Engineering Mathematics-I : For RTU (Subject Code: 102)

369 Pages·2011·4.79 MB·English

by  Babu Ram

Engineering Mathematics-I
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Engineering Mathematics-I
Semester I
Rajasthan Technical University
Paper Code: 102
BABU RAM
Former Dean, Faculty of Physical Sciences,
Maharshi Dayanand University, Rohtak
The publishers would like to acknowledge Dr Sunil Dutt Purohit of College of Technology and
Engineering, Udaipur, for his help in mapping the content of this book to the requirements of
Rajasthan Technical University, Kota.
Copyright © 2011 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 9788131759158
eISBN 9789332510265
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
In memory of
my parents
Smt. Manohari Devi and Sri Makhan Lal
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Contents
Preface ix 2.8 Circular Asymptotes 2.12
Roadmap to the Syllabus x 2.9 Concavity, Convexity and Singular
Points 2.13
Symbols and Basic Formulae xi
2.10 C urve Tracing (Cartesian
Equations) 2.17
1 Curvature 1.1 2.11 Curve Tracing (Polar Equations) 2.22
2.12 C urve Tracing (Parametric
1.1 Radius of Curvature of Intrinsic
Equations) 2.25
Curves 1.1
1.2 Radius of Curvature for Cartesian Exercises 2.26
Curves 1.2
1.3 Radius of Curvature for Parametric 3 Functions of Several Variables 3.1
Curves 1.6
3.1 Continuity of a Function of
1.4 R adius of Curvature for Pedal Curves 1.8
Two Variables 3.2
1.5 Radius of Curvature for Polar
3.2 Differentiability of a Function of
Curves 1.9
Two Variables 3.2
1.6 Radius of Curvature at the Origin 1.13
3.3 The Differential Coeffi cients 3.2
1.7 Center of Curvature 1.15
3.4 Distinction Between Derivatives
1.8 Evolutes and Involutes 1.16
and Differential Coeffi cients 3.3
1.9 Equation of the Circle of Curvature 1.16
3.5 Higher-order Partial Derivatives 3.3
1.10 Chords of Curvature Parallel
3.6 Envelopes and Evolutes 3.9
to the Coordinate Axes 1.19
3.7 Homogeneous Functions and Euler’s
1.11 Chord of Curvature in Polar
Theorem 3.11
Coordinates 1.19
3.8 Differentiation of Composite
1.12 Miscellaneous Examples 1.22
Functions 3.16
Exercises 1.28 3.9 Transformation from Cartesian
to Polar Coordinates and Vice Versa 3.20
3.10 Taylor’s Theorem for Functions
2 Asymptotes and Curve Tracing 2.1
of Several Variables 3.23
2.1 D etermination of Asymptotes When 3.11 E xtreme Values 3.27
the Equation of the Curve in Cartesian 3.12 Lagrange’s Method of Undetermined
Form is Given 2.1 Multipliers 3.34
2.2 The Asymptotes of the General Rational 3.13 Jacobians 3.38
Algebraic Curve 2.2 3.14 Properties of Jacobians 3.39
2.3 Asymptotes Parallel to Coordinate 3.15 Necessary and Suffi cient Conditions
Axes 2.3 for a Jacobian to Vanish 3.41
2.4 Working Rule for Finding Asymptotes 3.16 Differentiation Under the Integral
of Rational Algebraic Curve 2.3 Sign 3.42
2.5 Intersection of a Curve and Its 3.17 Approximation of Errors 3.46
Asymptotes 2.7 3.18 General Formula for Errors 3.47
2.6 Asymptotes by Expansion 2.10 3.19 Miscellaneous Examples 3.49
2.7 Asymptotes of the Polar Curves 2.10
Exercises 3.53
viii (cid:132) Contents
4 7.8 Volume and Surface Area as Double
Quadrature and Rectifi cation 4.1
Integrals 7.22
4.1 Quadrature 4.1 7.9 Triple Integrals and Their
4.2 Rectifi cation 4.8 Evaluation 7.29
Exercises 4.13 7.10 Change to Spherical Polar Coordinates
from Cartesian Coordinates in a Triple
Integral 7.34
5
Volumes and Surfaces of Solids of Revolution 5.1 7.11 Volume as a Triple Integral 7.37
7.12 Miscellaneous Examples 7.41
5.1 Volume of the Solid of Revolution
(Cartesian Equations) 5.1 Exercises 7.44
5.2 Volume of the Solid of Revolution
8
(Parametric Equations) 5.6 Ordinary Differential Equations 8.1
5.3 Volume of the Solid of Revolution
8.1 Defi nitions and Examples 8.1
(Polar Curves) 5.8
8.2 Formulation of Differential
5.4 Surface of the Solid of Revolution
Equation 8.2
(Cartesian Equations) 5.9
8.3 Solution of Differential Equation 8.4
5.5 Surface of the Solid of Revolution
8.4 D ifferential Equations
(Parametric Equations) 5.11
of First Order 8.4
5.6 Surface of the Solid of Revolution
8.5 Separable Equations 8.5
(Polar Curves) 5.13
8.6 Homogeneous Equations 8.8
Exercises 5.14 8.7 Equations Reducible to Homogeneous
Form 8.11
8.8 Linear Differential Equations 8.13
6
Beta and Gamma Functions 6.1
8.9 Equations Reducible to Linear Differential
6.1 Beta Function 6.1 Equations 8.14
6.2 Properties of Beta Function 6.1 8.10 Exact Differential Equation 8.15
6.3 Gamma Function 6.5 8.11 The Solution of Exact Differential
6.4 Properties of Gamma Function 6.5 Equation 8.17
6.5 Relation Between Beta and Gamma 8.12 Equations Reducible to Exact
Functions 6.6 Equation 8.19
6.6 Dirichlet’s and Liouville’s 8.13 Applications of First Order and First
Theorems 6.12 Degree Equations 8.26
6.7 Miscellaneous Examples 6.14 8.14 Linear Differential Equations 8.39
8.15 Solution of Homogeneous Linear
Exercises 6.15
Differential Equation with Constant
Coeffi cients 8.41
7 Multiple Integrals 7.1 8.16 Complete Solution of Linear
Differential Equation with Constant
7.1 Double Integrals 7.1
Coeffi cients 8.44
7.2 Properties of a Double Integral 7.2
8.17 Method of Variation of Parameters
7.3 Evaluation of Double Integrals
to Find Particular Integral 8.53
(Cartesian Coordinates) 7.2
8.18 Differential Equations with Variable
7.4 Evaluation of Double Integrals
Coeffi cients 8.56
(Polar Coordinates) 7.7
8.19 Miscellaneous Examples 8.68
7.5 Change of Variables in a Double
Integral 7.9 Exercises 8.80
7.6 Change of Order of Integration 7.13
7.7 Area Enclosed by Plane Curves Solved Question Papers Q.1
(Cartesian and Polar Coordinates) 7.19 Index I.1
Preface
All branches of engineering, technology and science require mathematics as a tool for the description
of their contents. Therefore, a thorough knowledge of the various topics in mathematics is essential
to pursue courses in these fi elds. The aim of this book is to provide students with a sound platform to
hone their skills in mathematics and its multifarious applications.
This edition has been prepared in accordance with the syllabus requirements of Engineering
M athematics-I, a compulsory paper taught during the fi rst semester in Rajasthan Technical University.
A roadmap to the syllabus has been included for the benefi t of students. Designed for classroom and
self-study sessions, the book uses simple and lucid language to explain concepts.
Several solved examples, fi gures, tables and exercises have been provided to enable students to
e nhance their problem-solving skills. Three solved university question papers have been appended
to the book for the benefi t of the students. Suggestions and feedback for improving the book further
are welcome.
Acknowledgements
My family members provided moral support during the preparation of this book. My son, Aman Kumar,
software engineer, Goldman Sachs, offered wise comments on some of the contents of the book. I am
thankful to Sushma S. Pradeep for excellently typing the manuscript.
Special thanks are due to Thomas Mathew Rajesh, Anita Yadav, Ravi Vishwakarma, M. E. Sethurajan,
M. Balakrishnan and Insiya Poonawala at Pearson Education for their constructive support.
BABU RAM