ENGINEERING MATHEMATICS–I

ENGINEERINGENGINEERINGENGINEERINGENGINEERINGENGINEERING M M M M MAAAAATHEMATHEMATHEMATHEMATHEMATICSTICSTICSTICSTICS–I–I–I–I–I

SUBJECT REVIEWERS

I would like to thank all those reviewers who took out time to review the manuscript andgave important suggestions. Their names are given below.

Dr. Abdul Majeed

Dr. Loka Pavani Osmania University College for Women, Hyderabad

Dr. Gitti Narsimlu Chaitanya Bharathi Institute of Tech., Hyderabad

C. Naga Anuradha Vasavi College of Engineering, Hyderabad

Dr. K. Ramesh Babu MVSR Engineering College, Hyderabad

Dr. G. Krishna Kumari Vidya Jyothi Institute of Tech, Hyderabad

Saroj M Revankar Stanley Engg College for Womens, Hyderabad

D. Swamy Methodist College of Engg and Technology, Hyderabad

Naveen Voruganti Methodist College of Engg and Technology, Hyderabad

M.A. Rawoof Sayeed Muffakham Jah College of Engg and Tech, Hyderabad

Syed Nisar Ahmed Deccan College of Engg and Tech, Hyderabad

Dr. T. Nagaiah Kakatiya University, Warangal

By

ENGINEERING

MATHEMATICS–I

Dr. Abdul Majeed N.P. BaliHead Dept. of Mathematics Former Principal

Muffakham Jah College of S.B. College, Gurugram

Engineering and Technology, Haryana

Banjara Hills, Hyderabad

Telangana

ForB.E., 1st Year/1st semester

Strictly according to the latest revised syllabi 2016FOR OSMANIA UNIVERSITY, HYDERABAD (TELANGANA)

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BOSTON (USA) ● ACCRA (GHANA) ● NAIROBI (KENYA)

ENGINEERING MATHEMATICS-I

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UNIT 1. LINEAR ALGEBRA .....................................................................1–71

1.1. Introduction ............................................................................................................................................... 11.2. Definitions .................................................................................................................................................. 11.3. Minors and Co-factors .............................................................................................................................. 21.4. Expansion of a Determinant ................................................................................................................... 31.5. Properties of Determinants ..................................................................................................................... 41.6. Definitions .................................................................................................................................................. 51.7. Matrix Multiplication ............................................................................................................................... 81.8. Properties of Matrix Multiplication ..................................................................................................... 101.9. Transpose of a Matrix ............................................................................................................................. 101.10. Properties of Transpose of a Matrix .................................................................................................... 111.11. Symmetric Matrix ................................................................................................................................... 111.12. Skew-symmetric Matrix (or Anti-symmetric Matrix) ...................................................................... 111.13. Orthogonal Matrix .................................................................................................................................. 111.14. Adjoint of a Square Matrix .................................................................................................................... 111.15. Singular and Non-singular Matrices ................................................................................................... 121.16. Inverse (or Reciprocal) of a Square Matrix ......................................................................................... 121.17. Elementary Transformations (or Operations) ................................................................................... 131.18. Elementary Matrices ............................................................................................................................... 131.19. Rank of a Matrix ...................................................................................................................................... 141.20. Method 2: Echelon Form or Triangular Form.................................................................................... 141.21. Method 3: Normal Form or Canonical Form (This Topic is for Student Reference Only) ........ 191.22. Solution of a System of Linear Equations ........................................................................................... 281.23. Characteristic Equation .......................................................................................................................... 391.24. Eigen Values or Characteristic Roots or Latent Roots ...................................................................... 391.25. Eigen Vectors or Characteristic Vectors or Latent Vectors .............................................................. 391.26. Cayley-Hamilton Theorem ................................................................................................................... 481.27. Reduction of a Matrix to Diagonal Form ............................................................................................ 541.28. Quadratic Forms ..................................................................................................................................... 611.29. Linear Transformation of a Quadratic Form ..................................................................................... 621.30. Canonical Form ....................................................................................................................................... 62

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CCCCCONTENTONTENTONTENTONTENTONTENTSSSSS

1.31. Index and Signature of the Quadratic Form ...................................................................................... 631.32. Definite, Semi-definite and Indefinite Real Quadratic Forms ........................................................ 631.33. Reduction to Canonical Form by Orthogonal Transformation ...................................................... 66

UNIT 2. INFINITE SERIES ...................................................................72–128

2.1. Introduction ............................................................................................................................................. 722.2. Sequence ................................................................................................................................................... 722.3. Range of a Sequence ............................................................................................................................... 722.4. Constant Sequence .................................................................................................................................. 732.5. Bounded and Unbounded Sequences ................................................................................................. 732.6. Convergent, Divergent and Oscillating Sequences .......................................................................... 732.7. Monotonic Sequences ............................................................................................................................. 742.8. Limit of a Sequence ................................................................................................................................. 752.9. Every Convergent Sequence is Bounded ............................................................................................ 752.10. Convergence of Monotonic Sequences ............................................................................................... 752.11. Infinite Series ........................................................................................................................................... 762.12. Series of Positive Terms ......................................................................................................................... 762.13. Alternating Series .................................................................................................................................... 762.14. Partial Sums ............................................................................................................................................. 762.15. Convergent, Divergent and Oscillating Series .................................................................................. 762.16. General Properties of a Series ............................................................................................................... 792.17. Absolute/Conditional Convergence of a Series .............................................................................. 1222.18. Working Rule for Testing a Series for Convergence ...................................................................... 128

UNIT 3. DIFFERENTIAL CALCULUS ................................................... 129–189

3.1. Introduction ........................................................................................................................................... 1293.2. Mean Value Theorems ......................................................................................................................... 1303.3. Curvature ............................................................................................................................................... 1543.4. Radius of Curvature ............................................................................................................................. 1543.5. Centre of Curvature .............................................................................................................................. 1643.6. Circle of Curvature ............................................................................................................................... 1643.7. Evolute .................................................................................................................................................... 1643.8. Envelope ................................................................................................................................................. 1693.9. Curve Tracing ........................................................................................................................................ 177

UNIT 4. FUNCTIONS OF SEVERAL VARIABLES ...................................... 190–262

4.1. Functions of Two Variables ................................................................................................................ 1904.2. Limit ........................................................................................................................................................ 1904.3. Continuity .............................................................................................................................................. 1934.4. Partial Derivatives of First Order ....................................................................................................... 1954.5. Partial Derivatives of Higher Order .................................................................................................. 1954.6. Composite Functions ............................................................................................................................ 2064.7. Total Derivative ..................................................................................................................................... 207

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4.8. Jacobians ................................................................................................................................................. 2154.9. Properties of Jacobians ......................................................................................................................... 2164.10. Taylor’s Theorem for a Function of Two Variables ........................................................................ 2204.11. Maxima and Minima of Functions of Two Variables ..................................................................... 2254.12. Conditions for f(x, y) to be Maximum or Minimum ....................................................................... 2274.13. Rule to Find the Extreme Values of a Function z = f(x, y) ............................................................. 2274.14. Double Integrals (This Topic is for Student Reference Only) ....................................................... 2384.15. Evaluation of Double Integrals (This Topic is for Student Reference Only) .............................. 2394.16. Evaluation of Double Integrals in Polar Co-ordinates

(This Topic is for Student Reference Only) ...................................................................................... 2464.17. Change of Order of Integration (This Topic is for Student Reference Only) ............................. 2494.18. Triple Integrals (This Topic is for Student Reference Only) ......................................................... 2524.19. Change of Variables .............................................................................................................................. 255

UNIT 5. VECTOR CALCULUS ............................................................ 263–333

5.1. Introduction ........................................................................................................................................... 2635.2. Types of Vectors .................................................................................................................................... 2635.3. Scalar Product (or Dot Product) of Two Vectors ............................................................................. 2645.4. Properties of Scalar Product of Two Vectors ................................................................................... 2645.5. Vector Product (or Cross Product) of Two Vectors ........................................................................ 2655.6. Properties of Vector Product of Two Vectors .................................................................................. 2665.7. Vector Functions ................................................................................................................................... 2665.8. Derivative of a Vector Function with Respect to a Scalar ............................................................. 2665.9. General Rules for Differentiation ....................................................................................................... 2675.10. Scalar and Vector Fields ...................................................................................................................... 2685.11. Gradient of a Scalar Field .................................................................................................................... 2695.12. Geometrical Interpretation of Gradient ............................................................................................ 2695.13. Directional Derivative .......................................................................................................................... 2705.14. Properties of Gradient .......................................................................................................................... 2705.15. Divergence of a Vector Point Function ............................................................................................. 2785.16. Curl of a Vector Point Function .......................................................................................................... 2785.17. Physical Interpretation of Divergence ............................................................................................... 2805.18. Physical Interpretation of Curl ........................................................................................................... 2815.19. Properties of Divergence and Curl .................................................................................................... 2815.20. Repeated Operations by ................................................................................................................... 2845.21. Line Integrals ......................................................................................................................................... 2945.22. Circulation .............................................................................................................................................. 2955.23. Work Done by a Force .......................................................................................................................... 2955.24. Surface Integrals .................................................................................................................................... 3005.25. Volume Integrals ................................................................................................................................... 3015.26. Divergence Theorem of Gauss (Relation between Surface and Volume Integrals) .................. 3085.27. Green’s Theorem in the Plane ............................................................................................................. 3185.28. Stoke’s Theorem (Relation between Line and Surface Integrals) ................................................. 324

EXAMINATION PAPERS .......................................................... 334–341

INDEX ............................................................................... 343–344

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AAAAACKNOCKNOCKNOCKNOCKNOWLEDGEMENTWLEDGEMENTWLEDGEMENTWLEDGEMENTWLEDGEMENTSSSSS

We are indebted to many of our former teachers and senior professors who have directly orindirectly helped us in preparing this book.

We have great pleasure to express our sincere thanks and deep sense of gratitude to Prof.T. Srinivas, Prof. Shabbir Ahmed and Prof. M.H. Muddebihal for their valuable suggestions andencouragement.

Our special and cordial thanks go to the highly esteemed Chairman, Hon’ble Secretary,Governing council members and Board of Governors of Muffakham Jah College of Engineeringand Technology, Hyderabad for their cooperation and motivation in preparing this book. We arevery much grateful to Dr. Basheer Ahmed, Advisor-cum-Director, MJCET for his kindencouragement.

We also thank the Principal, Dean (Academics), Dean (Administration), Heads of thedepartment and colleagues of MJCET for their great help and effort during the periods of preparingthe manuscript and publishing the book.

Lastly, we offer our love and gratitude to our beloved parents, family members for theircontinued encouragement and inspiration without which this book could not have taken the shape.

—Authors

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PPPPPREFREFREFREFREFAAAAACECECECECE TTTTTOOOOO THETHETHETHETHE S S S S SECONDECONDECONDECONDECOND E E E E EDITIONDITIONDITIONDITIONDITION

There is no dearth of books on Engineering Mathematics–I. The students find it difficult tosolve most of the problems in the exercises in the absence of an adequate number of solved examples.This inspired us to write the book Engineering Mathematics–I. The contents of the book are basedon revised syllabus prescribed by Osmania University. An outstanding and distinguishing featureof the book is the large number of workedout examples followed by well graded problems.Numerous examples and problems have been given from previous question papers of OsmaniaUniversity.

The selection, arrangement, and presentation of the text have been made with the greatestcare, based on past and present teaching experience. The presentation is detailed, to help readersavoid frequent references to other books. We have endeavoured to present the fundamental conceptsin a comprehensive and lucid manner.

Intimations of errors and suggestions for improvement will be highly appreciated andgratefully acknowledged.

—Authors

PPPPPREFREFREFREFREFAAAAACECECECECE TTTTTOOOOO THETHETHETHETHE F F F F FIRSTIRSTIRSTIRSTIRST E E E E EDITIONDITIONDITIONDITIONDITION

There is no dearth of books on Engineering Mathematics. The students find difficulty tosolve most of the problems in the exercises in the absence of an adequate number of solved examples.This inspired us to write this book Engineering Mathematics-I . The contents of the book is basedon revised syllabus prescribed by Osmania University. An outstanding and distinguishing featureof the book is the large number of typical solved examples followed by well graded problems.Numerous of examples and problems have been given from previous question papers of OsmaniaUniversity.

The selection, arrangement, and presentation of the text has been made with the greatestcare, based on past and present teaching experience. The presentation is detailed, to avoid readersby frequent references to details in other books. The examples are simple.

We have endeavoured to present the fundamental concepts in a comprehensive and lucidmanner.

Intimations of errors and suggestions for improvement will be highly appreciated andgratefully acknowledged.

—Authors

SSSSSYLLABUSYLLABUSYLLABUSYLLABUSYLLABUS

ENGINEERING MATHEMATICS–I

(COMMON TO ALL BRANCHES)

UNIT-I: Linear Algebra

Introduction to matrices, Elementary row and column operations, Rank of a Matrix, Echelonform, System of linear equations, Eigenvalues, Eigen vectors, Cayley-Hamilton theorem,Diagonalization, Quadratic forms, Signature and Index.

UNIT-II: Infinite Series

Sequences, Infinite series, Convergence and Divergence, P-Series test, Geometric series test,Comparison tests, D’Alembert’s Ratio test, Raabe’s test, Cauchy’s nth root test, Alternating series,Leibnitz’s test, Absolute convergence, Conditional convergence.

UNIT-III: Differential Calculus

Rolle’s theorem, Lagrange’s and Cauchy’s mean value theorems, Taylor’s series, Curvature,Radius of curvature, Envelopes, Evolutes and Involutes, Asymptotes of a curve, Curve sketching(Cartesian).

UNIT-IV: Functions of Several Variables

Limits and Continuity of Functions of two variables, Partial derivatives, Total differentialsand derivatives, Derivatives of composite and implicit functions, Higher order partial derivatives,Taylor’s theorem for functions of two variables, Maxima and minima of functions of two variables,Jacobian, Change of variables.

UNIT-V: Vector Calculus

Scalar and vector fields, Vector differentiation, Gradient of a scalar field, Directionalderivative, Divergence and Curl of a vector field, Line, Surface and Volume integrals, Green’stheorem in a plane, Gauss’s divergence theorem, Stoke’s theorem and their applications.

(x)

UNIT 1Linear Algebra

1.1. INTRODUCTION

Linear algebra comprises of the theory and applications of linear system of equations,linear transformations, eigen values and eigen vector problems. Determinants were firstintroduced for solving linear system of equations and have important engineering applicationsin systems of differential equations, electrical networks, eigen value problems and many more.Many complicated expressions occurring in electrical and mechanical systems can be simplifiedby expressing them in the form of determinants.

Matrices originated as mere stores of information but, at present, have found very wideapplication. They play a very vital role not only in mathematics but also in communicationtheory, network analysis, theory of structures, quantum mechanics, biology, sociology,economics, psychology, statistics etc.

Determinants and matrices play the key role in providing suitable criteria for testingthe consistency of a system of linear equations.

In this unit, we first deal with determinants and then matrices.

DETERMINANTS

1.2. DEFINITIONS

Let us eliminate x and y from the equations

a1x + b1y = 0 and a2x + b2y = 0

The eliminant is ab

ab

1

1

2

2 each F

HGIKJ

yx

or a1b2 – a2b1 = 0

which is conveniently written in the compact form a ba b

1 1

2 2 = 0

In other words, we have a ba b

1 1

2 2 = a1b2 – a2b1

The expression a ba b

1 1

2 2 is called a determinant of the second order. The numbers

a1, b1, a2, b2 are called the elements of the determinant. a1b2 – a2b1 is called the expansionor the value of the determinant.

1

2 ENGINEERING MATHEMATICS–I

We have

a b ca b ca b c

1 1 1

2 2 2

3 3 3

= a1(b2c3 – b3c2) + b1(c2a3 – c3a2) + c1(a2b3 – a3b2) ...(1)

The expression a b ca b ca b c

1 1 1

2 2 2

3 3 3

is called a determinant of the third order and the RHS

of (1) is called the expansion or the value of the determinant.A determinant of the nth order is denoted by

=

a a a aa a a a

a a a a

n

n

n n n nn

11 12 13 1

21 22 23 2

1 2 3

......

........................................................................

......

The elements a11, a22, a33, ......, ann (i.e., the elements aij with i = j) are called thediagonal elements. The diagonal through the left hand top corner along which the diagonalelements lie is called the leading or principal diagonal. Also a11, a22, a33, ......, ann iscalled the leading term.

1.3. MINORS AND CO-FACTORS

The minor of an element in a determinant is the determinant obtained by omittingthe row and the column in which that element lies.

For example, consider the determinant, = a b ca b ca b c

1 1 1

2 2 2

3 3 3

To find the minor of c2 which occurs in the second row and third column, we omit thesecond row and the third column.

Thus, minor of c2 = a ba b

1 1

3 3

Similarly, minor of a3 = b cb c

1 1

2 2

The co-factor of an element in a determinant is its minor with proper sign and isusually denoted by the corresponding capital letter. The co-factor of the element which lies inthe ith row and jth column is (– 1)i+j times the minor of the element.

Thus, the co-factor of c2 = (– 1)2+3 × minor of c2 i.e., C2 = – a ba b

1 1

3 3

Similarly, A3 = co-factor of a3 = (– 1)3+1 × minor of a3 = b cb c

1 1

2 2 .

LINEAR ALGEBRA 3

1.4. EXPANSION OF A DETERMINANT

(a) For a determinant of second order, we have seen that a

1

a2

b1

b2

= a1b2 – a2b1

Sign of a product remains unchanged with downward arrow while it changes with upwardarrow.

Thus,

2 53 4 = (– 2)(4) – (– 3)(5) = 7

(b) For a determinant of third order, we have seen that

a b ca b ca b c

1 1 1

2 2 2

3 3 3

= a1(b2c3 – b3c2) + b1(c2a3 – c3a2) + c1(a2b3 – a3b2)

= a1 b cb c

2 2

3 3 – b1

a ca c

2 2

3 3 + c1

a ba b

2 2

3 3 = a1A1 + b1B1 + c1C1

= the sum of the products of elements in the first row withtheir respective co-factors.

In fact, there is nothing particular about the first row. We can expand a determinant interms of any row or column as follows :

Multiply the elements of any row (or column) with their respective co-factors and addup.

Thus, =

a b ca b ca b c

1 1 1

2 2 2

3 3 3

= a1A1 + b1B1 + c1C1 (expanding by first row)= a2A2 + b2B2 + c2C2 (expanding by second row)= a3A3 + b3B3 + c3C3 (expanding by third row)= a1A1 + a2A2 + a3A3 (expanding by first column)= b1B1 + b2B2 + b3B3 (expanding by second column)= c1C1 + c2C2 + c3C3 (expanding by third column)

These expansions are called Laplace’s expansions and the technique can be adoptedfor determinants of any order.

However, the sum of the products of the elements of any row (or column) by the co-factors of any other row (or column) is zero.

Thus, a1A3 + b1B3 + c1C3 = a1 b cb c

1 1

2 2 – b1

a ca c

1 1

2 2 + c1

a ba b

1 1

2 2

= a1(b1c2 – b2c1) – b1(a1c2 – a2c1) + c1(a1b2 – a2b1)= a1b1c2 – a1b2c1 – a1b1c2 + a2b1c1 + a1b2c1 – a2b1c1 = 0

In general, aiAj + biBj + ciCj = if

ifi ji j

RST0

Engineering Mathematics I OsmaniaUniversity

Publisher : Laxmi Publications ISBN : 9789351382072Author : Dr Abdul Majeed,N.P.Bali

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