature:—(Cont.)in a rectangular parallelepiped,in a rectangular plate,in a rod,in a semiannular plate,in a semicircular plate,in a semi-infinite plate,in a sphere, ,in a wedge-shaped plate,Terminal velocity of a falling body, ,,Test point,Theory of distributions,Thermal conductivity,Thermal diffusivity,Three-dimensional Laplacian,Three-dimensional vector field, –-space (R ),Three-term recurrence relation,Threshold level,Time of death,TNB-frame,Torque,Torricelli, Evangelista,Torricelli’s law,Trace:of a matrix,of a plane,Tracer,Tractrix, ,Trajectories, orthogonal,Trajectory, , ,Transfer coefficients,Transfer function,Transfer matrix,Transform pair,Transformation,Transient solution, ,Transient term, ,Translation:and rotation,in the z-plane,Translation on the s-axis,Translation on the t-axis,Translation property for autonomousDEs,Translation theorems for Laplacetransform, ,Transpose of a matrix:definition of,properties of,Transverse vibrations,Traveling waves,Tree diagrams,Triangle inequality,Triangular matrix,Triangular wave,Tridiagonal matrix,Trigonometric functions, complex:definitions of,derivatives,Trigonometric identities,Trigonometric series, Index I-INDEXone-dimensional, – ,solution of, –two-dimensional,Weight function:of a linear system,orthogonality with respect to,Weighted average,Wire hanging under its own weight,, –Word:definition of,encoding,Work:as a dot product,as a line integral,Work done by a constant force,Wronskian:for a set of functions,for a set of solutions of ahomogeneous linear DE, ,for a set of solutions of ahomogeneous linear system,Wronskian determinant,Xx-coordinate of a point in -space,xy-plane,xz-plane,YYaw, –y-coordinate of a point in -space,Young’s modulus,yz-plane,Zz-axis in space, –z-coordinate of a point in -space,Zero matrix,Zero vector, ,Zero vector space,Zero-input response,Zeros:of Bessel functions,of complex cosine and sine, –of complex hyperbolic cosine and sine,**

**of a function,of order n,Zero-state response,z-plane,rules of differentiation,smooth,of three variables, ,of two variables, ,as velocity,Vector-valued functions,Vector space:axioms for a,basis for a,closure axioms for a,complex,dimension of,finite dimensional,infinite dimensional,inner product,linear dependence in a, ,linear independence in a, ,real,span of vectors in a,subspace of,trivial,zero,Velocity field,Velocity potential, complex,Velocity vector function,Verhulst, P. F.,Vertical component of a vector,Vibrating cantilever beam,Vibrating string,Vibrations, spring/mass systems, ,–Virga,Viscous damping,Voltage drops,Volterra integral equation,Volterra’s principle,Volume of a parallelepiped,Volume under a surface:using double integrals,using triple integrals,Von Mises, Richard,Vortex,Vortex point,WWater clock,Wave equation:derivation of the one-dimensionalequation,difference equation replacementfor,properties of, , ,right-hand rule,scalar multiple of, , ,scalar product,scalar triple product,as a solution of systems of linearDEs,span of,spanning set for,standard basis for, , ,subtraction of, ,sum of,tangent to a curve,terminal point of,triple product,in -space,in -space,unit, ,unit tangent to a curve,vector triple product,velocity,vertical component of,zero, , ,Vector differential operator,Vector equation for a curve,Vector equation for a line,Vector equation of a plane,Vector fields:and analyticity,conservative,curl of,definition of, –divergence of,flux of,gradient,irrotational,plane autonomous system of,rotational,solenoidal,three-dimensional, –two-dimensional, –velocity,Vector functions:as acceleration,continuity of,definition of,derivative of,differentiation of components,higher derivatives of,integrals of,limit of,**

**كلمة سر فك الضغط : books-world.netThe Unzip Password : books-world.net**

### تحميل

يجب عليك التسجيل في الموقع لكي تتمكن من التحميل

تسجيل | تسجيل الدخول