PDF ISBN: 9780819478290 | Print ISBN: 9781628410723
DESCRIPTION
As technology continues to move ahead, modern engineers and scientists are frequently faced with difficult mathematical problems that require an ever greater understanding of advanced concepts. Designed as a self-study text for practicing engineers and scientists, as well as a useful reference, the book takes the reader from ordinary differential equations to more sophisticated mathematics--Fourier analysis, vector and tensor analysis, complex variables, partial differential equations, and random processes. The emphasis is on the use of mathematical tools and techniques. The general exposition and choice of topics appeals to a wide audience of applied practitioners.
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1.4.1 Homogeneous equations: fundamental solution sets
1.4.2 Constant coefficient equations
1.4.3 Nonhomogeneous equations - Part I
1.4.4 Nonhomogeneous equations - Part II
1.4.5 Cauchy-Euler equations
1.5 Power Series Method
1.5.1 Review of power series
1.6 Solutions Near an Ordinary Point
1.6.1 Ordinary and singular points
1.6.2 General method for ordinary points
1.7 Legendre's Equation
1.7.1 Legendre polynomials: P(x)
1.7.2 Legendre functions of the second kind: Q(x)
1.8 Solutions Near a Singular Point
1.8.1 Method of Frobenius
1.9 Bessel's Equation
1.9.1 The gamma function: F(x)
1.9.2 Bessel functions of the first kind: J (x)
1.9.3 Bessel functions of the second kind: )ç(x)
1.9.4 Differential equations related to Bessel's equation
Suggested Reading
Exercises
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3.4.2 Homogeneous systems with constant coefficients
3.4.3 Stability of linear systems
3.4.4 Nonhomogeneous systems
3.5 Linear Vector Spaces
3.5.1 Linear independence and basis vectors
3.5.2 Inner product spaces
3.5.3 Orthonormal basis and the expansion theorem
3.5.4 Hilbert spaces
Suggested Reading
Exercises
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4.7 Integral Relations Between Line, Surface and Volume Integrals
4.7.1 Green's theorem in the plane
4.7.2 Harmonic functions
4.7.3 Divergence theorem and Stokes' theorem
4.8 Electromagnetic Theory
4.8.1 Maxwell's equations
4.8.2 Poisson's equation
4.8.3 Electromagnetic wave equation
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Exercises
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5.4.2 Contravariant base vectors: reciprocal basis
5.4.3 Metric tensor
5.5 Vector Algebra
5.5.1 Permutation symbols in rectilinear coordinates
5.5.2 Dot product
5.5.3 Cross product and mixed triple product
5.6 Relations Between Tensor Components
5.6.1 Raising and lowering indices
5.6.2 Physical components
5.7 Reduction of Tensors to Principal Axes
5.7.1 Two-dimensional case
5.7.2 Three-dimensional case
5.8 Tensor Calculus: Rectilinear Coordinates
5.8.1 Gradient, divergence, and curl
5.9 Curvilinear Coordinates
5.9.1 Differentials as tensors
5.9.2 Tensor fields and base vectors
5.9.3 Metric tensors
5.10 Tensor Calculus: Curvilinear Coordinates
5.10.1 Christoffel symbols
5.10.2 Orthogonal systems
5.10.3 Covariant derivative
5.10.4 Absolute derivative
5.11 Riemann-Christoffel Curvature Tensor
5.12 Applications
5.12.1 Gradient, divergence, and curl
5.12.2 Dynamics of a particle: Newton's second law
5.12.3 Dielectric tensor of an anisotropic medium
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Exercises
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6.6.3 Inverse trigonometric and hyperbolic functions
6.7 Mappings by Elementary Functions
6.7.1 Orthogonal families
6.7.2 Simple polynomials
6.7.3 Reciprocal mapping
6.7.4 Bilinear transformations
6.7.5 Conformal mapping
Exercises
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8.2.3 Even and odd functions: cosine and sine series
8.2.4 Nonperiodic functions: extensions to other intervals
8.3 Power Signals: Exponential Fourier Series
8.3.1 Parseval's theorem and the power spectrum
8.4 Eigenvalue Problems and Orthogonal Functions
8.4.1 Regular Sturm-Liouville systems
8.4.2 Generalized Fourier series
8.4.3 Periodic Sturm-Liouville systems
8.4.4 Singular Sturm-Liouville systems
8.5 Green's Function
8.5.1 One-sided Green's function
8.5.2 Boundary value problems
8.5.3 Bilinear formula
Suggested Reading
Exercises
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10.2.1 Table of transforms and operational properties
10.2.2 Inverse transforms - I
10.2.3 Inverse transforms - II
10.3 Initial Value Problems
10.3.1 Simple electric circuits
10.3.2 Impulse response function
10.3.3 Stability of linear systems
10.4 Hankel Transform
10.4.1 Operational properties and table of transforms
10.5 Mellin Transform
10.5.1 Operational properties and table of transforms
10.5.2 Complex variable methods
10.6 Applications Involving the Mellin Transform
10.6.1 Products of random variables
10.6.2 Electromagnetic wave propagation
10.7 Discrete Fourier Transform
10.7.1 Discrete transform pair
10.8 Z-transform
10.8.1 Operational properties
10.8.2 Difference equations
10.9 Walsh Transform
10.9.1 Walsh functions
10.9.2 Walsh series and the discrete Walsh transform
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Exercises
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11.2.2 Special cases of the Euler-Lagrange equation
11.3 Some Classical Variational Problems
11.3.1 Shortest arc connecting two points
11.3.2 Surface of revolution with minimum area
11.3.3 Brachistochrone problem
11.4 Variational Notation
11.4.1 Natural boundary conditions
11.5 Other Types of Functionals
11.5.1 Functionals with several dependent variables
11.5.2 Functionals with higher-order derivatives
11.5.3 Functionals with several independent variables
11.6 Isoperimetric Problems
11.6.1 Constraints and Lagrange multipliers
11.6.2 Sturm-Liouville problem
11.7 Rayleigh-Ritz Approximation Method
11.7.1 Eigenvalue problems
11.8 Hamilton's Principle
11.8.1 Generalized coordinates and Lagrange's equations
11.8.2 Linear theory of small oscillations
11.9 Static Equilibrium of Deformable Bodies
11.9.1 Deflections of an elastic string
11.9.2 Deflections of an elastic beam
11.10 Two-Dimensional Variational Problems
11.10.1 Forced vibrations of an elastic string
11.10.2 Equilibrium of a stretched membrane
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Exercises
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12.7.2 Scattering of plane waves by a circular cylinder
12.8 Transform Methods
12.8.1 Heat conduction on an infinite domain: Fourier transform
12.8.2 Heat cnduction on a semi-infinite domain: Laplace transform
12.8.3 Nonhomogeneous wave equation
12.8.4 Poisson integral formula for the half-plane
12.8.5 Axisymmetric Dirichlet problem for a half-space: Hankel transform
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Exercises
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13.2 Random Variables and Probability Distributions
13.2.1 Cumulative distribution function
13.2.2 Probability density function
13.2.3 Discrete random variables
13.3 Examples of Density Functions
13.3.1 Gaussian (or normal) distribution
13.3.2 Uniform distribution
13.3.3 Rayleigh distribution
13.3.4 Gamma distribution
13.4 Expected Values
13.4.1 Higher-order moments
13.4.2 Characteristic functions
13.5 Conditional Probability
13.5.1 Conditional CDF and PDF
13.5.2 Expected values
13.6 Functions of One Random Variable
13.6.1 Continuous random variables
13.6.2 Expected values
13.6.3 Characteristic function method
13.7 Two Random Variables
13.7.1 Joint distribution and density functions
13.7.2 Marginal density functions
13.7.3 Conditional distributions and densities
13.7.4 Independent random variables
13.7.5 Expected values
13.7.6 Moments and correlation
13.7.7 Bivariate Gaussian distribution
13.8 Functions of Two or More Random Variables
13.8.1 Sums of two random variables
13.8.2 Rician distribution
13.8.3 Products of random variables
13.8.4 Quotients of random variables
13.8.5 Two functions of two random variables
13.8.6 Sums of several random variables
13.9 Limit Distributions
13.9.1 Gaussian density function
13.9.2 Gamma density function
Suggested Reading
Exercises
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