Mathematical methods

437 Pages·2010·6.546 MB·English

by  Rukmangadachari, E

Mathematical
Methods
E. Rukmangadachari
Professor of Mathematics,
Department of Humanities and Sciences,
Malla Reddy Engineering College,
Secunderabad
RRuukkmmaannggaaddaacchhaarrii__FFMM..iinndddd ii 99//44//22000099 22::5577::1177 PPMM
Copyright © 2010 Dorling Kindersley (India) Pvt. Ltd.
Licensees of Pearson Education in South Asia
No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s prior
written consent.
This eBook may or may not include all assets that were part of the print version. The publisher reserves the
right to remove any material present in this eBook at any time.
ISBN 9788131725986
eISBN 9789332500853
Head Office: A-8(A), Sector 62, Knowledge Boulevard, 7th Floor, NOIDA 201 309, India
Registered Office: 11 Local Shopping Centre, Panchsheel Park, New Delhi 110 017, India
RRuukkmmaannggaaddaacchhaarrii__FFMM..iinndddd iiii 99//44//22000099 22::5577::1188 PPMM
To
my beloved father-in-law,
Pujyasri Nuthalapati Kumaraswamy,
who has been responsible for my spiritual progress
following
“sahaj marg.”
RRuukkmmaannggaaddaacchhaarrii__FFMM..iinndddd iiiiii 99//44//22000099 22::5577::1188 PPMM
About the Author
E. Rukmangadachari is former head of Computer Science and Engineering
as well as Humanities and Sciences at Malla Reddy Engineering College,
Secunderabad. Earlier, he was a reader in Mathematics (PG course) at
Government College, Rajahmundry. He is an M.A. from Osmania University,
Hyderabad, and an M.Phil. and Ph.D. degree holder from Sri Venkateswara
University, Tirupathi.
A recipient of the Andhra Pradesh State Meritorious Teachers’ Award in
1981, Professor Rukmangadachari has published over 40 research papers in
national and international journals. With a rich repertoire of over 45 years’
experience in teaching mathematics to undergraduate, postgraduate and engi-
neering students, he is currently the vice-president of the Andhra Pradesh
Society for Mathematical Sciences. An ace planner with fi ne managerial
skills, he was the organising secretary for the conduct of the 17th Congress of
the Andhra Pradesh Society for Mathematical Sciences, Hyderabad.
RRuukkmmaannggaaddaacchhaarrii__FFMM..iinndddd iivv 99//44//22000099 22::5577::1188 PPMM
CCoonntteennttss
About the Author iv 2
Eigenvalues and Eigenvectors 2-1
Preface ix
2.1 Introduction 2-1
1 2.2 Linear Transformation 2-1
Matrices and Linear Systems
2.3 Characteristic Value Problem 2-1
of Equations 1-1
Exercise 2.1 2-6
1.1 Introduction 1-1
2.4 Properties of Eigenvalues
1.2 Algebra of Matrices 1-3 and Eigenvectors 2-7
1.3 Matrix Multiplication 1-4 2.5 Cayley–Hamilton Theorem 2-9
1.4 Determinant of a Square Matrix 1-5 Exercise 2.2 2-12
1.5 Related Matrices 1-8 2.6 Reduction of a Square Matrix
1.6 Determinant-related Matrices 1-11 to Diagonal Form 2-14
1.7 Special Matrices 1-12 2.7 Powers of a Square Matrix A—
Finding of Modal Matrix P
Exercise 1.1 1-15 and Inverse Matrix A−1 2-18
1.8 Linear Systems of Equations 1-16
Exercise 2.3 2-23
1.9 Homogeneous (H) and
Nonhomogeneous (NH) Systems
3
Real and Complex Matrices 3-1
of Equations 1-16
1.10 Elementary Row and Column 3.1 Introduction 3-1
Operations (Transformations)
3.2 Orthogonal /Orthonormal System
for Matrices 1-17
of Vectors 3-1
Exercise 1.2 1-20
3.3 Real Matrices 3-1
1.11 Inversion of a Nonsingular Matrix 1-21
Exercise 3.1 3-6
Exercise 1.3 1-24
3.4 Complex Matrices 3-7
1.12 Rank of a Matrix 1-25
3.5 Properties of Hermitian,
1.13 Methods for Finding the Rank Skew-Hermitian and Unitary
of a Matrix 1-26 Matrices 3-8
Exercise 1.4 1-32 Exercise 3.2 3-14
1.14 Existence and Uniqueness
of Solutions of a System 4
Quadratic Forms 4-1
of Linear Equations 1-33
4.1 Introduction 4-1
1.15 Methods of Solution of NH
and H Equations 1-34 4.2 Quadratic Forms 4-1
1.16 Homogeneous System 4.3 Canonical Form (or) Sum
of Equations (H) 1-44 of the Squares Form 4-3
Exercise 1.5 1-46 4.4 Nature of Real Quadratic Forms 4-3
RRuukkmmaannggaaddaacchhaarrii__FFMM..iinndddd vv 99//44//22000099 22::5577::1199 PPMM
vi (cid:2) Contents
4.5 Reduction of a Quadratic Form 7 Curve Fitting 7-1
to Canonical Form 4-5
7.1 Introduction 7-1
4.6 Sylvestor’s Law of Inertia 4-6
7.2 Curve Fitting by the Method
4.7 Methods of Reduction of a
of Least Squares 7-2
Quadratic Form to a Canonical Form 4-6
7.3 Curvilinear (or Nonlinear)
Exercise 4.1 4-9
Regression 7-8
5 7.4 Curve fi tting by a Sum
Solution of Algebraic and
of Exponentials 7-10
Transcendental Equations 5-1
7.5 Weighted Least Squares
5.1 Introduction to Numerical Approximation 7-13
Methods 5-1 Exercise 7.1 7-14
5.2 Errors and their Computation 5-1
8
5.3 Formulas for Errors 5-2 Numerical Differentiation
5.4 Mathematical Pre-requisites 5-3 and Integration 8-1
5.5 Solution of Algebraic and 8.1 Introduction 8-1
Transcendental Equations 5-4
8.2 Errors in Numerical
5.6 Direct Methods of Solution 5-4 Differentiation 8-6
5.7 Numerical Methods of Solution 8.3 Maximum and Minimum
of Equations of the Form f (x) = 0 5-5 Values of a Tabulated Function 8-7
Exercise 5.1 5-20 Exercise 8.1 8-8
8.4 Numerical Integration:
6
Interpolation 6-1 Introduction 8-8
6.1 Introduction 6-1 Exercise 8.2 8-21
6.2 Interpolation with Equal 8.5 Cubic Splines 8-21
Intervals 6-2 Exercise 8.3 8-27
6.3 Symbolic Relations and
Separation of Symbols 6-4 9 Numerical Solution of Ordinary
Exercise 6.1 6-8 Differential Equations 9-1
6.4 Interpolation 6-8
9.1 Introduction 9-1
6.5 Interpolation Formulas
9.2 Methods of Solution 9-1
For Equal Intervals 6-9
9.3 Predictor–Corrector Methods 9-17
Exercise 6.2 6-13
Exercise 9.1 9-21
6.6 Interpolation with Unequal
Intervals 6-14
10
6.7 Properties Satisfi ed by D′ 6-15 Fourier Series 10-1
6.8 Divided Difference Interpolation 10.1 Introduction 10-1
Formula 6-16
10.2 Periodic Functions, Properties 10-1
6.9 Inverse Interpolation Using
10.3 Classifi able Functions—Even
Lagrange’s Interpolation
and Odd Functions 10-2
Formula 6-17
10.4 Fourier Series, Fourier
6.10 Central Difference Formulas 6-21
Coeffi cients and Euler’s
Exercise 6.3 6-26 Formulae in (a, a + 2p) 10-3
RRuukkmmaannggaaddaacchhaarrii__FFMM..iinndddd vvii 99//44//22000099 22::5577::1199 PPMM
Contents (cid:2) vii
10.5 Dirichlet’s Conditions for 12 Partial Differential Equations 12-1
Fourier Series Expansion
of a Function 10-4 12.1 Introduction 12-1
10.6 Fourier Series Expansions: 12.2 Order, Linearity and
Even/Odd Functions 10-5 Homogeneity of a Partial
Differential Equation 12-1
10.7 Simply-Defined and
Multiply-(Piecewise) Defined 12.3 Origin of Partial Differential
Functions 10-7 Equation 12-2
Exercise 10.1 10-18 12.4 Formation of Partial
Differential Equation by
10.8 Change of Interval: Fourier
Series in Interval (a, a + 2l ) 10-19 Elimination of Two Arbitrary
Constants 12-3
Exercise 10.2 10-23
Exercise 12.1 12-4
10.9 Fourier Series Expansions
12.5 Formation of Partial
of Even and Odd Functions
in (−l, l ) 10-24 Differential Equations by
Elimination of Arbitrary
Exercise 10.3 10-26 Functions 12-5
10.10 Half-range Fourier Sine/ Exercise 12.2 12-7
Cosine Series: Odd and Even
12.6 Classification of
Periodic Continuations 10-26
First-order Partial
Exercise 10.4 10-33 Differential Equations 12-7
10.11 Root Mean Square (RMS) 12.7 Classifi cation of Solutions
Value of a Function 10-34 of First-order Partial
Exercise 10.5 10-36 Differential Equation 12-8
12.8 Equations Solvable by Direct
11
Fourier Integral Transforms 11-1
Integration 12-9
11.1 Introduction 11-1 Exercise 12.3 12-10
11.2 Integral Transforms 11-1 12.9 Quasi-linear Equations
of First Order 12-11
11.3 Fourier Integral Theorem 11-1
12.10 Solution of Linear,
11.4 Fourier Integral in Complex
Semi-linear and
Form 11-2
Quasi-linear Equations 12-11
11.5 Fourier Transform of f (x) 11-3
Exercise 12.4 12-17
11.6 Finite Fourier Sine Transform
12.11 Nonlinear Equations
and Finite Fourier Cosine
of First Order 12-18
Transform (FFCT) 11-4
Exercise 12.5 12-22
11.7 Convolution Theorem
for Fourier Transforms 11-5 12.12 Euler’s Method of Separation
of Variables 12-22
11.8 Properties of Fourier
Transform 11-6 Exercise 12.6 12-25
Exercise 11.1 11-15 12.13 Classifi cation of Second-
order Partial Differential
11.9 Parseval’s Identity for Fourier
Equations 12-25
Transforms 11-16
Exercise 12.7 12-33
11.10 Parseval’s Identities for Fourier
Sine and Cosine Transforms 11-17 Exercise 12.8 12-42
Exercise 11.2 11-18 Exercise 12.9 12-46
RRuukkmmaannggaaddaacchhaarrii__FFMM..iinndddd vviiii 99//44//22000099 22::5577::1199 PPMM
viii (cid:2) Contents
13 Z-Transforms and Solution 13.8 Method for Solving a Linear
Difference Equation with
of Difference Equations 13-1 Constant Coeffi cients 13-18
13.1 Introduction 13-1 Exercise 13.3 13-21
13.2 Z-Transform: Definition 13-1
Question Bank A-1
13.3 Z-Transforms of Some
Standard Functions Multiple Choice Questions A-1
(Special Sequences) 13-4
Fill in the Blanks A-35
13.4 Recurrence Formula
Match the Following A-51
for the Sequence of a Power
True or False Statements A-57
of Natural Numbers 13-5
13.5 Properties of Z-Transforms 13-6
Solved Question Papers A-61
Exercise 13.1 13-11
13.6 Inverse Z-Transform 13-11 Bibliography B-1
Exercise 13.2 13-16
Index I-1
13.7 Application of Z-Transforms:
Solution of a Difference
Equation; by Z-Transform 13-17
RRuukkmmaannggaaddaacchhaarrii__FFMM..iinndddd vviiiiii 99//44//22000099 22::5577::1199 PPMM
Preface
I am pleased to present this book on Mathematical Methods to the fi rst year B.Tech. students of Jawaharlal
Nehru Technological Universities (JNTU) at Hyderabad, Anantapur and Kakinada. Written in a simple, lucid
and easy-to-understand manner, the book conforms to the syllabus prescribed for JNTU, effective from
September 2009.
The concepts have been discussed with focus on clarity and coherence, supported by illustrations for
better comprehension. Over 370 well-chosen examples are worked out in the book to enable students under-
stand the fundamentals and the principles governing each topic.
Adequate exercises given at the end of each chapter, with answers and hints furnished as needed, pro-
vide students an insight into the methods of solving the problems with ingenuity. Model questions from past
University Examinations have been included in examples and exercises. A vast, answer-appended Question
Bank comprising Multiple Choice Questions, Fill in the Blanks, Match the Following and True or False
Statements serves to help the student in effortless recapitulation of the subject. In addition to helping students
to enhance their knowledge of the subject, these pedagogical elements also help them to prepare for their
mid-term examinations.
Suggestions for the improvement of the book are welcome and will be gratefully acknowledged.
Acknowledgements
I express my deep sense of gratitude to Sri Ch. Malla Reddy, Chairman, and Sri Ch. Mahender Reddy,
Secretary, Malla Reddy Group of Institutions (MRGI), whose patronage has given me an opportunity to write
this book.
I am also thankful to Prof. R. Madan Mohan, Director (Academics); Col G. Ram Reddy, Director
(Administration), MRGI; and Dr R. K. Murthy, Principal, Malla Reddy Engineering College, Secunderabad,
for their kindness, guidance, and encouragement.
E. RUKMANGADACHARI
RRuukkmmaannggaaddaacchhaarrii__FFMM..iinndddd iixx 99//44//22000099 22::5577::1199 PPMM