كتاب Advanced Engineering Mathematics 7ed

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Advanced Engineering Mathematics 7ed
Dennis G. Zill

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Contents
Preface
Ordinary Differential Equations
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
Higher-Order Differential Equations
Theory of Linear Equations
Initial-Value and Boundary-Value Problems
Homogeneous Equations
Nonhomogeneous Equations
Reduction of Order
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 DEs
Chapter in Review
The Laplace Transform
Definition of the Laplace Transform
Inverse Transforms 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
Dirac Delta Function
Systems of Linear Differential Equations
Chapter in Review
Series Solutions of Linear 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 ReviewPART
Vectors, Matrices, and Vector Calculus
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
Matrices
Matrix Algebra
Systems of Linear 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
Eigenvalue Problem
Powers of Matrices
Orthogonal Matrices
Approximation of Eigenvalues
Diagonalization
LU-Factorization
Cryptography
Error-Correcting Code
Method of Least Squares
Discrete Compartmental Models
Chapter in Review
Vector Calculus
Vector Functions
Motion on a Curve
Curvature
Partial Derivatives
Directional Derivative
Tangent Planes and Normal Lines
Curl and Divergence
Line Integrals
Independence of Path
Double Integrals
Double Integrals in Polar Coordinates
Green’s Theorem
PART
Surface Integrals
Stokes’ Theorem
Triple Integrals
Divergence Theorem
Change of Variables in Multiple Integrals
Chapter in Review
Systems of Differential Equations
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
Partial Differential Equations
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
PART
Laplace’s Equation
Nonhomogeneous Boundary-Value Problems
Orthogonal Series Expansions
Higher-Dimensional Problems
Chapter in Review
Boundary-Value Problems in Other Coordinate Systems
Polar Coordinates
Cylindrical Coordinates
Spherical Coordinates
Chapter in Review
Integral Transforms
Error Function
Laplace Transform
Fourier Integral
Fourier Transforms
Finite Fourier Transforms
Fast Fourier Transform
Chapter in Review
Numerical Solutions of Partial Differential Equations
Laplace’s Equation
Heat Equation
Wave Equation
Chapter in Review
Complex Analysis
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
Integration in the Complex Plane
Contour Integrals
Cauchy–Goursat Theorem
Independence of Path
Cauchy’s Integral Formulas
Chapter in Review
Series and Residues
Sequences and Series
Taylor Series .
A B C D
Laurent Series
Zeros and Poles
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
Integral-Defined Functions
Derivative and Integral Formulas
Laplace Transforms
Conformal Mappings
Answers to Selected Odd-Numbered Problems
Index
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, John Couch,
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, ,
Aging spring equation, solution of,
Agnew, Ralph Palmer, ,
Air resistance:
linear, , ,
nonlinear, , ,
projectile motion with, ,
projectile motion with no, ,
proportional to square of velocity, , ,
proportional to velocity, , , –
Airy, Sir George Biddell,
Airy functions of the first and second kind, APP-
Airy’s differential equation:
definition of, ,
solution as power series, –
solution in terms of Bessel functions,
various forms of, –
Allee, Warder Clyde,
Allee effect,
Aliasing, –
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, ,
Angers suspension bridge,
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,
catenary,
caught pendulum,
chemical reactions, – , –
column bending under its 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,
hanging chain,
hard spring,
harvesting,
heart pacemaker, ,
hitting bottom,
hole drilled through the Earth,
immigr
ation, ,
infus
io
n of a drug,
le
aking tanks,
linear spring,
lifting a heavy rope,
lo
gistic 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, , – , –
pulsating sphere,
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, – ,
telephone wires,
temperature in an annular cooling fin,
temperature in a circular cylinder, –
temperature in a circular plate,
temperature between concentric cylinders,
temperature between concentric spheres, ,
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 of Syracuse,
Archimedes’ principle,
Area:
of a parallelogram,
of a surface,
of a triangle,
Area as a double integral, , ,
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:
column vector form of,
definition of,
row vector form of,
Auxiliary equation:
for a Cauchy–Euler equation,
for a linear equation with constant coefficients, ,
rational roots of,
repeated complex roots of,
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,
Bashforth, Francis,
Basis of a vector space:
definition of,
standard, –
BC,
Beams:
axis of symmetry,
cantilever,
clamped,
deflection curve of,
elastic curve of,
embedded,
free,
flexural rigidity of,
simply supported,
static deflection of a homogeneous beam, – , –
use of the Laplace transform, –
Beats,
Bell curve,
Bending of a circular plate,
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 c,
of order ½, – of order –½,
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 ν,
of order ν,
parametric form of,
parametric form of modified equation,
series solution of, –
Beta function:
definition of, APP-
expressed in terms of the gamma function, APP-
Biharmonic function,
Binary string of length n,
Binormal,
Bits,
Boundary conditions (BC):
homogeneous, ,
mixed,
nonhomogeneous,
for an ordinary differential equation, , , ,
for a partial differential equation, –
periodic,
separated,
time dependent, –
time independent, –
Boundary points, ,
Boundary of a set,
Boundary-value problem (BVP):
in
cylindrical coordinates,
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, , – , – , , , , , – , –
with periodic boundary conditions, , ,
in
polar coordinates,
in
rectangular coordinates,
regular,
rotating string, –
second-order, ,
singular,
in
spherical coordinates, –
two point,
Bounding theorem for contour integrals, –
Branch of the complex logarithm,
Branch cut,
Branch point,
Branch point of an electrical network, ,
Broughton suspension bridge,
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, –
generalization of,
method of solution, –
reduction to constant coefficients,
regular singular point of,
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,
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, ,
of a power series,
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 messages, –
Chemical reactions:
first-order, –
second-order, , , , –
Chemostat,
Cholesky, André-Louis,
Cholesky’s method,
Christoffel, Elwin Bruno,
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 differential equations:
by linearity,
by order, –
by type,
Classification of linear partial differential equations by type,
Classifying critical points, , – ,
Clepsydra,
Clockwise (negative) direction on a simple closed curve,
Closed curve, ,
Closed disk,
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(s):
definition of, –
expansion of a determinant by, –
a property of,
Coil spring,
Column bending under its own weight,
Column vector, ,
Column vector form of an autonomous system,
Commutative laws of complex numbers,
Commutator,
Compartmental analysis,
Compartmental models,
Compatibility condition,
Competition models, – , ,
Competitive interaction,
Complementary error function:
definition of, , , APP-
graph of,
Complementary function:
for a linear differential equation,
for a system of linear differential equations, ,
Complete set of functions,
Completing the square,
Complex conjugate,
Complex eigenvalues of a matrix,
Complex form of Fourier integral,
Complex form of Fourier series,
Complex function:
analytic, , , ,
continuous, ,
definition of,
derivative of,
differentiable, –
domain of,
entire, , ,
exponential,
fundamental region,
hyperbolic,
inve
rse hyperbolic,
inve
rse trigonometric,
limit of,
lo
garithmic,
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 integral,
Complex line integrals:
definition of,
evaluation of, , ,
properties of, –
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,
ima
ginary part of,
ima
ginary unit,
int
eger powers of,
lo
garithm of,
magnitude 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,
ima
ginary 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,
Component of a vector on another vector,
Components of a vector, , ,
Conformal mapping,
Conformal mapping and the Dirichlet problem,
Conformal mappings, table of, APP-
Conjugate of a complex number, ,
Conjugate complex roots, , – , ,
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 of differentiation, , APP-
Constants of a linear system,
Constructing an orthogonal basis:
for R ,
for R ,
for R ,
Continuing numerical method,
Continuity of a complex function,
Continuity equation, –
Continuity of a vector function,
Continuous compound interest,
nContour:
definition of,
ind
ented,
Contour integral:
bounding theorem for, –
definition of,
evaluation of, – , ,
fundamental theorem for,
ind
ependent of the path, – ,
for the inverse Laplace transform,
properties of,
Contourplot,
Convergence:
of a complex geometric series,
of a complex sequence,
of a complex series, –
of a Fourier integral,
of a Fourier series,
of a Fourier–Bessel series,
of a Fourier–Legendre series,
of an improper integral,
of a power series, ,
Convergent improper integral,
Convolution, properties of, –
Convolution integral, , , –
Convolution theorem:
for the Fourier transform, , –
inve
rse 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 integral function, APP-
Cosine series,
Cosine series in two variables,
Coulomb (C),
Coulomb’s law,
Counterclockwise (positive) direction on a simple closed curve,
Coupled pendulums,
Coupled spring/mass system, –
Coupled systems,
Cover-up method,
Cramer, Gabriel,
Cramer’s rule,
Crank, John,
Crank–Nicolson method, –
Criterion for an exact differential,
Critical loads,
Critical points of an autonomous first-order differential equation:
asymptotically stable,
attractor,
definition of,
is
olated,
repeller,
semi-stable,
unstable,
Critical points for autonomous linear systems:
attractor,
center,
classifying,
definition of,
degenerate stable node, –
degenerate unstable node, –
lo
cally 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:
alternative form of,
component form 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,
great circles,
of intersection,
parallel,
parametric,
piecewise smooth,
positive direction on,
simple closed,
smooth,
Curvilinear motion in the plane,
Cycle of a plane autonomous system,
Cycloid,
Cylinder functions,
Cylindrical coordinates:
conversion to rectangular coordinates,
definition of,
Laplacian in,
triple integrals in,
Cylindrical wedge,
D
D’Alembert, Jean-Baptiste le Rond,
D’Alembert’s solution,
Da Vinci, Leonardo,
Damped amplitude,
Damped motion, , –
Damping, linear,
Damping constant,
Damping factor,
Daphnia,
Daughter isotope,
DE,
De Moivre, Abraham,
De Moivre’s formula,
Decay, radioactive,
Decay constant,
Decoding a message, – Definite integral, definition of,
Deflation, method of,
Deflection of a cantilever beam,
Deflection curve of a beam,
Deflection of a vertical column,
Deformation of contours,
Degenerate nodes:
stable, –
unstable, –
Del operator, –
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 of an integral defined function, APP- , APP-
Derivative and integral formulas, APP- , APP-
Derivative of a Laplace transform,
Derivative of real function, notation for,
Derivative of a sum of functions, APP-
Derivative of vector function, definition of,
Descartes, René,
Determinant(s):
of a × matrix,
of a × matrix,
cofactors of, –
definition of,
evaluating by row reduction, –
expansion by cofactors, – ,
of a matrix product,
minor of,
of order n,
as a product of eigenvalues,
properties of, –
of a transpose,
of a triangular matrix,
with two identical rows,
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, ,
exact,
form of a first-order ODE,
of a function of several variables,
nth-order operator,
operator,
recurrence relation,
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, , – , , ,
imp
licit solution of,
Laguerre’s, ,
Legendre’s, ,
linear first-order, ,
linear second-order,
as a mathematical model, –
modified Bessel’s, –
nonhomogeneous, , ,
nonlinear, , , – , –
with nonpolynomial coefficients,
normal form of,
notation for,
order of,
ordinary,
ordinary point of,
parametric Bessel,
parametric modified Bessel,
particular solution of, , ,
piecewise linear,
with polynomial coefficients, – ,
Riccati’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, , , , , – ,
written as a first-order system, –
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:linearity property of,
nth order,
Differential recurrence relation,
Differentiation of vector functions, rules of,
Diffusion equation, , –
Dimension of a vector space, ,
Dirac, Paul Adrien Maurice,
Dirac delta function:
definition of, ,
Laplace transform of,
Direction angles,
Direction cosines,
Direction field, ,
Direction numbers of a line,
Direction vector of a line,
Directional derivative:
computing,
definition of,
for functions of three variables,
for functions of two variables, –
maximum values of, –
Dirichlet, Johann Peter Gustav Lejeune,
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 signal,
Discretization error,
Distance formula,
Distance from a point to a line,
Distributions, theory of,
Distributive law:
for complex numbers,
for matrices,
Divergence theorem,
Divergence of a vector field:
definition of,
physical interpretation of, ,
Divergent improper integral,
Division of two complex numbers, ,
Domain:
of a complex function,
in
the complex plane,
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,
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,
DSolve command,
Duffing, Georg,
Duffing’s differential equation,
Dulac negative criterion,
Dynamical system:
definition of, ,
response of,
state of,
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, , , , – , – , –
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, –
Equilibrium point,
Equilibrium position of a spring/mass system,
Equilibrium solution, ,
Equipotential curves,
Error(s):
absolute,
discretization,
formula,
global truncation,
lo
cal truncation,
percentage relative,
relative,
round-off,
sum of square,
Error function:
definition of, , , APP-
graph of,
Error-correcting code, –
Error-detecting code, ,
Escape velocity,
Essential singularity,
Euclidean inner product,
Euler, Leonhard, , APP-
Euler load,
Euler’s constant,
Euler’s formula,
Euler’s improved method,
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 of Fourier transforms,
Existence of Laplace transform, Existence of power series solutions,
Existence of a unique solution, – , ,
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 integral,
Exponential form of a Fourier series,
Exponential function:
definition of,
derivative of,
fundamental region of,
period of,
properties of, –
Exponential integral function, APP-
Exponential order,
Exponents of a singularity,
Exterior Dirichlet problem,
Exterior point,
External force,
Extreme displacement,
F
Factorial function, APP-
Falling bodies, mathematical models of, – ,
Falling chain, –
Falling raindrops, ,
Family of solutions,
Farads (f),
Fast Fourier transform,
Fast Fourier transform, computing with,
Fehlberg, Erwin,
Fibonacci,
Fibonacci sequence,
Fick, Adolf Eugen,
Fick’s law of diffusion,
Filtered signals,
Finite difference approximations, – , , ,
Finite difference equation,
Finite difference method:
explicit, ,
imp
licit,
Finite differences,
Finite dimensional vector space,
Finite Fourier transforms:
cosine,
inve
rse of,
operational properties of, ,
sine,
First buckling mode,
First harmonic,
First moments,
First normal mode,
First octant,
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, First shifting theorem,
First standing wave,
First translation theorem:
form of,
inve
rse form of,
Five-point approximation of the Laplacian,
Flexural rigidity,
Flow:
of fluid around a corner,
of fluid around a cylinder,
of heat,
steady-state fluid,
Fluctuating population,
Flux and Cauchy’s integral formula,
Flux through a surface,
Flux of a vector field,
Folia of Descartes, ,
Forced electrical vibrations,
Forced motion:
with damping,
without damping,
Forcing function, ,
Forgetfulness,
Formula error,
Forward difference,
Fossil, age of,
Fourier, Jean-Baptiste Joseph,
Fourier coefficients,
Fourier cosine transform:
definition of,
operational properties of, –
Fourier integral:
complex form,
conditions for convergence,
cosine form,
definition of,
exponential form,
sine form,
Fourier integrals,
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:
complex (exponential) form of,
definitions of,
existence of,
finite,
linearity 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-end conditions of a beam, ,
Free motion of a spring/mass system:
damped, – undamped,
Free vectors,
Frequency of free vibrations,
Frequency filtering,
Frequency response curve,
Frequency spectrum,
Fresnel, Augustin-Jean,
Fresnel cosine integral function, APP-
Fresnel sine integral function, , , APP-
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):
Airy, APP-
beta, APP-
complementary,
complementary error, , APP-
of a complex variable,
continuous,
cosine integral, APP-
defined by an integral, , , APP-
differentiable,
directional derivative of, –
domain of,
driving, , ,
error, , – , APP-
even,
exponential,
exponential integral, APP-
factorial, APP-
forcing, , , –
Fresnel cosine integral, APP-
Fresnel sine integral, , APP-
gamma, , , APP-
generalized,
generalized factorial, APP-
gradient of,
graph of,
harmonic, , – ,
hyperbolic,
inne
r product of,
int
egral defined,
inp
ut, ,
lo
garithmic,
lo
garithmic integral, APP-
odd,
orthogonal,
output, ,
partial derivative of, –
periodic,
polynomial,
potential, ,
power, ,
range of,
rational,
of a real variable,
response, ,
sine integral, , , APP-
vs. solution,
stream,
of three variables,
trigonometric,
as a two-dimensional flow,
of two variables,
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, G
g, ,
Galileo Galilei, ,
Gamma function:
asymptotes of graph, APP-
definition of, , APP-
graph of, APP-
recursion formula for, , APP-
Gauss, Johann Carl Friedrich, , APP-
Gauss’ law, ,
Gauss’ theorem,
Gaussian elimination,
Gauss–Jordan elimination,
Gauss–Seidel iteration, ,
General form of a linear ordinary differential equation,
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, –
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, Josiah Willard,
Gibbs phenomenon,
Global truncation error,
Globally stable critical point,
Gompertz, Benjamin,
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, – ,
Graph(s):
of the complementary error function,
of the error function,
of a function of two variables,
of the gamma function, , , APP-
of level curves,
of level surfaces,
of a plane,
Great circles,
Green, George, ,
Green’s function:
for a boundary-value problem, –
for an initial-value problem,
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,
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.,
Hanging chain,
Hard spring,
Harmonic conjugate functions,
Harmonic function, , – ,
Harmonic function, transformation theorem for,
Harmonic functions and the Dirichlet problem,
Harvesting,
Hawking, Stephen,
Hawking radiation,
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,
Heaviside layer,
Helmholtz, Hermann Ludwig Ferdinand von,
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,
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, Józef Maria,
Hooke, Robert,
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, , – , APP-
divergent, , –
Improved Euler’s method,
Improved Euler’s method, truncation errors for,
Impulse response,
Incompressible flow, ,
Incompressible fluid,
Inconsistent system of linear equations,
Indefinite integral, ,
Indented contours,
Independence of path:
definition of, ,
test for, , , ,
Independent variables,
Indicial equation,
Indicial roots,
Inductance, ,
Infinite-dimensional vector space, –
Infinite linearly dependent set,
Infinite linearly independent set,
Infinite series of complex numbers:
absolute convergence,
convergence of, –
definition of,
geometric,
necessary condition for convergence,
nth term test for divergence,
sum of, –
Infusion of a drug,
Initial conditions (IC), ,
Initial-value problem (IVP):
definition of, ,
first-order, ,
nth-order, ,
second-order,
for systems of linear differential equations, unique solution of,
Inner partition,
Inner product:
definition of, , , ,
properties of, ,
space, ,
of two column matrices,
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, , , – , , APP-
Integral equation,
Integral transform:
definition of,
finite Fourier cosine,
finite Fourier sine,
Fourier,
Fourier cosine,
Fourier sine,
inve
rse,
kernel of, ,
Laplace, ,
pair,
Integral transforms of derivatives:
finite Fourier cosine,
finite Fourier sine,
Fourier,
Fourier cosine,
Fourier sine,
Laplace, –
Integral of a vector function,
Integrating factor, , –
Integration along a curve, –
Integration by parts,
Integrodifferential equation,
Interest, compounded continuously,
Interior mesh points,
Interior point, ,
Interior point of a set in the complex plane,
Interpolating function,
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 of finite Fourier cosine transform,
Inverse of finite Fourier sine transform,
Inverse hyperbolic functions:
derivatives of,
as logarithms,
Inverse integral transform:
Fourier,
Fourier cosine,
Fourier sine,
Laplace, ,
Inverse of a matrix:
by the adjoint method, – definition of,
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
Jacobi, Carl Gustav Jacob,
Jacobian determinant,
Jacobian matrix,
Joukowski airfoil,
Joukowski transformation,
K
Kepler’s first law of planetary motion,
Kernel of an integral transform, ,
Kinetic friction,
Kirchhoff, Gustav Robert,
Kirchhoff’s first law,
Kirchhoff’s point and loop rules,
Kirchhoff’s second law, , ,
Kutta, Martin Wilhelm,
L
Lagrange, Joseph-Louis,
Lagrange’s identity,
Laguerre, Edmond,
Laguerre polynomials,
Laguerre’s differential equation, ,
Laplace, Pierre-Simon Marquis de,
Laplace transform:
alternative form of second translation theorem, –
behavior as s → θ
change of scale theorem for,
conditions for existence,
convolution theorem for, –
definition of, ,
of a derivative, –
derivatives of a,
of differential equations, –
differentiation of, –
of Dirac delta function,
of an integral, inve
rse of, ,
linearity of,
linearity of the inverse transform,
and the matrix exponential,
operational properties of, , , , , , , , , , , ,
of a partial derivative,
of a periodic function,
sufficient conditions for existence of,
of systems of ordinary differential equations, –
tables of, , APP-
translation theorems for, ,
of unit step function,
Laplace’s partial differential equation:
in
cylindrical coordinates, –
difference equation replacement for,
maximum principle for,
in
polar coordinates, ,
solution of, –
in
three dimensions,
in
two rectangular coordinates, , – ,
Laplacian:
in
cylindrical coordinates,
definition of,
in
polar coordinates,
in
rectangular coordinates,
in
spherical coordinates,
in
three dimensions, ,
Lascaux cave paintings, dating of,
Latitude,
Lattice points,
Laurent, Pierre Alphonse,
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,
LC-series circuit, integrodifferential equation of,
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,
Rodrigues’ formula for,
Legendre’s differential equation:
associated,
of order n,
series solution of, –
Leibniz, Gottfried Wilhelm,
Leibniz notation,
Leibniz’s rule, , APP-
Length of a space curve,
Length of a vector:
in
-space,
in
n-space,
Leonardo of Pisa,
Leonardo da Vinci,
Level curves, ,
Level of resolution of a mathematical model,
Level surfaces,
L’Hôpital’s rule, , , ,
Liber Abaci, Libby, Willard Frank, –
Libby half-life,
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,
definition of,
evaluation of, – ,
fundamental theorem for,
ind
ependent of the path,
in
the plane,
in
space,
as work,
Line segment,
Lineal element,
Linear algebraic equations, systems of, –
Linear combination of vectors,
Linear damping,
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,
int
egrating 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, , ,
complementary function for, –
with constant coefficients,
definition,
first-order, ,
general solution of, , , , – , –
higher-order, , , – ,
homogeneous, , ,
ind
ic
ia
l equation for,
infinit
e series solutions for, ,
initia
lvalue problems for, , , , , – ,
nonhomogeneous, – , , , –
nth-order,
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 regression,
Linear second-order partial differential equation,
Linear second-order partial differential equations:
classification of,
homogeneous,
nonhomogeneous,
solution of,
superposition principle for,
Linear spring,
Linear system of algebraic equations:
consistent,
definition of,
elementary operations on,
homogeneous,
inc
onsistent,
nonhomogeneous,
Linear system of first-order differential equations:
definition of, , ,
homogeneous,
matrix form of,
nonhomogeneous,
normal form of,
solution of,
Linear transform, , ,
Linearity:
of a differential operator,
of the Fourier transforms,
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, Joseph,
Liouville’s theorem,
Lissajous curve, ,
Loblolly pines, bending in high winds, –
Local linear approximation, ,
Local truncation error,
Locally stable critical point,
Lockheed Electra,
Logarithm of a complex number:
branch cut for,
branch of,
definition of,
derivative of,
principal branch,
principal value of,
properties of,
Logarithmic integral function, APP-
Logistic curve,
Logistic equation:
definition of, ,
modifications of, –
solution of,
Logistic function,
Logistic growth, –
Longitude, Loop rule, Kirchhoff’s,
Losing a solution, –
Lotka, Alfred James,
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, , –
int
egrodifferential equation of,
LR-series circuit, differential equation of,
LU-decomposition of a matrix,
LU-factorization of a matrix, –
M M
aclaurin, Colin,
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 Robert,
Malthusian model,
Mapping, ,
Mapping, conformal,
Marching soldiers,
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,
anticommute,
associative law, ,
augmented, ,
banded,
characteristic equation of,
coefficient,
column vector,
commutative law,
decomposed,
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, –
factored,
fundamental,
identity,
inve
rse of, , –
inve
rtible,
Jacobian, – lo
wer triangular, ,
LU-factorization of, –
main diagonal entries of,
multiplication,
multiplicative identity,
multiplicative inverse,
nilpotent, , ,
nonsingular, ,
nullspace of,
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 differential equations,
Matrix form of a system of linear 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, –
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,
M vector space,
Möbius strip,
Modeling process, steps in,
Modifications of the logistic equation, –
Modified Bessel equation:
general solution of,
of order ν,
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,
Moulton, Forest Ray,
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,
N
n-dimensional vector, –
Negative criterion, ,
Negative direction on a curve,
Negative of a vector,
Neighborhood,
Networks, – ,
Neumann, Carl Gottfried,
Neumann condition,
Neumann function,
Neumann problem:
for a circular plate,
for a rectangle,
Newton, Sir 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, , , , –
Nicolson, Phyllis,
Nilpotent matrix, , ,
Nodal line,
Nodes:
of a plane autonomous system, – ,
m,nof a standing wave,
Nonconservative force,
Nonelementary integral, , , APP-
Nonhomogeneous boundary condition,
Nonhomogeneous boundary-value problem, , , ,
Nonhomogeneous linear differential equation:
boundary-value problem for,
definition of,
general solution of,
initia
lvalue problem for,
ordinary, – ,
partial,
particular solution of,
Nonhomogeneous systems of linear differential equations:
complementary function of,
definition of,
general solution of,
initia
lvalue problem for,
matrix form of,
normal form of,
particular solution of,
solution vector of,
Nonhomogeneous systems of linear equations,
Nonisolated singular point,
Nonlinear air resistance,
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,
Norm:
of a column vector (matrix),
of a function,
of a partition, ,
square,
of a vector, , ,
Normal components 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,
NSolve command,
n-space (R
coordinates relative to an orthonormal basis,
dimension of,
dot (or inner) product in,
le
ngth (or norm) in,
orthogonal vectors in,
standard basis for,
unit vector in,
vector in,
zero vector in,
nth-order differential operator,
nth-order differential equation expressed as a system, –
nth-order initial-value problem, ,
nth-order ordinary differential equation, – , ,
nth root of a nonzero complex number, –
nth roots of unity,
nth term test for divergence,
Nullcline,
nNullspace 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–Nicolson method, –
deflation method,
difference equation replacement,
errors in, , – ,
Euler’s method, – , , , ,
finite-difference methods, – , , , ,
Gauss–Seidel iteration,
imp
roved Euler’s method,
inve
rse power method,
multistep method,
power method,
predictor–corrector methods, ,
Runge–Kutta methods, , , – ,
Runge–Kutta–Fehlberg method,
shooting method,
single-step method,
stability of,
stable,
starting method,
unstable,
using the tangent line,
Numerical solution curve,
Numerical solutions of partial differential equations,
Numerical solver,
Numerical values of Bessel functions,
O
Octants,
Odd function:
definition of,
properties of,
ODE,
Ohm, Georg Simon,
Ohms (Σ),
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, exponential,
Order of a differential equation,
Order of integration, , –
Order of a matrix,
Order of a Runge–Kutta method, –
Ordered n-tuple, ,
Ordered pair, , , ,
Ordered triple, , ,
Ordinary differential equation,
Ordinary point of an ordinary differential equation:
definition of, –
solution about,
Orientable surface:of a closed,
definition of,
Orientation of a surface:
downward,
inw
ard,
outward,
upward,
Orthogonal basis for a vector space, –
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 with respect to a weight function,
Orthogonal series expansion, –
Orthogonal set of functions, –
Orthogonal surfaces at a point,
Orthogonal trajectories,
Orthogonal vectors,
Orthogonally diagonalizable matrix, –
Orthonormal basis:
definition of,
for R ,
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 equations,
Overtones,
P
Pacemaker, heart, ,
Parabolic partial differential equation,
Parallel vectors:
criterion for,
definition of,
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 ν,
in
self-adjoint form,
Parametric form of modified Bessel equation of order ν,
Parent isotope,
Paris Guns, –
Parity,
Parity check bits,
Parity check code,
Parity check equations,
Parity check matrix,
Parity error,
Partial derivatives:
nChain 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, , ,
separation of variables, –
solution of,
Partial fractions, ,
Particular solution:
of Bessel’s equation, –
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, Wolfgang Ernst,
Pauli spin matrices,
PDE,
Pendulum:
ballistic,
double,
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 hyperbolic sine and cosine, ,
of the complex 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 first-order differential equations, – Phase-plane method,
Physical pendulum,
Piecewise-continuous function

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