Modern Advanced Mathematics for Engineers

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Edition: 1st
Format: Hardcover
Pub. Date: 2001-04-02
Publisher(s): Wiley-Interscience
List Price: $222.87

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Summary

A convenient single source for vital mathematical concepts, written by engineers and for engineers.Builds a strong foundation in modern applied mathematics for engineering students, and offers them a concise and comprehensive treatment that summarizes and unifies their mathematical knowledge using a system focused on basic concepts rather than exhaustive theorems and proofs.The authors provide several levels of explanation and exercises involving increasing degrees of mathematical difficulty to recall and develop basic topics such as calculus, determinants, Gaussian elimination, differential equations, and functions of a complex variable. They include an assortment of examples ranging from simple illustrations to highly involved problems as well as a number of applications that demonstrate the concepts and methods discussed throughout the book. This broad treatment also offers:* Key mathematical tools needed by engineers working in communications, semiconductor device simulation, and control theory * Concise coverage of fundamental concepts such as sets, mappings, and linearity * Thorough discussion of topics such as distance, inner product, and orthogonality * Essentials of operator equations, theory of approximations, transform methods, and partial differential equationsIt makes an excellent companion to less general engineering texts and a useful reference for practitioners.

Author Biography

VLADIMIR V. MITIN and DMITRI A. ROMANOV are both Professors in the Department of Electrical and Computer Engineering at Wayne State University.

Table of Contents

Dedication v
Preface vii
The Basics of Set Theory
1(18)
Introduction
1(1)
Logical Connectives --- Notation
2(1)
Introduction of Sets
2(1)
Specifications of Sets
3(1)
Examples of Sets
4(1)
Basic Sets of Numbers
4(1)
Paradoxes
5(1)
Axioms of Set Theory
6(7)
Sample Problems
13(2)
Summary of Chapter 1
15(1)
Problems
15(1)
Further Reading
16(3)
Relations and Mappings
19(22)
Relations --- The Naive Approach
19(1)
Ordered n-tuple
20(1)
An n-tuple Versus a Set
21(1)
Cross-Product Set and Its Properties
22(3)
Sample Problem I
25(2)
n-ary Relation
27(2)
Binary Relations --- Terminology
29(1)
Mappings --- Classification
30(6)
Sample Problem II
36(2)
Summary of Chapter 2
38(1)
Problems
39(1)
Further Reading
39(2)
Mathematical Logic
41(26)
Logical Objects and Logical Statements
41(1)
Logical-Statement and Truth-Value Mappings
42(2)
Logical Connectives and Logical Formulas
44(1)
Adequate Set of Connectives
44(1)
Truth Values of Complex Logical Formulas
45(4)
Tautologies and Contradictions
49(1)
Sample Problems I
50(2)
Fundamental Tautologies. De Morgan Laws. Proof Formats
52(5)
Properties of Connectives. An Axiomatic Approach to Mathematical Logic
57(1)
Sample Problems II
58(1)
Boolean Algebra. The Analogy Between Mathematical Logic and Set Theory
59(2)
Sample Problems III
61(2)
Summary of Chapter 3
63(1)
Problems
64(1)
Further Reading
65(2)
Algebraic Structures: Group through Linear Space
67(28)
Classification of Algebraic Structures
67(6)
Algebra of Complex Numbers
73(2)
Geometric Representation of Complex Numbers
75(2)
Linear Space: Axioms and Examples
77(3)
Linear Combination and Span. Linear Mappings
80(1)
Linear Subspaces
80(2)
Linear Independence
82(1)
Basis and Dimension of a Linear Space
82(3)
Decomposition as Mapping. Isomorphism of Finite-Dimensional Linear Spaces
85(1)
Sample Problems
86(4)
Summary of Chapter 4
90(2)
Problems
92(1)
Further Reading
92(3)
Linear Mappings and Matrices
95(32)
Linear Mappings and Column Vectors
95(1)
Matrices
95(2)
The Matrix Product
97(2)
Matrix Algebra
99(2)
Transpose of a Matrix
101(1)
Fundamental Subspaces of Column Vectors Associated With a Given Matrix
102(1)
The Echelon Matrix. LU-decomposition
103(3)
The Fundamental Theorem of Linear Algebra
106(2)
Sample Problem I
108(2)
Eigenvalues, Eigenvectors, and the Characteristic Equation
110(1)
Matrix Diagonalization
111(1)
Functions of Matrices
112(5)
Sample Problem II
117(1)
Multiple Eigenvalues. The Jordan Form
118(4)
Summary of Chapter 5
122(1)
Problems
123(1)
Further Reading
124(3)
Metrics and Topological Properties
127(20)
Necessity of Distance Concept
127(1)
Metric Mapping and Distance: Definition and Axioms
127(2)
Metric Spaces
129(3)
Sample Problem
132(1)
Proximity. Limits in Metric Spaces
133(1)
Cauchy Sequences
134(1)
Open Ball: A Geometric Interpretation
134(1)
Examples of Cauchy Sequences
135(2)
Open and Closed Sets. Completeness vs. Closedness
137(3)
Continuous mapping
140(1)
Lipschitz (Bounded) Mapping
141(1)
Example of a Contraction Mapping
141(1)
Summary of Chapter 6
142(1)
Problems
143(1)
Further Reading
144(3)
Banach and Hilbert Spaces
147(24)
Introduction: The Great Alliance of Linearity and Metrics
147(1)
Norm Mapping. Banach Space
147(2)
Sample Problem I
149(1)
Finite-Dimensional Subspaces in Banach Space
150(2)
Inner-Product Mapping. Hilbert Space
152(2)
Sample Problems II
154(1)
The Cauchy-Schwarz Inequality
155(2)
Inner Product and Matrices. Hermitian Matrices
157(2)
Continuity of the Inner-Product Mapping
159(1)
Orthogonality
160(1)
Sample Problem III
160(2)
Eigenvectors of Hermitian Matrices
162(1)
The Projection Theorem
163(4)
Summary of Chapter 7
167(1)
Problems
168(1)
Further Reading
169(2)
Orthonormal Bases and Fourier Series
171(28)
Orthonormal Basis
171(1)
The Gram - Schmidt Orthonormalization
172(4)
Sample Problem I
176(2)
Sample Problem II
178(2)
Bessel's Inequality
180(1)
The Projection Decomposition
181(3)
Sample Problem III
184(2)
Complete Orthonormal Sequences
186(4)
The Fourier Series
190(1)
Properties of the Fourier Series
191(1)
Sample Problem IV
192(2)
Sample Problem V
194(2)
Summary of Chapter 8
196(1)
Problems
197(1)
Further Reading
197(2)
Operator Equations
199(36)
Inverse Mappings and Operator Equations
199(1)
Three Generic Problems and Their Equivalence
200(1)
Fixed-Point Problem. The Contraction Mapping Theorem
201(3)
Successive Approximations
204(3)
Application to Differential Equations
207(5)
Sample Problem I
212(3)
Residual Principle and Residual Approximation
215(6)
Incompletely Specified Equations. Least-Square Approximation
221(5)
Problem of Uniqueness. Pseudo-Inverse Mapping
226(2)
Sample Problem II
228(2)
Summary of Chapter 9
230(1)
Problems
231(1)
Further Reading
232(3)
Fourier and Laplace Transforms
235(38)
Fourier Integral
235(2)
The Fourier Transform as a Mapping
237(2)
Digression. Properties of the δ-function
239(2)
Back to Fourier Transform Mapping
241(1)
Properties of the Fourier Transform
241(5)
Application to Linear Systems. The Convolution Product
246(4)
Sample Problem I: Smoothing Operator
250(1)
Residue Theory
251(5)
From Fourier Transform to Laplace Transform
256(2)
Properties of the Laplace Transform
258(4)
Basic Laplace Transforms
262(2)
The Laplace Transform Table
264(1)
Sample Problems II
265(2)
Laplace Transform of the Convolution Product
267(1)
Applications of the Laplace Transform to Solving Differential Equations
268(1)
Sample Problem III
269(1)
Summary of Chapter 10
269(1)
Problems
270(1)
Further Reading
271(2)
Partial Differential Equations
273(33)
A Short Introduction
273(1)
Definitions
274(1)
Canonical Forms
274(6)
The Parabolic Case
280(1)
Canonical Forms --- A Summary
281(1)
Sample Problem I
282(3)
The Wave Equation. D'Alembert's Formula
285(3)
The Diffusion Equation. Transform Methods
288(4)
The Elliptic Case: Poisson's Equation
292(7)
A Very Brief Comment in Defense of the Laplace Transform
299(1)
Sample Problem II
300(2)
Summary of Chapter II
302(1)
Problems
303(1)
Further Reading
303(3)
Topic Index 306

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