Engineering Mathematics - 1

400 Pages·2013·9.372 MB·English

by  T. Veerarajan

Engineering Mathematics-1
About the Author
T Veerarajan is the retired Dean, Department of
Mathematics, Velammal College of Engineering and
Technology, Viraganoor, Madurai, Tamil Nadu. A Gold
Medalist from Madras University, he has had a brilliant
academic career all through. He has 50 years of teaching
experience at undergraduate and postgraduate levels in
various established Engineering Colleges in Tamil Nadu
including Anna University, Chennai.
Engineering Mathematics
T Veerarajan
Dean, Department of Mathematics, (Retd.)
Velammal College of Engineering & Technology
Viraganoor, Madurai,
Tamil Nadu
McGraw Hill Education (India) Private Limited
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Engineering Mathematics-1
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Contents
Preface xi
1. Matrices I-1.3
1.1 Introduction I-1.3
1.2 Vectors I-1.4
1.3 Linear Dependence and Linear Independence of Vectors I-1.5
1.4 Methods of Testing Linear Dependence or
Independence of a Set of Vectors I-1.5
1.5 Consistency of a System of Linear Algebraic Equations I-1.5
Worked Example 1(a) I-1.7
Exercise 1(a) I-1.24
1.6 Eigenvalues and Eigenvectors I-1.27
Worked Example 1(b) I-1.32
Exercise 1(b) I-1.44
1.7 Cayley – Hamilton Theorem I-1.46
1.8 Property I-1.47
1.9 Calculation of Powers of a Matrix A I-1.49
1.10 Diagonalisation by Orthogonal Transformation or
Orthogonal Reduction I-1.49
Worked Example 1(c) I-1.49
Exercise 1(c) I-1.62
1.11 Quadratic Forms I-1.64
Worked Example 1(d) I-1.67
Exercise 1(d) I-1.74
Answers I-1.75
2. Sequences and Series I-2.1
I-2.1
2.2 General Properties of Series I-2.3
Worked Example 2(a) I-2.8
Exercise 2(a) I-2.21
Worked Example 2(b) I-2.23
Exercise 2(b) 1-2.36
2.3 Alternating Series I-2.38
Worked Example 2(c) I-2.42
Exercise 2(c) I-2.47
vi Contents
2.4 Sequences and Series I-2.49
Exercise 2(d) I-2.50
Worked Example 2(d) I-2.52
Exercise 2(e) I-2.54
Answers I-2.54
3. Application of Differential Calculus I-3.1
3.1 Curvature and Radius of Curvature I-3.1
3.2 Centre and Circle of Curvature I-3.6
Worked Example 3(a) I-3.7
Exercise 3(a) I-3.20
3.3 Evolutes and Envelopes I-3.21
Worked Example 3(b) I-3.23
3.4 Aliter I-3.33
Exercise 3(b) I-3.38
Answers I-3.40
4. Differential Calculus of Several Variables I-4.1
4.1 Introduction I-4.1
4.2 Total Differentiation I-4.1
Worked Example 4(a) I-4.4
Exercise 4(a) I-4.20
4.3 Jacobians I-4.27
4.4 Differentiation Under the Integral Sign I-4.31
Worked Example 4(b) I-4.33
Exercise 4(b) I-4.48
4.5 Maxima and Minima of Functions of Two Variables I-4.50
Worked Example 4(c) I-4.52
Exercise 4(c) I-4.65
Answers I-4.67
5. Multiple Integrals I-5.1
5.1 Introduction I-5.1
5.2 Evaluation of Double and Triple Integrals I-5.1
5.3 Region of Integration I-5.2
Worked Example 5(a) I-5.3
Exercise 5(a) I-5.15
5.4 Change of Order of Integration in a Double Integral I-5.17
5.5 Plane Area at Double Integral I-5.18
Worked Example 5(b) I-5.21
Exercise 5(b) I-5.38
5.6 Line Integral I-5.41
5.7 Surface Integral I-5.43
5.8 Volume Integral I-5.44
Contents vii
Worked Example 5(c) I-5.45
Exercise 5(c) I-5.57
5.9 Gamma and Beta Functions I-5.59
Worked Example 5(d) I-5.63
Exercise 5(d) I-5.77
Answers I-5.79
Appendix A: Partial Differentiation I-A.1–21
Appendix B: Solutions to the January 2012 Question Paper I-B.1–9
Appendix C: Solutions to the January 2013 Question Paper I-C.1–13
Appendix D: Solutions to the Model Question Paper I I-D.1–9
Appendix E: Solutions to the Model Question Paper II I-E.1–9

Preface
of my book on Engineering Mathematics-1 by the students and teachers throughout
the country.
course on Engineering Mathematics offered to the students of engineering. The
contents have been covered in adequate depth for the semester 1 Syllabi of various
universities/ deemed universities across the country.
The book offers a balanced coverage of both theory and problems. Lucid writing
style supported by step-by-step solutions to all problems enhances understanding
of the concepts. It has an excellent pedagogy with 1212 unsolved problems, 539
solved problems, 669 short answer questions and 914 descriptive questions. For
appendix. Additionally, the book is accompanied by a website which provides useful
supplements including recent solved question papers.
I hope that the book will be received by both the faculty and the students as
enthusiastically as the earlier editions of the book and my other books. Critical
evaluation and suggestions for the improvement of the book will be highly appreciated
and acknowledged.
T VEERARAJAN
Publisher’s Note:
McGraw Hill Education (India) Private Limited looks forward to receiving
from teachers and students their valuable views, comments and suggestions
for improvements, all of which may be sent to [email protected],
mentioning the title and author’s name.