IT training intelligent routines solving mathematical analysis with MATLAB, mathcad, mathematica and maple anastassiou iatan 2012 07 28

se-The second chapter is dedicated to the power series, which are lar cases of series of functions and that have an important role for somepractical applications; for example, using the

E-mail: Lakhmi.jain@unisa.edu.au

For further volumes:

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Intelligent Routines

Solving Mathematical Analysis with Matlab, Mathcad, Mathematica and Maple

123

ISSN 1868-4394 e-ISSN 1868-4408

ISBN 978-3-642-28474-8 e-ISBN 978-3-642-28475-5

DOI 10.1007/978-3-642-28475-5

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3

”Nihil est in intellectu, quod non prius fuerit in sensu.”

John Locke

”Les beaux (grands) esprits se rencontrent.”

Voltaire

”Men should be what they seem,

Or those that be not, would they might seem none.”

Shakespeare, Othello III 3

”Science needs a man’s whole life And even if you had two lives, theywould not be enough It is great passion and strong effort that sciencedemands to men ”

I P Pavlov

”Speech is external thought, and thought internal speech.”

A Rivarol

”Nemo dat quod non habet.”

Latin expression

”Scientia nihil aliud est quam veritatis imago.”

Bacon, Novum Organon

Real Analysis is a discipline of intensive study in many institutions of highereducation, because it contains useful concepts and fundamental results inthe study of mathematics and physics, of the technical disciplines andgeometry.

This book is the first one of the kind that solves mathematical analysisproblems with all four related main software Matlab, Mathcad, Mathemat-ica and Maple

Besides the fundamental theoretical notions, the book contains manyexercises, solved both mathematically and by computer, using: Matlab 7.9,Mathcad 14, Mathematica 8 or Maple 15 programming languages

Due to the diversity of the concepts that the book contains, it is addressednot only to the students of the Engineering or Mathematics faculties butalso to the students at the master’s and PhD levels, which study RealAnalysis, Differential Equations and Computer Science

The book is divided into nine chapters, which illustrate the application ofthe mathematical concepts using the computer The introductory section ofeach chapter presents concisely, the fundamental concepts and the elementsrequired to solve the problems contained in that chapter Each chapterfinishes with some problems left to be solved by the readers of the bookand can verified for the correctness of their calculations using a specificsoftware such as Matlab, Mathcad, Mathematica or Maple

The first chapter presents some basic concepts about the theory of quences and series of numbers

se-The second chapter is dedicated to the power series, which are lar cases of series of functions and that have an important role for somepractical applications; for example, using the power series we can find theapproximate values of some functions so we can appreciate the precision of

particu-a computing method

In the third chapter are treated some elements of the differentiationtheory of functions.

The fourth chapter presents some elements of Vector Analysis with plications to physics and differential geometry

ap-The fifth chapter presents some notions of implicit functions and tremes of functions of one or more variables

ex-Chapter six is dedicated to integral calculus, which is useful to solve ous geometric problems and to mathematical formulation of some conceptsfrom physics

vari-Seventh chapter deals with the study of the differential equations andsystems of differential equations that model the physical processes.The chapter eight deals with the line and double integrals The line inte-gral is a generalization of the simple integral and allows the understanding

of some concepts from physics and engineering; the double integral has

a meaning analogous to that of the simple integral: like the simple nite integral is the area bordered by a curve, the double integral can beinterpreted as the volume bounded by a surface

defi-The last chapter is dedicated to the triple and surface integral calculus.Although it is not possible a geometric interpretation of the triple integral,mechanically speaking, this integral can be interpreted as a mass, beingconsidered as the distribution of the density in the respective space Thesurface integral is a generalization of the double integral in some planedomains, as the line integral generalizes the simple definite integral.This work was supported by the strategic grant POSDRU/89/1.5/S/

58852, Project “Postdoctoral programme for training scientific researchers”cofinanced by the European Social Fund within the Sectorial OperationalProgram Human Resources Development 2007-2013

The authors would like to thank Professor Razvan Mezei of Lander versity, South Carolina, USA for checking the final manuscript of our book.January 10, 2012

Uni-George Anastassiou,

MemphisUSA

Iuliana IatanBucharestRomania

1 Sequences and Series of Numbers 1

1.1 Cauchy Sequences 1

1.2 Fundamental Concepts 3

1.2.1 Convergent Series 3

1.2.1.1 Cauchy’s Test 5

1.2.2 Divergent Series 7

1.2.3 Operations on Convergent Series 11

1.3 Tests for Convergence of Alternating Series 14

1.4 Tests of Convergence and Divergence of Positive Series 16

1.4.1 The Comparison Test I 16

1.4.2 The Root Test 19

1.4.3 The Ratio Test 22

1.4.4 The Raabe’s and Duhamel’s Test 26

1.4.5 The Comparison Test II 29

1.4.6 The Comparison Test III 30

1.5 Absolutely Convergent and Semi-convergent Series 31

1.6 Problems 34

2 Power Series 41

2.1 Region of Convergence 41

2.2 Taylor and Mac Laurin Series 49

2.2.1 Expanding a Function in a Power Series 49

2.3 Sum of a Power Series 60

2.4 Problems 65

3 Differentiation Theory of the Functions 71

3.1 Partial Derivatives and Differentiable Functions of Several Variables 71

3.1.1 Partial Derivatives 71

3.1.2 The Total Differential of a Function 83

3.1.3 Applying the Total Differential of a Function to Approximate Calculations 90

3.1.4 The Functional Determinant 93

3.1.5 Homogeneous Functions 99

3.2 Derivation and Differentiation of Composite Functions of Several Variables 102

3.3 Change of Variables 119

3.4 Taylor’s Formula for Functions of Two Variables 126

3.5 Problems 143

4 Fundamentals of Field Theory 157

4.1 Derivative in a Given Direction of a Function 157

4.2 Differential Operators 162

4.3 Problems 179

5 Implicit Functions 187

5.1 Derivative of Implicit Functions 187

5.2 Differentiation of Implicit Functions 193

5.3 Systems of Implicit Functions 203

5.4 Functional Dependence 209

5.5 Extreme Value of a Function of Several Variables Conditional Extremum 214

5.6 Problems 229

6 Terminology about Integral Calculus 245

6.1 Indefinite Integrals 245

6.1.1 Integrals of Rational Functions 245

6.1.2 Reducible Integrals to Integrals of Rational Functions 251

6.1.2.1 Integrating Trigonometric Functions 251

6.1.2.2 Integrating Certain Irrational Functions 252

6.2 Some Applications of the Definite Integrals in Geometry and Physics 260

6.2.1 The Area under a Curve 260

6.2.2 The Area between by Two Curves 265

6.2.3 Arc Length of a Curve 269

6.2.4 Area of a Surface of Revolution 274

6.2.5 Volumes of Solids 276

6.2.6 Centre of Gravity 277

7 Equations and Systems of Linear Ordinary

Differential Equations 317

7.1 Successive Approximation Method 317

7.2 First Order Differential Equations Solvable by Quadratures 320

7.2.1 First Order Differential Equations with Separable Variables 322

7.2.2 First Order Homogeneous Differential Equations 324

7.2.3 Equations with Reduce to Homogeneous Equations 326

7.2.4 First Order Linear Differential Equations 330

7.2.5 Exact Differential Equations 332

7.2.6 Bernoulli’s Equation 339

7.2.7 Riccati’s Equation 340

7.2.8 Lagrange’s Equation 342

7.2.9 Clairaut’s Equation 346

7.3 Higher Order Differential Equations 349

7.3.1 Homogeneous Linear Differential Equations with Constant Coefficients 349

7.3.2 Non-homogeneous Linear Differential Equations with Constant Coefficients 357

7.3.2.1 The Method of Variation of Constants 357

7.3.2.2 The Method of the Undetermined Coefficients 360

7.3.3 Euler’s Equation 367

7.3.4 Homogeneous Systems of Differential Equations with Constant Coefficients 369

7.3.5 Method of Characteristic Equation 371

7.3.6 Elimination Method 373

7.4 Non-homogeneous Systems of Differential Equations with Constant Coefficients 377

7.5 Problems 380

8 Line and Double Integral Calculus 395

8.1 Line Integrals of the First Type 395

8.1.1 Applications of Line Integral of the First Type 396

8.2 Line Integrals of the Second Type 406

8.3 Calculus Way of the Double Integrals 421

8.4 Applications of the Double Integral 430

8.4.1 Computing Areas 430

8.4.2 Mass of a Plane Plate 431

8.4.3 Coordinates the Centre of Gravity of a Plane Plate 433

8.4.4 Moments of Inertia of a Plane Plate 436

8.4.5 Computing Volumes 438

8.5 Change of Variables in Double Integrals 441

8.5.1 Change of Variables in Polar Coordinates 441

8.5.2 Change of Variables in Generalized Polar Coordinates 445

8.6 Riemann-Green Formula 447

8.7 Problems 454

9 Triple and Surface Integral Calculus 475

9.1 Calculus Way of the Triple Integrals 475

9.2 Change of Variables in Triple Integrals 477

9.2.1 Change of Variables in Spherical Coordinates 477

9.2.2 Change of Variables in Cylindrical Coordinates 480

9.3 Applications of the Triple Integrals 485

9.3.1 Mass of a Solid 485

9.3.2 Volume of a Solid 493

9.3.3 Centre of Gravity 499

9.3.4 Moments of Inertia 509

9.4 Surface Integral of the First Type 512

9.5 Surface Integral of the Second Type 520

9.5.1 Flux of a Vector Field 524

9.5.2 Gauss- Ostrogradski Formula 529

9.5.3 Stokes Formula 540

9.6 Problems 551

References 573

List of Symbols 577

Index 579

1.1 Cauchy Sequences

Definition 1.1 (see [41], p.13) A sequence (x n)n ∈N is convergent if (∃)

a ∈ R such that (∀) ε > 0, (∃) n ε ∈ N such that |x n − a| < ε, (∀) n  n ε

Definition 1.2 (see [41], p.23) A sequence (x n)n ∈N is called a Cauchy

sequence if the terms of the sequence, eventually all become arbitrarily

close to one another, i.e if for (∀) ε > 0, (∃) n ε ∈ N such that (∀) n  n ε

G.A Anastassiou and I.F Iatan: Intelligent Routines, ISRL 39, pp 1–40.

springerlink.com  Springer-Verlag Berlin Heidelberg 2013c

We shall use the definition of convergence of the sequence y n = n+11 ,

n ∈ N ∗ in 0 to determine the rank n ε ∈ N such that (∀) n  n εwe have

ε − 1

+ 1;

we achieve

n >

1

ε − 1 + 1∈ N, such that (∀) n  n ε we have

|x n+p − x n | < ε; according to the Definition 1.2 it results that

|x n+p − x n | < ε; according to the Definition 1.2 it results that



k=1

1

or using Mathematica 8:

ln[1]:=Sum[1/(16*nˆ2 - 8*n - 3), {n, 1, Infinity}]

Out[1]=14

or with Maple 15:

Proposition 1.7 (see [41], p 36) The necessary and sufficient condition

for the convergence of the series n ≥1 a n is: (∀) ε > 0, (∃) n ε ∈ N such

that:

|a +· · · + a | < ε, (∀) n  n , ( ∀) p ∈ N.

Example 1.8 Use the Cauchy’s test for testing the convergence of the

Definition 1.9 (see [41], p 30) If the limit limn →∞ S n does not exist (or

it is infinite), the series is then called divergent.

Proposition 1.10 (see [15], p 93) If the terms of a series sequence is not

convergent to 0, the series is divergent, namely if for the series n ≥1 a n

We can prove that using Matlab 7.9:

using the Definition 1.9 it results that the series diverges

We can also deduce that using Matlab 7.9:

Hence, based on the Proposition 1.10 it follows that the series diverges.

It will also result that in Matlab 7.9:

one deduces that the series is divergent.

The same result can be achieved using Matlab 7.9:

= a = 0,

therefore the series is divergent

We can also obtain that using Matlab 7.9:

1.2.3 Operations on Convergent Series

Proposition 1.12 (see [15], p 95) Let n ≥1 a n and n ≥1 b n be two

convergent series, which have the sums A and B respectively and let be

α ∈ R Then:

a) the series n ≥1 αa n converges and it has the sum αA, namely a vergent series may be multiplied term by term by any number α;

con-b) the sum (difference) of the two convergent series is a convergent series

n ≥1 (a n ± b n ) and it has the sum A ± B.

Example 1.13 Use the operations on convergent series to compute

the sum of the series:

1

n + 2

+

1

1

n + 2

+

n ≥1

1

Applying the Proposition 1.12 it follows the series n ≥1 αa n will alsoconverges and

hence, it results that the series n ≥1 v n converges

Applying the Proposition 1.12 it follows that the series n ≥1 (u n + v n)will also converge and it has the sum

U + V = 3

4 + 1 =

7

4.b) We shall have

n ≥1

14

n

n ≥1

12

1.3 Tests for Convergence of Alternating Series

Definition 1.14 (see [15], p 94) An alternating series is an infinite

series of the form n ≥1(−1) n+1

a n , where all the a n are non-negative,

namely a n > 0 for all n ∈ N.

Proposition 1.15 (Leibnitz’s test, see [15], p 94) If the sequence (a n)n

is monotone decreasing and it equals 0 as n approaches infinity, then the

n →∞

27n2+ 36n + 11 (3n + 1) (3n + 2) (3n + 3) = 0;

therefore, using the Leibnitz’s test it follows that the alternating series isconvergent

c) We can notice that

1.4 Tests of Convergence and Divergence of Positive Series

Definition 1.17 (see [15], p 93) The series n ≥1 a n is with positive

terms if a n > 0, ( ∀) n ∈ N.

1.4.1 The Comparison Test I

Proposition 1.18 ( The comparison test I, see [15], p 93): Let n ≥1 a n

and n ≥1 b n be two positive series such that a n ≤ b n Then if:

a) n ≥1 b n converges it follows that n ≥1 a n also converges;

b) n ≥1 a n diverges it results that n ≥1 b n also diverges

Example 1.19 Test the convergence of the positive series:

n + √ n) · n

2· n 1/2 · n=

1

2· n 3/2;hence



n ≥1

√

n + 1 − √ n n

Then we have the following:

A)If λ < 1, then the series n ≥1 a n is convergent;

B) If λ > 1, then the series n ≥1 a n is divergent;

C) If λ = 1, then the series n ≥1 a n may be convergent or it may bedivergent, namely we do not have a definite conclusion

Example 1.21 Discuss the convergence of the positive series:

1.4.3 The Ratio Test

Proposition 1.22 (The ratio test, see [15], p 94): Let n ≥1 a n be a

positive series such that a n = 0 for any n ≥ 1 Assume that:

Then we have the following:

A)If λ < 1, then the series n ≥1 a n is convergent;

B) If λ > 1, then the series n ≥1 a n is divergent;

C) If λ = 1, then the series n ≥1 a n may be convergent or it may bedivergent, namely we do not have a definite conclusion

Example 1.23 Discuss the convergence of the positive series:

or with Mathematica 8:

or in Maple 15:

b) We shall notice that

We don’ t have a definite conclusion for a = 1.

1.4.4 The Raabe’s and Duhamel’s Test

Proposition 1.24 (The Raabe’s and Duhamel’s test, see [15], p 94):

Let n ≥1 a n be a positive series Assume that:

Then we have the following:

A)If λ > 1, then the series n ≥1 a n is convergent;

b) One deduces that: