A TEXTBOOK OF

ENGINEERING MATHEMATICS

A TEXTBOOK OF

ENGINEERING

MATHEMATICSFor

B. Tech. I semester

(Common to All Branches)(According to the Latest Syllabus Prescribed by APJ Abdul Kalam

Kerala Technological University, Kerala)

By

N.P. Bali Dr. Remadevi.SFormerly, Principal Prof. and Head of Department of Mathematics

S.B. College, Gurugram Model Engineering College, CochinHaryana Kerala

UNIVERSITY SCIENCE PRESS(An Imprint of Laxmi Publications Pvt. Ltd.)

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A TEXTBOOK OF ENGINEERING MATHEMATICS

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First Edition : 2016ISBN 978-93-85750-42-7

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CONTENTSPreface ... (vi)Syllabus ... (vii)–(viii)Chapters Pages

MODULE–I

(Single Variable Calculus and Infinite Series)(15 Marks)

1. Hyperbolic Functions ... 3

2. Infinite Series ... 9

3. Power Series ... 44

MODULE–II

(Three Dimensional Space and Functions of More Than One Variable)(15 Marks)

4. Three Dimensional Space ... 85

5. Functions of Two or More Variables ... 95

MODULE–III

(Partial Derivatives and its Applications)(15 Marks)

6. Partial Differentiation and Applications ... 101

MODULE–IV

(Calculus of Vector Valued Functions)(15 Marks)

7. Calculus of Vector Valued Functions ... 145

MODULE–V

(Multiple Integrals)(20 Marks)

8. Multiple Integrals ... 173

MODULE–VI

(Vector Integration)(20 Marks)

9. Vector Integration ... 213

Appendix–I: List of Important Formulae ... 257Appendix–II: Some Important Curves ... 266

( v )

PREFACEThis book is a modified form of the book ‘‘A Textbook of Engineering Mathematics’’,

First Year for Cochin University of Science and Technology which is a part of the original book‘‘A Textbook of Engineering Mathematics’’ (with 28 chapters and covering the syllabi ofEngineering courses of all semesters of all the Indian universities) running its eighth editionand very well received by the students and teachers of all Indian Universities.

The present form of the book is divided into 6 modules containing 9 chapters and coversentire portion according to latest syllabus prescribed by Kerala Technological University (KTU)for the B Tech First semester MA 101 Calculus paper.

There is no dearth of books on Engineering Mathematics but the students find it difficultto solve most of the problems in the exercise in the absence of adequate number of solvedexamples. Anoutstanding and distinguishing feature of the book is the large number of typicalsolved examples followed by well-graded problems.

We have endeavoured to present the fundamental concepts in a comprehensive andlucid manner. We are indebted to all authors, Indian and Foreign, whose works we havefrequently consulted.

All efforts have been made to keep the book free from errors. Answers to all exerciseshave been checked. All suggestions for improvement will be highly appreciated and gratefullyacknowledged.

—Authors

( vi )

SYLLABUSFor B. Tech. Semester I

Kerala Technological University (KTU)

Course No. MA 101L.T.P. Credits3-1-0-4

Course ObjectivesIn this course the students are introduced to some basic tools in Mathematics which areuseful in modelling and analysing physical phenomena involving continuous changes ofvariables or parameters. The differential and integral calculus of functions of one ormore variables and of vector functions taught in this course have applications across allbranches of engineering. This course will also provide basic training in plotting andvisualising graphs of functions and intuitively understanding their properties usingappropriate software packages.

Module I: Single Variable Calculus and Infinite SeriesIntroduction: Hyperbolic functions and inverses–derivatives and integrals.Basic ideas of infinite series and convergence. Convergence tests-comparison, ratio,root tests (without proof). Absolute convergence. Maclaurins series–Taylor series–radiusof convergence.

Module II: Three Dimensional Space and Functions of More than One VariableThree dimensional space: Quadric surfaces, Rectangular, Cylindrical and sphericalcoordinates, Relation between coordinate systems.Equation of surfaces in cylindrical and spherical coordinate systems.Functions of two or more variables: Graphs of functions of two variables–level curvesand surfaces–Limits and continuity.

Module III: Partial Derivatives and its ApplicationsPartial derivatives–Partial derivatives of functions of more than two variables-higherorder partial derivatives–differentiability, differentials and local linearity.The chain rule–Maxima and Minima of functions of two variables–extreme value theorem(without proof)–relative extrema.

Module IV: Calculus of Vector Valued FunctionsIntroduction to vector valued functions–parametric curves in 3-space. Limits andcontinuity–derivatives–tangent lines–derivative of dot and cross product–definiteintegrals of vector valued functions.Change of parameter–arc length–unit tangent–normal–velocity–acceleration and speed-Normal and tangential components of acceleration.Directional derivatives and gradients–tangent planes and normal vectors.

( vii )

Module V: Multiple IntegralsDouble integrals–Evaluation of double integrals–Double integrals in non-rectangularcoordinates–reversing the order of integration.Area calculated as double integral–Double integrals in polar coordinates.Triple integrals–volume calculated as a triple integral–triple integrals in cylindricaland spherical coordinates. Converting triple integrals from rectangular to cylindricalcoordinates–converting triple integrals from rectangular to spherical coordinates–changeof variables in multiple integrals–Jacobians (applications of results only).

Module VI: Vector IntegrationVector and scalar fields–Gradient fields–conservative fields and potential functions–divergence and curl–the operator–the Laplacian 2.Line integrals–work as a line integral–independence of path–conservative vector field.Green’s Theorem (without proof–only for simply connected region in plane), surfaceintegrals, Flux integral–Divergence Theorem (without proof), Stokes’ Theorem (withoutproof).

( viii )

MODULE–I

Single Variable Calculus

and Infinite Series(15 Marks)

3

1Hyperbolic Functions

1.1. HYPERBOLIC FUNCTIONS

Definition. The hyperbolic functions have similar names to the trigonometric functions,but they are defined in terms of the exponential functions. Hyperbolic sine and cosine functionsare defined as

sinh x =

2

x xe e

cosh x =

2

x xe e

tanh x =sinhcosh

xx

=

x x

x x

e e

e e

coth x =1

tanh x =

x x

x x

e e

e e

sech x =1

cosh x =

2

x xe e

cosech x =1

sinh x =

2x xe e

3

2

1

–1 1

2

1

–1 1

–1

–2

cosh x sinh x tanh x

2

1

–1 1

–1

–2

sech x

2

1

–1 1

–1

–2cosech x

2

1

–1 1

–1

–2coth x

2

1

–1 1

–1

–2

Hyperbolic functions

4 A TEXTBOOK OF ENGINEERING MATHEMATICS

1.2. RELATION CONNECTING TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS IS

Results: (i) sin ix = i sinh x (ii) cos ix = cosh x

(iii) tan ix = i tanh x

Proof. (i) We have sin =

2

i ie ei

Put = ix

sin ix = ( ) ( )

2

i ix i ixe ei

= ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦(– )

2 2

x x x xe e e ei i = i sinh x

(ii) cos =

2

i ie e

Put = ix, cos ix = 2 2( )

cosh2 2

i x i x x xe e e e x

(iii) tan =

sincos

Put = ix, tan ix = sin sinhcos cosh

ix i xix x

= i tanh x. [Using results (i) and (ii)]

Notes:

(i) sinh x and cosh x are periodic functions of period 2i.

(ii) cosh x is an even function.

(iii) sinh x is an odd function.

(iv) sinh 0 = 0, cosh 0 = 1, tanh 0 = 0

(v) The range of sinh x is ( – , )

(vi) The range of cosh x is (1, ).

1.3. FUNDAMENTAL FORMULAS

(i) cosh2 x – sinh2 x = 1

Proof. We have sin2 + cos2 = 1

Put = ix, (sin ix)2 + (cos ix)2 = 1

(i sinh x)2 + (cosh x)2 = 1

– sinh2 x + cosh2 x = 1 cosh2 x – sinh2 x = 1

(ii) sech2 x + tanh2 x = 1

(iii) coth2 x – cosech2 x = 1

(iv) sinh (x ± y) = sinh x cosh y ± cosh x sinh y

(v) tanh 3x =

3

3

3 tanh tanh

1 3 tanh

x x

x

HYPERBOLIC FUNCTIONS 5

Proof. Same as (i)First write down the respective result in trigonometry in , then put = ix and apply

results in 1.2.

Inverse Hyperbolic Functions

If sinh u = x, then u is called the hyperbolic sine inverse of x and is written asu = sinh–1 x

Similarly we can define cosh–1 x, tanh–1 x etc.

ILLUSTRATIVE EXAMPLES

Example 1. Prove that sinh–1 x = log (x + 2x + 1 ).

Sol. Let u = sinh–1 x.

x = sinh u =

2

u ue e

2x = eu – 1ue

= 2 1u

u

e

e e2u – 2xeu – 1 = 0

Treating this equation as a quadratic equation in eu, we get

eu = 22 4 4

2x x

= x ± 2 1x

Taking the positive sign

eu = x + 2 1x u = log (x + 2 1x )

sinh–1 x = log (x + 2 1x ).

Example 2. Prove that tanh–1 x = 12

log ⎛ ⎞⎜ ⎟⎝ ⎠

1 + x1 x

.

Sol. Let u = tanh–1 x

x = tanh u =

u u

u u

e ee e

.

Using Componendo-Dividendo rule,

1 +1

xx

=

2

2

u

u

e

e = e2u

u =12

log ⎛ ⎞⎜ ⎟⎝ ⎠

1 +1

xx

.

A Textbook Of EngineeringMathematics Sem-I (Kerla

Technological University) Kerla

Publisher : Laxmi Publications ISBN : 9789385750427Author : N. P. Bali, DrRemadevi

Type the URL : http://www.kopykitab.com/product/12185

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