Essentials of Mathematical Methods in Science and Engineering 1st Edition by Selçuk Bayin ISBN 9781119580287 1119580285

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ISBN 10: 1119580285
ISBN 13: 9781119580287
Author: Selçuk Bayin

A comprehensive introduction to the multidisciplinary applications of mathematical methods, revised and updated

The second edition of Essentials of Mathematical Methods in Science and Engineering offers an introduction to the key mathematical concepts of advanced calculus, differential equations, complex analysis, and introductory mathematical physics for students in engineering and physics research. The book’s approachable style is designed in a modular format with each chapter covering a subject thoroughly and thus can be read independently.

This updated second edition includes two new and extensive chapters that cover practical linear algebra and applications of linear algebra as well as a computer file that includes Matlab codes. To enhance understanding of the material presented, the text contains a collection of exercises at the end of each chapter. The author offers a coherent treatment of the topics with a style that makes the essential mathematical skills easily accessible to a multidisciplinary audience. This important text:

•    Includes derivations with sufficient detail so that the reader can follow them without searching for results in other parts of the book

•    Puts the emphasis on the analytic techniques

•    Contains two new chapters that explore linear algebra and its applications

•    Includes Matlab codes that the readers can use to practice with the methods introduced in the book

Written for students in science and engineering, this new edition of Essentials of Mathematical Methods in Science and Engineering maintains all the successful features of the first edition and includes new information.

Essentials of Mathematical Methods in Science and Engineering 1st Edition Table of contents:

CHAPTER 1: FUNCTIONAL ANALYSIS

1.1 CONCEPT OF FUNCTION

1.2 CONTINUITY AND LIMITS

1.3 PARTIAL DIFFERENTIATION

1.4 TOTAL DIFFERENTIAL

1.5 TAYLOR SERIES

1.6 MAXIMA AND MINIMA OF FUNCTIONS

1.7 EXTREMA OF FUNCTIONS WITH CONDITIONS

1.8 DERIVATIVES AND DIFFERENTIALS OF COMPOSITE FUNCTIONS

1.9 IMPLICIT FUNCTION THEOREM

1.10 INVERSE FUNCTIONS

1.11 INTEGRAL CALCULUS AND THE DEFINITE INTEGRAL

1.12 RIEMANN INTEGRAL

1.13 IMPROPER INTEGRALS

1.14 CAUCHY PRINCIPAL VALUE INTEGRALS

1.15 INTEGRALS INVOLVING A PARAMETER

1.16 LIMITS OF INTEGRATION DEPENDING ON A PARAMETER

1.17 DOUBLE INTEGRALS

1.18 PROPERTIES OF DOUBLE INTEGRALS

1.19 TRIPLE AND MULTIPLE INTEGRALS

REFERENCES

PROBLEMS

CHAPTER 2: VECTOR ANALYSIS

2.1 VECTOR ALGEBRA: GEOMETRIC METHOD

2.2 VECTOR ALGEBRA: COORDINATE REPRESENTATION

2.3 LINES AND PLANES

2.4 VECTOR DIFFERENTIAL CALCULUS

2.5 GRADIENT OPERATOR

2.6 DIVERGENCE AND CURL OPERATORS

2.7 VECTOR INTEGRAL CALCULUS IN TWO DIMENSIONS

2.8 CURL OPERATOR AND STOKES’S THEOREM

2.9 MIXED OPERATIONS WITH THE DEL OPERATOR

2.10 POTENTIAL THEORY

REFERENCES

PROBLEMS

CHAPTER 3: GENERALIZED COORDINATES AND TENSORS

3.1 TRANSFORMATIONS BETWEEN CARTESIAN COORDINATES

3.2 CARTESIAN TENSORS

3.3 GENERALIZED COORDINATES

3.4 GENERAL TENSORS

3.5 DIFFERENTIAL OPERATORS IN GENERALIZED COORDINATES

3.6 ORTHOGONAL GENERALIZED COORDINATES

REFERENCES

PROBLEMS

CHAPTER 4: DETERMINANTS AND MATRICES

4.1 BASIC DEFINITIONS

4.2 OPERATIONS WITH MATRICES

4.3 SUBMATRIX AND PARTITIONED MATRICES

4.4 SYSTEMS OF LINEAR EQUATIONS

4.5 GAUSS’S METHOD OF ELIMINATION

4.6 DETERMINANTS

4.7 PROPERTIES OF DETERMINANTS

4.8 CRAMER’S RULE

4.9 INVERSE OF A MATRIX

4.10 HOMOGENEOUS LINEAR EQUATIONS

REFERENCES

PROBLEMS

CHAPTER 5: LINEAR ALGEBRA

5.1 FIELDS AND VECTOR SPACES

5.2 LINEAR COMBINATIONS, GENERATORS, AND BASES

5.3 COMPONENTS

5.4 LINEAR TRANSFORMATIONS

5.5 MATRIX REPRESENTATION OF TRANSFORMATIONS

5.6 ALGEBRA OF TRANSFORMATIONS

5.7 CHANGE OF BASIS

5.8 INVARIANTS UNDER SIMILARITY TRANSFORMATIONS

5.9 EIGENVALUES AND EIGENVECTORS

5.10 MOMENT OF INERTIA TENSOR

5.11 INNER PRODUCT SPACES

5.12 THE INNER PRODUCT

5.13 ORTHOGONALITY AND COMPLETENESS

5.14 GRAM–SCHMIDT ORTHOGONALIZATION

5.15 EIGENVALUE PROBLEM FOR REAL SYMMETRIC MATRICES

5.16 PRESENCE OF DEGENERATE EIGENVALUES

5.17 QUADRATIC FORMS

5.18 HERMITIAN MATRICES

5.19 MATRIX REPRESENTATION OF HERMITIAN OPERATORS

5.20 FUNCTIONS OF MATRICES

5.21 FUNCTION SPACE AND HILBERT SPACE

5.22 DIRAC’S BRA AND KET VECTORS

REFERENCES

PROBLEMS

CHAPTER 6: PRACTICAL LINEAR ALGEBRA

6.1 SYSTEMS OF LINEAR EQUATIONS

6.2 NUMERICAL METHODS OF LINEAR ALGEBRA

REFERENCES

PROBLEMS

CHAPTER 7: APPLICATIONS OF LINEAR ALGEBRA

7.1 CHEMISTRY AND CHEMICAL ENGINEERING

7.2 LINEAR PROGRAMMING

7.3 LEONTIEF INPUT–OUTPUT MODEL OF ECONOMY

7.4 APPLICATIONS TO GEOMETRY

7.5 ELIMINATION THEORY

7.6 CODING THEORY

7.7 CRYPTOGRAPHY

7.8 GRAPH THEORY

REFERENCES

PROBLEMS

CHAPTER 8: SEQUENCES AND SERIES

8.1 SEQUENCES

8.2 INFINITE SERIES

8.3 ABSOLUTE AND CONDITIONAL CONVERGENCE

8.4 OPERATIONS WITH SERIES

8.5 SEQUENCES AND SERIES OF FUNCTIONS

8.6 ‐TEST FOR UNIFORM CONVERGENCE

8.7 PROPERTIES OF UNIFORMLY CONVERGENT SERIES

8.8 POWER SERIES

8.9 TAYLOR SERIES AND MACLAURIN SERIES

8.10 INDETERMINATE FORMS AND SERIES

REFERENCES

PROBLEMS

CHAPTER 9: COMPLEX NUMBERS AND FUNCTIONS

9.1 THE ALGEBRA OF COMPLEX NUMBERS

9.2 ROOTS OF A COMPLEX NUMBER

9.3 INFINITY AND THE EXTENDED COMPLEX PLANE

9.4 COMPLEX FUNCTIONS

9.5 LIMITS AND CONTINUITY

9.6 DIFFERENTIATION IN THE COMPLEX PLANE

9.7 ANALYTIC FUNCTIONS

9.8 HARMONIC FUNCTIONS

9.9 BASIC DIFFERENTIATION FORMULAS

9.10 ELEMENTARY FUNCTIONS

REFERENCES

PROBLEMS

CHAPTER 10: COMPLEX ANALYSIS

10.1 CONTOUR INTEGRALS

10.2 TYPES OF CONTOURS

10.3 THE CAUCHY–GOURSAT THEOREM

10.4 INDEFINITE INTEGRALS

10.5 SIMPLY AND MULTIPLY CONNECTED DOMAINS

10.6 THE CAUCHY INTEGRAL FORMULA

10.7 DERIVATIVES OF ANALYTIC FUNCTIONS

10.8 COMPLEX POWER SERIES

10.9 CONVERGENCE OF POWER SERIES

10.10 CLASSIFICATION OF SINGULAR POINTS

10.11 RESIDUE THEOREM

REFERENCES

PROBLEMS

CHAPTER 11: ORDINARY DIFFERENTIAL EQUATIONS

11.1 BASIC DEFINITIONS FOR ORDINARY DIFFERENTIAL EQUATIONS

11.2 FIRST‐ORDER DIFFERENTIAL EQUATIONS

11.3 SECOND‐ORDER DIFFERENTIAL EQUATIONS

11.4 LINEAR DIFFERENTIAL EQUATIONS OF HIGHER ORDER

11.5 INITIAL VALUE PROBLEM AND UNIQUENESS OF THE SOLUTION

11.6 SERIES SOLUTIONS: FROBENIUS METHOD

REFERENCES

PROBLEMS

CHAPTER 12: SECOND‐ORDER DIFFERENTIAL EQUATIONS AND SPECIAL FUNCTIONS

12.1 LEGENDRE EQUATION

12.2 HERMITE EQUATION

12.3 LAGUERRE EQUATION

REFERENCES

PROBLEMS

CHAPTER 13: BESSEL’S EQUATION AND BESSEL FUNCTIONS

13.1 BESSEL’S EQUATION AND ITS SERIES SOLUTION

13.2 ORTHOGONALITY AND THE ROOTS OF BESSEL FUNCTIONS

REFERENCES

PROBLEMS

CHAPTER 14: PARTIAL DIFFERENTIAL EQUATIONS AND SEPARATION OF VARIABLES

14.1 SEPARATION OF VARIABLES IN CARTESIAN COORDINATES

14.2 SEPARATION OF VARIABLES IN SPHERICAL COORDINATES

14.3 SEPARATION OF VARIABLES IN CYLINDRICAL COORDINATES

REFERENCES

PROBLEMS

CHAPTER 15: FOURIER SERIES

15.1 ORTHOGONAL SYSTEMS OF FUNCTIONS

15.2 FOURIER SERIES

15.3 EXPONENTIAL FORM OF THE FOURIER SERIES

15.4 CONVERGENCE OF FOURIER SERIES

15.5 SUFFICIENT CONDITIONS FOR CONVERGENCE

15.6 THE FUNDAMENTAL THEOREM

15.7 UNIQUENESS OF FOURIER SERIES

15.8 EXAMPLES OF FOURIER SERIES

15.9 FOURIER SINE AND COSINE SERIES

15.10 CHANGE OF INTERVAL

15.11 INTEGRATION AND DIFFERENTIATION OF FOURIER SERIES

REFERENCES

PROBLEMS

CHAPTER 16: FOURIER AND LAPLACE TRANSFORMS

16.1 TYPES OF SIGNALS

16.2 SPECTRAL ANALYSIS AND FOURIER TRANSFORMS

16.3 CORRELATION WITH COSINES AND SINES

16.4 CORRELATION FUNCTIONS AND FOURIER TRANSFORMS

16.5 INVERSE FOURIER TRANSFORM

16.6 FREQUENCY SPECTRUMS

16.7 DIRAC‐DELTA FUNCTION

16.8 A CASE WITH TWO COSINES

16.9 GENERAL FOURIER TRANSFORMS AND THEIR PROPERTIES

16.10 BASIC DEFINITION OF LAPLACE TRANSFORM

16.11 DIFFERENTIAL EQUATIONS AND LAPLACE TRANSFORMS

16.12 TRANSFER FUNCTIONS AND SIGNAL PROCESSORS

16.13 CONNECTION OF SIGNAL PROCESSORS

REFERENCES

PROBLEMS

CHAPTER 17: CALCULUS of VARIATIONS

17.1 A SIMPLE CASE

17.2 VARIATIONAL ANALYSIS

17.3 ALTERNATE FORM OF EULER EQUATION

17.4 VARIATIONAL NOTATION

17.5 A MORE GENERAL CASE

17.6 HAMILTON’S PRINCIPLE

17.7 LAGRANGE’S EQUATIONS OF MOTION

17.8 DEFINITION OF LAGRANGIAN

17.9 PRESENCE OF CONSTRAINTS IN DYNAMICAL SYSTEMS

17.10 CONSERVATION LAWS

REFERENCES

PROBLEMS

CHAPTER 18: PROBABILITY THEORY AND DISTRIBUTIONS

18.1 INTRODUCTION TO PROBABILITY THEORY

18.2 PERMUTATIONS AND COMBINATIONS

18.3 APPLICATIONS TO STATISTICAL MECHANICS

18.4 STATISTICAL MECHANICS AND THERMODYNAMICS

18.5 RANDOM VARIABLES AND DISTRIBUTIONS

18.6 DISTRIBUTION FUNCTIONS AND PROBABILITY

18.7 EXAMPLES OF CONTINUOUS DISTRIBUTIONS

18.8 DISCRETE PROBABILITY DISTRIBUTIONS

18.9 FUNDAMENTAL THEOREM OF AVERAGES

18.10 MOMENTS OF DISTRIBUTION FUNCTIONS

18.11 CHEBYSHEV’S THEOREM

18.12 LAW OF LARGE NUMBERS

REFERENCES

PROBLEMS

CHAPTER 19: INFORMATION THEORY

19.1 ELEMENTS OF INFORMATION PROCESSING MECHANISMS

19.2 CLASSICAL INFORMATION THEORY

19.3 QUANTUM INFORMATION THEORY

REFERENCES

PROBLEMS

Further Reading

MATHEMATICAL METHODS TEXTBOOKS:

MATHEMATICAL METHODS WITH COMPUTERS:

CALCULUS/ADVANCED CALCULUS:

LINEAR ALGEBRA AND ITS APPLICATIONS:

COMPLEX CALCULUS:

DIFFERENTIAL EQUATIONS:

CALCULUS OF VARIATIONS:

FOURIER SERIES, INTEGRAL TRANSFORMS AND SIGNAL PROCESSING:

SERIES AND SPECIAL FUNCTIONS:

MATHEMATICAL TABLES:

CLASSICAL MECHANICS:

QUANTUM MECHANICS:

ELECTROMAGNETIC THEORY:

PROBABILITY THEORY:

INFORMATION THEORY:

INDEX

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