Engineering Mathematics - 1

Dr. Pandurangappa C.Professor

Department of MathematicsUniversity B.D.T. College of Engineering

(A Constituent College of Visvesvaraya Technological University)Davangere - 577 004

Karnataka.

SANGUINE

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Sanguine Technical PublishersBangalore

2014

EngineeringMathematics - 1

Second Edition

Engineering Mathematics - 1, 2nd Edition

Dr. Pandurangappa. C.

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Preface

This is a textbook for Engineering Mathematics–I (2nd edition) for Engineering Students ofVisveswaraya Technological University, introduced from the year 2014–2015. The topicsincluded are Differential Calculus, Vector Calculus, Integral Calculus, Differential Equationsand Linear Algebra.

Each topic contains several solved problems to elucidate the concepts. Some of these exampleshave appeared in the various University examinations. The problems are solved in simple stepsas can be understood by students. Assignment problems with answers appear at the end ofeach topic. Both theory and problems have been explained using necessary diagrams.

Each topic is presented with care and it is hoped that the book will meet the requirements ofthe students and the teachers.

Suggestions for further improvement of the book are welcome.

I am indebted to my professors, colleagues and students, who have directly or indirectlycontributed to this work. I express my thanks with gratitude to Dr. Venugopal K. R,UVCE, Dr. T. Basavaraju, Dr. Shivaprasad. B. Dandagi, UBDTCE , Dr. Ejanul Haque,Dr. Lakshmi Narayanachari. K, SVIT, Dr. M. Sankar, EPCET, Dr. D.V. Chandrashekar,VKIT, Dr. Hariprasad SSSMCE, Dr. C. Vinay, JSSATE, Dr. G. S. Prasad, RRCE,Dr. M. Suresha, VIT, Dr. Siddalinga Prasad, SIT, Mr. Krishna, Sri K. G. Jagadeesh, Dr.K. M. Niranjan, Dr. Ravichandra Nayakar, Dr. Mahesh, Smt. Mamatha, UBDTCE for theiradvice and help while preparing this book.

I also express my sincere thanks to my wife Dr. Bharathi. K and my children Rithvik. P.R andManasa. P.R for their continuous support and encouragement.

I express my sincere thanks to Sanguine Technical Publishers, Bangalore for publishing thebook in such an excellent form, and Interline Publishing, Bangalore for online publication ofthis book.

I also express my sincere thanks to Mr. Sathish for keen interesting in typesetting the book.

September 2014 AUTHOR

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Foreword

It gives me great pleasure in writing a foreword to this book on Engineering Mathematics byDr. Pandurangappa C. I know the author for over two decades as my student and has earneda good reputation as a teacher.

The results and properties on specific topics have been stated in the form of theorems andfollowed by the proofs. This really helps the students to understand the meaning of the resultsand properties. A large number of solved examples on each topic are also given. In eachexercise, a large number of graded problems are given for rigorous practice.

In addition, this book also contains the required preliminaries that the students have studiedearlier. This helps the students to brush up the requisite preliminaries and glance through thecurrent topics for the examination purposes.

I am confident that the above book will be of great help to the teachers also in selecting thegraded problems from worked out examples and exercises for effective teaching.

Dr. M. Venkatachalappa

Professor (Retired),Department of Mathematics,Bangalore University,Bangalore.

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Contents

Prelimenaries xiii

1 Differential Calculus-I 1

1.1 Introduction 1

1.2 Successive Differentiation 1

1.2.1 nth Derivatives of Some Standard Functions 2

1.2.2 Leibnitz’s Theorem 31

1.3 Polar Curves 49

1.3.1 Angle Between Radius Vector and the Tangent to the Polar Curve 50

1.3.2 Angle Between two Polar Curves 56

1.3.3 Length of the Perpendicular from the Pole to Tangent 66

1.3.4 Pedal Equation (p-r equation) 67

1.4 Derivatives of Arc Length 75

1.4.1 Derivative of arc length in Cartesian form 76

1.4.2 Derivative of arc length in parametric form 77

1.4.3 Derivative of arc length in polar form 78

1.5 Curvature 89

1.5.1 Radius of Curvature 89

1.5.2 Radius of curvature in Cartesian form 90

1.5.3 Radius of curvature in parametric form 91

1.5.4 Radius of curvature in Pedal form 92

1.5.5 Radius of curvature in polar form 93

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2 Differential Calculus-II 123

2.1 Introduction 123

2.2 Taylor’s Theorem for a Function of Single Variable 123

2.3 Maclaurin’s series 125

2.4 Indeterminate Forms 144

2.4.1 L’Hospital’s Rule 145

2.5 Partial Differentiation 178

2.5.1 Homogeneous Functions 198

2.5.2 Total Differential and Total Derivative 213

2.5.3 Differentiation of implicit functions 214

2.5.4 Differentiation of Composite functions 214

2.5.5 Jacobians 230

2.6 Maxima and Minima 249

3 Vector Calculus 259

3.1 Introduction 259

3.2 Vector Differentiation 259

3.2.1 Tangent Vector to the Space Curve 263

3.2.2 Normal Vector to the Space Curve 263

3.2.3 Application to Space Curve 263

3.2.4 Tangential and Normal Components of Acceleration 264

3.3 The vector differential operator ‘Del’ or ‘Nabla’ 276

3.3.1 Gradient of a Scalar Function 276

3.3.2 Divergence of a Vector Function 278

3.3.3 Curl of a Vector Function 278

3.4 Vector Identities 323

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Contents | xi

3.5 Leibnitz rule for differentiation under the integral sign 331

3.6 Curve Tracing 345

3.6.1 Procedure for tracing the Cartesian curve 346

3.6.2 Procedure for tracing curves in parametric form x = f (t), y = g(t) 355

3.6.3 Procedure for tracing the polar curve 356

4 Integral Calculus 365

4.1 Introduction 365

4.2 Reduction Formulae 365

4.2.1 Reduction Formula for∫

sinn xdx 365

4.2.2 Reduction Formula for∫

cosn xdx 367

4.2.3 Reduction Formula for∫

sinm x cosn x dx 376

5 Differential Equations 385

5.1 Introduction 385

5.2 Separation of Variables 388

5.3 Homogeneous Equations 403

5.4 Equations Reducible to Homogeneous 418

5.5 Linear Differential Equations 428

5.6 Bernoulli’s Equation 445

5.7 Exact Differential Equations 456

5.8 Equations Reducible to Exact Equations 465

5.9 Orthogonal Trajectories 474

5.9.1 Orthogonal trajectories in Cartesian form 475

5.9.2 Orthogonal trajectories in Polar form 485

5.10 Newton’s Law of Cooling 492

5.11 Flow of Electricity 498

5.12 Laws of Decay and Growth 507

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6 Linear Algebra 517

6.1 Introduction 517

6.2 Elementary Transformations of a Matrix 517

6.2.1 Echelon form 517

6.2.2 Normal form of a matrix 518

6.2.3 Rank of a matrix 518

6.3 Consistency of System of Linear Equations 528

6.4 Solution of system of linear equations Guass elimination method 529

6.4.1 Homogeneous system 529

6.4.2 Non Homogeneous System 529

6.5 Gauss – Seidel Iteration Method 538

6.6 LU-Decomposition Method 545

6.7 Linear transformations 563

6.7.1 Orthogonal Transformation 564

6.8 Similarity of matrices 572

6.9 Diagonalisation of a matrix 573

6.9.1 Calculation of powers of a matrix A 574

6.10 Rayleigh – Power Method 593

6.11 Quadratic Forms 600

6.11.1 Linear Transformation of a Quadratic Form 602

6.11.2 Canonical Form of a Quadratic Form 603

6.11.3 Orthogonal Transformation 603

6.11.4 Nature of Quadratic Forms 604

Index 617

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Prelimenaries – Short Notes

Theory of indices

i. am.an = am+n ii.am

an= am−n

iii. (am)n = amn iv. (ab)m = ambm

v.(ab

)m = am

bmvi. a−m = 1

am

vii. a0 = 1[a �= ∞] viii. n√a = a

1n

ix. If am = an and a �= 1, then m = n

Logarithms

If am = x, then mis called the logarithm of x base a, i.e., if am = x ⇒ m = loga x or

i. loga (mn) = loga m+ loga n ii. loga(mn

) = loga m− loga niii. loga (m)

n = n loga m iv. logb a = (loge a

)/(loge b

)v. loga a = 1 vi. loga 1 = 0vii. loge 0 = −∞

Progressions

Arithmetic progression (AP)

A sequence in which the difference between any two consecutive terms is always constant iscalled an Arithmetic Progression (AP).

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i.e., a, a + d, a + 2d, a + 3d, . . . is an AP.

tn = nth term = a + (n− 1) d

sn = sum of n terms = n

2{2a + (n− 1) d}

Geometric progression (GP)

A sequence in which the ratio of any two consecutive terms is always constant is called aGeometric Progression (GP).

i.e., a, ar , ar2, ar3, . . . is a GP

tn = nth term = arn−1

sn = sum of n terms = a (1 − rn)

1 − r

Mathematical induction

Using mathematical induction we can prove certain results valid for positive integral valuesof n. This process consists of three parts namely;

(i) To prove the result is true for n = 1

(ii) Assume the result is true for n = k

(iii) To prove the result is true for n = k + 1

Hence, we can say that the result is true for all the integral values of n.

Permutations and combinations

Arrangement of n different things taken r at a time is called the number of Permutations. Itis denoted and defined by

npr = n (n− 1) (n− 2) · · · (n− r + 1) = n !(n− r) !

Selection of n different things taken r at a time is called number of combinations. It is denotedand defined by

ncr = n !r ! (n− r) !

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Prelimenaries – Short Notes | xv

Binomial theorem

(a + b)n = an + nc1 an−1b + nc2a

n−2b2 + nc3an−3b3 + · · · + bn

Partial fractions

Conversion of a single proper fraction into sum of two or more proper fractions is calledPartial Fraction.

Let f (x) and g(x) be two polynomials of some degree, then the fraction f (x)/g(x) is calledrational fraction.

If the degree of f (x) is less than the degree of g(x), then the rational fraction f (x)/g(x) iscalled proper fraction, otherwise it is improper fraction.

Example

(i)x + 3

3x2 + 2x + 1proper fraction.

(ii)x2 + 2x + 1

(x − 1) (x + 2)improper fraction.

(iii)x3 + 3x + 5

x2 + x + 1improper fraction.

Rules of making the partial fractions

(i) To each non-repeated linear factors in the denominator i.e.,

f (x)

(a1x + b1) (a2x + b2) · · · (anx + bn)= A1

a1x + b1

+ A2

a2x + b2+ · · · + An

anx + bn

(ii) To each repeated linear factors in the denominator, i.e.,

f (x)

(ax + b)n= A1

ax + b+ A2

(ax + b)2+ · · · + An

(ax + b)n

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(iii) To each non-repeated quadratic factors in the denominator i.e.,

f (x)(a1x2 + b1x + c1

) (a2x2 + b2x + c2

) · · · (anx2 + bnx + cn)

= A1x + B1

a1x2 + b1x + c1+ A2x + B2

a2x2 + b2x + c2+ · · · + Anx + Bn

anx2 + bnx + cn

(iv) To each repeated quadratic factors in the denominator i.e.,

f (x)

(ax2 + bx + c)n= A1x + B1

ax2 + bx + c+ A2x + B2

(ax2 + bx + c)2+ · · · + Anx + Bn

(ax2 + bx + c)n

where A1, A2, . . . Anand B1, B2, . . . Bn are constants to be determined.

Illustrations

1.2x + 1

(x + 1) (x + 2)= A

x + 1+ B

x + 2

2.x + 1

(x + 3)2 (2x + 1)= A

x + 3+ B

(x + 3)2+ C

2x + 1

3.1

(x − 1)(x2 + 2x + 3

) = A

x − 1+ Bx + C

x2 + 2x + 3

4.1(

x2 + 1) (x2 + 2x + 5

)2 = Ax + B

x2 + 1+ Cx +D

x2 + 2x + 5+ Ex + F(

x2 + 2x + 5)2

Partial fraction can be applied only for the proper fractions. If the fraction is improper fractionwe divide and rewrite using the known concept as shown here

f (x)

g(x)= Quotient + Remainder

Divisor= Q(x)+ F(x)

g(x)

where F(x)/g(x) is proper fraction.

Trigonometry

Consider the right-angled triangle ABC of which ACB = θ . The side AB is called theopposite side, BC is the adjacent side and AC is the hypotenuse side

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Prelimenaries – Short Notes | xvii

All the six trigonometric ratios are defined as follows

sin θ = AB

OAcos θ = OB

OA

tan θ = AB

OBsec θ = OA

OB

cos ec θ = OA

ABcot θ = OB

AB

Reciprocal relations

1. sin θ = 1

cosec θ2. cos θ = 1

sec θ3. tan θ = 1

cot θ

4. tan θ = sin θ

cos θ5. cot θ = cos θ

sin θ

Identities

4. sin2 θ + cos2 θ = 1

5. 1 + tan2 θ = sec2 θ

6. 1 + cot2 θ = cosec2θ

Trigonometric ratios of standard angles

θ 0◦ 30◦ 45◦ 60◦ 90◦

sin θ 0 12

1√2

√3

2 1

cos θ 1√

32

1√2

12 0

tan θ 0 1√3

1√

3 ∞

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Note: The values of sine ratio are got as follows

(i) Write the numbers 0, 1, 2, 3, 4

(ii) Divide each by 4 0/4, 1/4, 2/4, 3/4, 4/4

(iii) Take the square root for each number√

0/4,√

1/4,√

2/4,√

3/4,√

4/4

Final values are 0, 1/2, 1/√

2,√

3/2, 1

Write these numbers for sine ratio of the angles 0◦, 30◦, 45◦, 60◦ and 90◦ respectively.Reverse them for cosine ratio and get the value of other ratios by using the following formulae.

tan θ = sin θ

cos θcot θ = cos θ

sin θ

sec θ = 1

cos θcosec θ = 1

sin θ

Allied angles

Signs of trignometrical ratios

A − S − T − C rule

A = All positive in Ist quadrant

S = sin and its reciprocal cosec are positive in IInd quadrant

T = tan and its reciprocal cot are positive in IIIrd quadrant

C = cos and its reciprocal sec are positive in IVth quadrant

Angles like nπ2 ± θ are all called allied angles to θ where n is any positive integer.

Example 90 − θ , 90 + θ , 180 − θ , 270 + θ etc

Trigonometric ratios of allied angles

Case (i): For 90 ± θ , 270 ± θ

sin → cos tan → cotcos → sin cot → tan

sec → coseccosec → sec

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Prelimenaries – Short Notes | xix

Case (ii): For 0 ± θ , 180 ± θ , 360 ± θ

sin → sin tan → tancos → cos cot → cot

sec → seccosec → cosec

Illustrations

(i) sin (90 − θ)

According to case (i) sin → cos and 90−θ is in Ist quadrant, in Ist quadrant sin is positive.sin (90 − θ) = cos θ

(ii) tan (180 − θ)

Using case (ii) tan → tan and 180 − θ is in IInd quadrant, in the IInd quadrant tan isnegative. tan (180 − θ) = − tan θ

(iii) cos (−θ) = cos θ

(iv) cos (90 + θ) = − sin θ

(v) cot (270 + θ) = − tan θ

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Compound angle formulae

1. sin (A+ B) = sinA cosB + cosA sinB

sin (A− B) = sinA cosB − cosA sinB

2. cos (A+ B) = cosA cosB − sinA sinB

cos (A− B) = cosA cosB + sinA sinB

3. tan (A+ B) = tanA+ tanB

1 − tanA tanB

tan (A− B) = tanA− tanB

1 + tanA tanB

Multiple and sub multiple angle formulae

4. sin 2A = 2 sinA cosA or sinA = 2 sin (A/2) cos (A/2)

5. cos 2A = cos2A− sin2A or cosA = cos2 (A/2)− sin2 (A/2)

= 1 − 2 sin2A = 1 − 2 sin2 (A/2)

= 2 cos2A− 1 = 2 cos2 (A/2)− 1

6. sin 3A = 3 sinA− 4 sin3A

7. cos 3A = 4 cos3A− 3 cosA

8. sin2A = 1 − cos 2A

2

9. cos2A = 1 + cos 2A

210. sin3A = 1

4 [3 sinA− sin 3A]

11. cos3A = 14 [3 cosA+ cos 3A]

Transformation formulaeConversion from product into sum or difference

12. sinA cosB = 12 [sin (A+ B)+ sin (A− B)]

13. cosA sinB = 12 [sin (A+ B)− sin (A− B)]

14. cosA cosB = 12 [cos (A+ B)+ cos (A− B)]

15. sinA sinB = 12 [cos (A− B)− cos (A+ B)]

Engineering Mathematics-1

Publisher : Sanguine Publishers ISBN : 9789383506224 Author : Dr.Pandurangappa C.

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