Engineering Mathematics, Notas de estudo de Engenharia Mecânica

Baixe Engineering Mathematics e outras Notas de estudo em PDF para Engenharia Mecânica, somente na Docsity!

Copyright © 2009, New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers

All rights reserved.

No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher. All inquiries should be emailed to [email protected]

PUBLISHING FOR ONE WORLD NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.newagepublishers.com

ISBN (13) : 978-81-224-2882-

Acknowledgement

It gives us a great pleasure to present this book ENGINEERING MATHEMATICS-II as per the latest syllabus and question pattern of VTU effective from 2008-2009. Let us take this opportunity to thank one and all who have actually given me all kinds of support directly and indirectly for bringing up my textbook.

We whole heartedly thank our Chairman Mr. S. Narasaraju Garu, Executive Director, Mr. S. Ramesh Raju Garu, Director Prof. Basavaraju, Principal Dr. T. Krishnan, HOD Dr. K. Mallikarjun, Dr. P.V.K. Perumal, Dr. M. Surekha, The Oxford College of Engineering, Bangalore. We would like to thank the other members of our Department, Prof. K. Bharathi, Prof. G. Padhmasudha, Mr. Ravikumar, Mr. Sivashankar and other staffs of The Oxford College of Engineering, Bangalore for the assistance they provided at all levels for bringing out this textbook successfully. We must acknowledge Prof. M. Govindaiah, Principal, Prof. K.V. Narayana, Reader, Department of Mathematics, Vivekananda First Grade Degree College, Bangalore are the ones who truly made a difference in our life and inspired us a lot.

We must acknowledge HOD Prof. K. Rangasamy, Mr. C. Rangaraju, Dr. S. Murthy, Department of Mathematics, Govt. Arts College (Men), Krishnagiri. We are also grateful to Dr. A.V. Satyanarayana, Vice-Principal of R.L. Jalappa Institute of Technology, Doddaballapur, Prof. A.S. Hariprasad, Sai Vidya Institute of Technology, Prof. V.K. Ravi, Mr. T. Saravanan, Bangalore college of Engineering and Technology, Prof. L. Satish, Raja Rajeshwari College of Engineering, Prof. M.R. Ramesh, S.S.E.T., Bangalore. A very special thanks goes out to Mr. K.R. Venkataraj and Bros., our well wisher friend Mr. N. Aswathanarayana Setty, Mr. D. Srinivas Murthy without whose motivation and encouragement this could not have been completed. We express our sincere gratitude to Managing Director, New Age International (P) Limited, and Bangalore Division Marketing Manager Mr. Sudharshan for their suggestions and provisions of the font materials evaluated in this study.

We would also like to thank our friends and students for exchanges of knowledge, skills during our course of time writing this book. AUTHORS

Dedicated to

my dear parents,

Shiridi Sai Baba,

my dear loving son Monish Sri Sai G

and my wife and best friend S. Mamatha

—— ——— A. Ganesh

&

my dear parents,

and my wife S. Geetha

—— ———G. Balasubramanian

QUESTION PAPER LAYOUT

Engineering Mathematics-II

O6MAT

PART-A PART-B

4 Qns. 4 Qns.

Units-1, 2, 3, 4 Units-5, 6, 7, 8

1 Qn. from each unit 1 Qn. from each unit

To answer five full questions choosing at leasttwo questions from each part

Time: 3 Hrs. Max. Marks: 100

Unit/Qn. No. Topics Unit/Qn. No. Topics

1. DIFFERENTIAL CALCULUS-I Radius of Curvature: Cartesian curve Parametric curve, Pedal curve, Polar curve and some fundamental theorems.
2. DIFFERENTIAL CALCULUS-II Taylor’s, Maclaurin’s Maxima and Minima for a function of two variables.
3. INTEGRAL CALCULUS-II Double and triple integral, Beta and Gamma functions.
4. VECTOR INTEGRATION AND ORTHOGONAL CURVILINEAR COORDINATES 5. DIFFERENTIAL EQUATIONS-I Linear differential equation with constant coefficients, Solution of homogeneous and non homogene- ous linear D.E., Inverse differential operator and the Particular Integral (P.I.) Method of undetermined coefficients. 6. DIFFERENTIAL EQUATIONS-II Method of variation of parameters, Solutions of Cauchy's homogeneous linear equation and Legendre’s linear equation, Solution of initial and Boundary value problems. 7. LAPLACE TRANSFORMS Periodic function, Unit step function (Heaviside function), Unit impulses function. 8. INVERSE LAPLACE TRANSFORMS Applications of Laplace transforms.

This page

intentionally left

blank

8.4 Inverse Laplace Transforms of the Functions of the Form F s

• UNIT I Differential Calculus–I 1– ACKNOWLEDGEMENT ( vi )
  • 1.1 Introduction
  • 1.2 Radius of Curvature
    • 1.2.1 Radius of Curvature in Cartesian Form
    • 1.2.2 Radius of Curvature in Parametric Form
    • Worked Out Examples
    • Exercise 1.1
    • 1.2.3 Radius of Curvature in Pedal Form
    • 1.2.4 Radius of Curvature in Polar Form
    • Worked Out Examples
    • Exercise 1.2
  • 1.3 Some Fundamental Theorem
    • 1.3.1 Rolle’s Theorem
    • 1.3.2 Lagrange’s Mean Value Theorem
    • 1.3.3 Cauchy’s Mean Value Theorem
    • 1.3.4 Taylor’s Theorem
    • Worked Out Examples
    • Exercise 1.3
    • Additional Problems (from Previous Years VTU Exams.)
    • Objective Questions
• UNIT II Differential Calculus–II 61–
  • 2.1 Indeterminate Forms
    • Worked Out Examples
    • Exercise 2.1
    • 2.1.2 Indeterminate Forms ∞ – ∞ and 0 × ∞
    • Worked Out Examples
    • Exercises 2.2
    • 2.1.3 Indeterminate Forms 0^0 , 1∞, ∞^0 , 0∞
    • Worked Out Examples
    • Exercise 2.3
  • 2.2 Taylor’s Theorem for Functions of Two Variables
    • Worked Out Examples ( xii )
    • Exercise 2.4
  • 2.3 Maxima and Minima of Functions of Two Variables
    • 2.3.1 Necessary and Sufficient Conditions for Maxima and Minima
    • Worked Out Examples
    • Exercise 2.5
  • 2.4 Lagrange’s Method of Undetermined Multipliers
    • Working Rules
    • Worked Out Examples
    • Exercise 2.6
    • Additional Problems (from Previous Years VTU Exams.)
    • Objective Questions
• UNIT III Integral Calculus 105–
  • 3.1 Introduction
  • 3.2 Multiple Integrals
  • 3.3 Double Integrals
    • Worked Out Examples
    • Exercise 3.1
    • 3.3.1 Evaluation of a Double Integral by Changing the Order of Integration
    • 3.3.2 Evaluation of a Double Integral by Change of Variables
    • 3.3.3 Applications to Area and Volume
    • Worked Out Examples
    • Type 1. Evaluation over a given region
    • Type 2. Evaluation of a double integral by changing the order of integration
    • Type 3. Evaluation by changing into polars
    • Type 4. Applications of double and triple integrals
    • Exercise 3.2
  • 3.4 Beta and Gamma Functions
    • 3.4.1 Definitions
    • 3.4.2 Properties of Beta and Gamma Functions
    • 3.4.3 Relationship between Beta and Gamma functions
    • Worked Out Examples
    • Exercise 3.3
    • Additional Problems (From Previous Years VTU Exams.)
    • Objective Questions
• UNIT IV Vector Integration and Orthogonal Curvilinear Coordinates 166–
  • 4.1 Introduction
  • 4.2 Vector Integration
    • 4.2.1 Vector Line Integral
    • Worked Out Examples
    • Exercise 4.1
  • 4.3 Integral Theorem ( xiii )
    • 4.3.1. Green’s Theorem in a Plane
    • 4.3.2. Surface Integral and Volume Integral
    • 4.3.3. Stoke’s Theorem
    • 4.3.4. Gauss Divergence Theorem
    • Worked Out Examples
    • Exercise 4.2
  • 4.4 Orthogonal Curvilinear Coordinates
    • 4.4.1 Definition
    • 4.4.2 Unit Tangent and Unit Normal Vectors
    • 4.4.3. The Differential Operators
    • Worked Out Examples
    • Exercise 4.3
    • 4.4.4. Divergence of a Vector
    • Worked out Examples
    • Exercise 4.4
    • 4.4.5. Curl of a Vector
    • Worked Out Examples
    • Exercise 4.5
    • 4.4.6. Expression for Laplacian ∇^2 ψ
    • 4.4.7. Particular Coordinate System
    • Worked Out Examples
    • Exercise 4.6
    • Additional Problems (From Previous Years VTU Exams.)
    • Objective Questions
• UNIT V Differential Equations–I 214–
  • 5.1 Introduction
    • Constant Coefficients 5.2 Linear Differential Equations of Second and Higher Order with
  • 5.3 Solution of a Homogeneous Second Order Linear Differential Equation
    • Worked Out Examples
    • Exercise 5.1
  • 5.4 Inverse Differential Operator and Particular Integral
  • 5.5 Special Forms of x
    • Worked Out Examples
    • Exercise 5.2
    • Exercise 5.3
    • Exercise 5.4
  • 5.6 Method of Undetermined Coefficients
    • Worked Out Examples
    • Exercise 5.5
  • 5.7 Solution of Simultaneous Differential equations ( xiv )
    • Worked Out Examples
    • Exercise 5.6
    • Additional Problems (From Previous Years VTU Exams.)
    • Objective Questions
• UNIT VI Differential Equations–I 280—
  • 6.1 Method of Variation of Parameters
    • Worked Out Examples
    • Exercise 6.1
    • Linear Equation 6.2 Solution of Cauchy’s Homogeneous Linear Equation and Lengendre’s
    • Worked Out Examples
    • Exercise 6.2
  • 6.3 Solution of Initial and Boundary Value Problems
    • Worked Out Examples
    • Exercise 6.3
    • Additional Problems (From Previous Years VTU Exams.)
    • Objective Questions
• UNIT VII Laplace Transforms 321—
  • 7.1 Introduction
  • 7.2 Definition
  • 7.3 Properties of Laplace Transforms
    • 7.3.1 Laplace Transforms of Some Standard Functions
    • Worked out Examples
    • Exercise 7.1
    • 7.3.2 Laplace Transforms of the form eat f ( t )
    • Worked Out Examples
    • Exercise 7.2
    • 7.3.3 Laplace Transforms of the form t n f ( t ) where n is a positive integer
    • Worked out Examples
    • Exercise 7.3
  • 7.4 Laplace Transforms of Periodic Functions
    • Worked Out Examples
    • Exercise 7.3
  • 7.5 Laplace Transforms of Unit Step Function and Unit Impulse Function
    • Unit Step Function (Heaviside Function)
    • 7.5.1 Properties Associated with the Unit Step Function
    • 7.5.2 Laplace Transform of the Unit Impulse Function ( xv )
    • Exercise 7.4
    • Additional Problems (From Previous Years VTU Exams.)
    • Objective Questions
• UNIT VIII Inverse Laplace Transforms 369–
  • 8.1 Introduction
  • 8.2 Inverse Laplace Transforms of Some Standard Functions
    • Worked Out Examples
  • 8.3 Inverse Laplace Transforms using Partial Fractions
    • Exercise 8.1
      • a f s
    • Worked Out Example
    • Exercise 8.2
  • 8.5 Convolution Theorem
    • Worked Out Examples
    • Exercise 8.3
  • 8.6 Laplace Transforms of the Derivatives
  • 8.7 Solution of Linear Differential Equations
    • Worked Out Examples
    • Solution of Simultaneous Differential Equations
    • Exercise 8.4
  • 8.8 Applications of Laplace Transforms
    • Worked Out Examples
    • Exercise 8.5
    • Additional Problems (From Previous Years VTU Exams.)
    • Objective Questions
• Model Question Paper–I 427–
• Model Question Paper–II 441–

This page

intentionally left

blank

2 ENGINEERING MATHEMATICS—II

The sign of

d

ds

indicates the convexity and concavity of the curve in the neighbourhood of

the point. Many authors take ρ =

ds

d ψ and discard negative sign if computed value is negative.

∴ Radius of curvature ρ =

k

1.2.1 Radius of Curvature in Cartesian Form

Suppose the Cartesian equation of the curve C is given by y = f ( x ) and A be a fixed point on it. Let

P ( x , y ) be a given point on C such that arc AP = s.

Then we know that

dy

dx = tan^ ψ^ ...(1)

where ψ is the angle made by the tangent to the curve C at P with the x -axis and

ds

dx =^

2

1 2

• F HG^

I KJ

R S | T|^

U V | W|

dy

dx

Differentiating (1) w.r.t x , we get

d y

dx

2

2 =^ sec^

2 ψ ⋅ d ψ

dx

= 1 +^ tan^2 ψ^ ⋅

ψ d i

d ds

ds dx

2 2

1 2

• F HG^

I KJ

L

N

M M

O

Q

P P

• F HG^

I KJ

L

N

M M

O

Q

P P

dy

dx

dy

ρ dx

[By using the (1) and (2)]

2

3 2

• F HG^

I KJ

R S

| T|^

U V

| W|

dy

dx

Therefore, ρ =

2

3 2

2 2

• F HG^

I KJ

R S

| T|^

U V

| W|

dy

dx

d y

dx

where y 1 =

dy

dx and^ y^2 =^

d y

dx

2

DIFFERENTIAL CALCULUS—I 3

Equation (3) becomes,

3 2

2

• y y

o t

This is the Cartesian form of the radius of curvature of the curve y = f ( x ) at P ( x , y ) on it.

1.2.2 Radius of Curvature in Parametric Form

Let x = f ( t ) and y = g ( t ) be the Parametric equations of a curve C and P ( x , y ) be a given point

on it.

Then

dy

dx =^

dy dt

dx dt ...(4)

and

d y

dx

2

2 =^

d dt

dy dt dx dt

dt dx

R S T

U V W

dx

dt

d y

dt

dy

dt

d x

dt

dx

dt

dx

dt

F HG^

I KJ^

2 2

2 2 2

d y

dx

2

dx

dt

d y

dt

dy

dt

d x

dt

dx

dt

F HG^

I KJ

2 2

2 2 3

Substituting the values of

dy

dx and^

d y

dx

2

2 in the Cartesian form of the radius of curvature of the

curve y = f ( x ) [Eqn. (3)]

1 2

3 2

2

2

3 2

2 2

• F HG^

I KJ

R S

| T|^

U V

| y W|

y

dy

dx

d y

dx

o t

2

3 2

2 2

2 2

3

• F HG^

I KJ

R S

| T|^

U V

| W|

R S T

U V W

F HG^

I KJ

dy dt

dx dt

dx

dt

d y

dt

dy

dt

d x

dt

dx

dt

• /

dx

dt

dy

dt

dx

dt

d y

dt

dy

dt

d x

dt

F HG^

I KJ^

• F HG^

I KJ

R S

| T|^

U V

| W|

2 2

3 2

2 2

2 2