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 |  موضوع: كتاب Advanced Engineering Mathematic 6ed الجمعة 11 فبراير 2022, 2:34 am | |
| 
أخواني في الله
أحضرت لكم كتاب
Advanced Engineering Mathematic 6ed
Dennis G. Zill
Loyola Marymount University
و المحتوى كما يلي :
Contents
Preface xiii
Introduction to Differential Equations
. Definitions and Terminology
. Initial-Value Problems
. Differential Equations as Mathematical Models
Chapter in Review
First-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 DEs
Chapter in Review
PART Ordinary Differential Equations
iv Contents
Higher-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 Equations
Chapter in Review
The 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 Equations
Chapter 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
Dennis Hallinan/Alamy Images Image State/Alamy Stock PhotoContents vii
. Double Integrals in Polar Coordinates
. Green’s Theorem
. Surface Integrals
. Stokes’ Theorem
. Triple Integrals
. Divergence Theorem
. Change of Variables in Multiple Integrals
Chapter in Review
Systems 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 Exponential
Chapter in Review
Systems 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 Stability
Chapter in Review
PART Systems of Differential Equations
kochanowski/Shutterstock Mike Kiev/Collection/iStock/Getty Images Plusviii Contents
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
Corbis Premium RF/Alamy Images science photo/Shutterstock
Palette /ShutterStock, Inc.Contents ix
Integral Transform Method
. Error Function
. Applications of the Laplace Transform
. Fourier Integral
. Fourier Transforms
. Fast Fourier Transform
Chapter in Review
Numerical Solutions of Partial Differential Equations
. Laplace’s Equation
. Heat Equation
. Wave Equation
Chapter in Review
Functions 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 Functions
Chapter in Review
PART Complex Analysis
Peteri/ShutterStock, Inc. Design Pics Inc./Alamy Images enigmatico/ShutterStock, Inc.x Contents
Integration in the Complex Plane
. Contour Integrals
. Cauchy–Goursat Theorem
. Independence of the Path
. Cauchy’s Integral Formulas
Chapter in Review
Series and Residues
. Sequences and Series
. Taylor Series
. Laurent Series
. Zeros and Poles
. Residues and Residue Theorem
. Evaluation of Real Integrals
Chapter in Review
Conformal Mappings
. Complex Functions as Mappings
. Conformal Mappings
. Linear Fractional Transformations
. Schwarz–Christoffel Transformations
. Poisson Integral Formulas
. Applications
Chapter in Review
Appendices
I 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-
Index
Index
I-
A
Absolute 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 translation
theorem,
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- Index
INDEX
Applications of differential
equations:—(Cont.)
caught pendulum,
chemical reactions, – ,
column bending under it’s own
weight,
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 a
chemostat,
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 constant
force, , –
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 concentric
cylinders, ,
temperature between concentric
spheres,
temperature in a circular plate,
temperature in a cylinder,
– ,
temperature in a quarter-circular
plate,
temperature in a ring,
temperature in a semiannular plate,
temperature in a semicircular plate,
temperature in a sphere, ,
– ,
temperature in a wedge-shaped
plate,
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 differential
equation,
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 differential
equations,
Auxiliary equation:
for a Cauchy–Euler equation,
for a linear equation with constant
coefficients,
rational roots of,
Axis of symmetry of a beam,
B
Back 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-
INDEX
Cauchy’s integral formula for
derivatives,
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 differential
equations:
by linearity, ,
by order, ,
by type,
Classification of linear partial differential
equations 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 contour
integrals,
Boxcar function,
Branch cut,
Branch of the complex logarithm,
Branch point,
Branch point of an electrical
network,
Buckling modes,
Buckling of a tapered column,
Buckling of a thin vertical column,
,
Buoyant force,
BVP,
C
Calculation 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 multiply
connected domains,
Cauchy principal value of an
integral,
Cauchy–Riemann equations,
Cauchy–Schwarz inequality,
Cauchy’s inequality,
Cauchy’s integral formula,
embedded,
free,
simply supported,
static deflection of a homogeneous
beam, –
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 terms
of, –
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 modified
equation,
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- Index
INDEX
Constants 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 a
basis,
Coordinates of a vector relative to an
orthonormal 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 another
vector,
Conformal mapping,
Conformal mapping and the Dirichlet
problem, –
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 linear
equations,
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 linear
system,
Cofactor,
Cofactor expansion of a determinant,
–
Column bending under its own
weight,
Column vector, ,
Commutative laws of complex
numbers,
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 differential
equations, ,
Complete set of functions,
Completing the square,
Complex eigenvalues of a
matrix,
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-
INDEX
expansion 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 linear
system 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 homogeneous
coefficients,
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 rectangular
coordinates,
definition of,
Laplacian in,
triple integrals in,
Cylindrical functions,
Cylindrical wedge,
D
D’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 hyperbolic
functions,
of complex inverse trigonometric
functions,
of the complex logarithm
function,
of complex trigonometric
functions,
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, definition
of,
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 linear
systems:
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 autonomous
systems:
asymptotically stable,
classifying,
stability criteria for,
stable,
unstable,
Critical speeds,
Critically damped electrical circuit,
Critically damped spring/mass
system,
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- Index
INDEX
change of variables,
definition of,
evaluation of,
as an iterated integral,
in polar coordinates,
properties of,
reversing the order of integration
in,
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, ,
E
Ecosystem, states of,
Effective spring constant,
Effective weight,
Eigenfunctions:
of a boundary-value problem, ,
of a Sturm-Liouville problem,
–
Eigenvalues of a boundary-value
problem, – , –
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 conformal
mapping,
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 for
homogeneous linear,
time dependent,
time independent,
wave, , – , – ,
Differential form, ,
Differential operator, nth order,
Differential recurrence relation,
–
Differentiation of vector functions, rules
of,
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-
INDEX
Fast Fourier transform, computing
with,
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’s
equation,
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 differential
equations,
for systems of differential
equations,
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 Fourier
series,
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,
F
Falling 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 linear
systems,
Elementary row operations on a matrix:
definition of,
notation for,
Elimination method(s):
for a system of algebraic equations,
– ,
for a system of ordinary differential
equations,
Elliptic partial differential
equation,
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/mass
system,
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- Index
INDEX
of a homogeneous system of linear
differential equations,
of a linear first-order equation,
of linear higher-order equations,
–
of modified Bessel’s equation,
of a nonhomogeneous linear
differential equation,
of a nonhomogeneous system of linear
differential equations,
of parametric form of Bessel’s
equation,
of parametric form of modified
Bessel’s equation,
of a second-order Cauchy–Euler
equation, –
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 orthogonalization
process, – ,
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 differential
equation,
for a second-order differential
operator,
Green’s identities,
Green’s theorem in the plane,
Green’s theorem in -space,
Growth and decay, , –
Growth constant,
Growth rate, relative,
H
Half-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 complex
exponential 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 differential
equation,
General solution:
of Bessel’s equation, ,
definition of, , , ,
of a homogeneous linear differential
equation,
of a homogeneous second-order linear
differential 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 differential
equations,
for systems of differential
equations,
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-
INDEX
Independent 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 for
convergence,
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 differential
equations,
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, compounded
continuously,
Interior point,
Interior mesh points,
Interior point of a set in the complex
plane,
Interpolating function,
Homogeneous systems of linear
algebraic equations:
definition of, ,
matrix form of,
nontrivial solutions of,
properties of,
trivial solution of,
Homogeneous systems of linear
differential 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 differential
equation,
I
IC,
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 a
transformation,
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 linear
equations,
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, transformation
theorem for,
Harmonic functions and the Dirichlet
problem,
Harvesting,
Heart pacemaker, model for, ,
Heat equation:
derivation of one-dimensional
equation,
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 differential
equation,
Helix:
circular,
elliptical,
pitch of,
Henrys (h),
Hermite, Charles,
Hermite polynomials,
Hermite’s differential equation,
,
Higher-order ordinary differential
equations, , ,
Hinged end of a beam,
Hitting bottom,
Hole through the Earth,
Homogeneous boundary conditions,
, ,
Homogeneous boundary-value problem,
,
Homogeneous first-order differential
equation:
definition of,
solution of,
Homogeneous function,
Homogeneous linear differential
equation:
ordinary, ,
partial, I- Index
INDEX
Leaking 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 mathematical
model,
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 complex
variable:
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, ,
L
Lagrange’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 differential
equations,
tables of, , APP-
translation theorems for, ,
of unit step function,
Laplace’s equation, , , ,
– , , ,
Laplace’s partial differential equation:
in cylindrical coordinates,
difference equation replacement
for,
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 mechanical
energy,
Law of mass action,
Law of universal gravitation,
Laws of exponents for complex
numbers,
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,
J
Jacobian determinant,
Jacobian matrix,
Joukowski airfoil,
Joukowski transformation,
K
Kepler’s first law of planetary
motion,
Kernel of an integral transform,
Kinetic friction,
Kirchhoff’s first law, Index I-
INDEX
LR-series circuit, differential equation
of,
LU-decomposition of a matrix,
LU-factorization of a matrix, –
M
Maclaurin 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 differential
equations, –
Linearly dependent set of functions,
Linearly independent set of
functions,
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 of
convergence,
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 differential
equations:
classification of,
homogeneous, I- Index
INDEX
Negative 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 boundary
condition,
Nonhomogeneous boundary-value
problem, , , ,
Nonhomogeneous linear differential
equation:
definition of,
general solution of,
initial-value problem for,
ordinary, ,
partial,
particular solution of,
Nonhomogeneous systems of algebraic
equations,
Nonhomogeneous systems of linear
differential 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 differential
equation, ,
Nonlinear oscillations,
Nonlinear pendulum, , –
Nonlinear spring,
Nonlinear systems of differential
equations, ,
Nonoriented surface,
Nonpolynomial coefficients,
Nonsingular matrix, , ,
Nontrivial solution,
for nonhomogeneous systems of linear
DEs,
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 linear
DEs, –
Method of variation of parameters:
for nonhomogeneous linear DEs, ,
–
for nonhomogeneous systems of linear
DEs, –
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 undetermined
coefficients,
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 linear
algebraic equations,
Matrix form of a system of linear
differential 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 linear
DEs, Index I-
INDEX
Orthogonal 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 a
subspace,
Orthogonal series expansion, –
Orthogonal set of functions,
Orthogonal with respect to a weight
function,
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 linear
algebraic equations,
Overtones,
P
Pacemaker, heart, ,
Parabolic partial differential
equation,
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 in
space,
Parametric form of Bessel equation:
of order n,
of order n,
in self-adjoint form,
Parametric form of modified Bessel
equation 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 Bessel
functions,
O
Octants,
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 Laplace
transform, , , , , ,
, , , ,
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 differential
equation:
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-order
equations,
Normal line to a surface,
Normal modes,
Normal plane,
Normal vector to a plane,
Normalization of a vector, ,
Normalized eigenvector,
Normalized set of orthogonal
functions,
Notation for derivatives,
n-parameter family of solutions,
n-space (Rn):
coordinates relative to an orthonormal
basis,
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 expressed
as 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 a
linear 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- Index
INDEX
Poisson integral formula:
for unit disk,
for upper half-plane, –
Poisson’s partial differential
equation,
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 complex
number,
of the complex hyperbolic sine and
cosine,
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 autonomous
system,
Phase angle,
Phase line,
Phase plane, , ,
Phase portrait:
for first-order differential
equations,
for systems of two linear first-order
differential 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 ordinary
differential 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 a
plane,
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, linear
second 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 linear
DEs, , , ,
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 exponential
function,
of the complex sine and cosine, Index I-
INDEX
zero-input,
zero-state,
Rest point,
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