Practical Mathematica

Practical Mathematica

Dr. Pragati Gautam Kamala Nehru College, University of Delhi Swapnil Verma Kamala Nehru College, University of Delhi

Ane Books Pvt. Ltd. New Delhi i Chennai

Practical Mathematica Dr. Pragati Gautam and Swapnil Verma

© Authors First Edition : 2019 , Reprint : 2020 Price: ` 795/Pages: xii + 340 Size: 8.5” × 11”

Published by

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ISBN : 978-93-88264-80-8 $OOULJKWVUHVHUYHG1RSDUWRIWKLVERRNPD\EHUHSURGXFHGLQDQ\IRUPLQFOXGLQJSKRWRFRS\LQJPLFUR¿OPVSKRWRSULQWVVWRUDJHLQDQ\UHWULHYDO system, transmission in any permanent or temporary form, without the prior written consent of the publisher. Printed at: Repro Knowledgecast Limited, Thane.

Syllabus of Practical Paper C1 – Calculus Practicals, Semester -1(CBCS) Plotting of graphs of function of type (greatest integer function), ( even and odd positive integer), ( even and odd positive integer), ( a positive integer) , discuss the effect of and on the graph, Plotting the graphs of polynomial of degree 4 and 5, the derivative graph, the second derivative graph and comparing them. Sketching parametric curves, tracing of conics in Cartesian coordinates, obtaining surface of revolution of curves, sketching of ellipsoid, hyperboloid of one and two sheets, elliptic cone, elliptic paraboloid, hyperbolic paraboloid XVLQJ&DUWHVLDQFRRUGLQDWHVWR¿QGQXPEHUVEHWZHHQWZRUHDOQXPEHUVDQGSORWWLQJRI¿QLWHDQGLQ¿QLWHVXEVHW of R, matrix operations (addition, multiplication, inverse, transpose, determinant, rank, eigenvectors, eigenvalues, &KDUDFWHULVWLFHTXDWLRQDQGYHUL¿FDWLRQRI&D\OH\+DPLOWRQHTXDWLRQV\VWHPRIOLQHDUHTXDWLRQV  *UDSKRI+\SHUEROLFIXQFWLRQVFRPSXWDWLRQRIOLPLWGLIIHUHQWLDWLRQDQGLQWHJUDWLRQRIYHFWRUIXQFWLRQVFRPSOH[ numbers and their representations, operations like addition, multiplication, division, modulus, Graphical representation of polar form.

C4 - Differential Equations Practicals, Semester -2(CBCS) Plotting of second order solution family of differential equation, Plotting of third order solution family of differential equation. Growth model (exponential case only), Decay model (exponential case only), Lake pollution model (with constant/ VHDVRQDOÀRZDQGSROOXWLRQFRQFHQWUDWLRQ &DVHRIVLQJOHFROGSLOODQGDFRXUVHRIFROGSLOOV/LPLWHGJURZWKRI population (with and without harvesting), Predatory -prey model (Basic Volterra Model, with density dependence, HIIHFWRI''7WZRSUH\RQHSUHGDWRU (SLGHPLFPRGHORILQÀXHQ]DEDVLFHSLGHPLFPRGHOFRQWDJLRXVIRUOLIH disease with carriers). Battle model (basic battle model, jungle warfare, long range weapons). Plotting of recursive sequences,study the convergence of sequences through plotting,verify Bolzano Weierstrass theorem through plotting of sequences and hence identify convergent subsequences from the plot, study the FRQYHUJHQFHGLYHUJHQFHRILQ¿QLWHVHULHVE\SORWWLQJWKHLUVHTXHQFHVRISDUWLDOVXP&DXFK\¶VURRWWHVWE\SORWWLQJ nth roots, Ratio test by plotting the ratio of nth and n+1th term, study the convergence of sequences through plotting.

PREFACE  :KHQHYHUDQ\RQHRIXVVHHVDQHZERRNRQ3UDFWLFDOLQ0DWKHPDWLFVWKH¿UVWTXHVWLRQWKDWFRPHVWRPLQG is why yet another book on the topic is required . Of course there is a plethora of titles on Practical in mathematics, from old classics to latest versions, which offer the students a wide range of choices; yet the needs, demands and expectations of student are as different as their socio-economic background, academic requirements and goals are. Therefore no matter how hard a book has been worked upon, it is next to impossible to be able to GHOLYHUDWH[WERRNWRHYHU\RQH¶VQHHGV0RVWVWXGHQWVHVSHFLDOO\LQ,QGLDWKHUHIRUHVWLOOUHO\ODUJHO\XSRQWKHQRWHV provided by their teachers to study the topics of the prescribed syllabi. The problem of being able to choose a VXLWDEOHWH[WERRNRI0DWKHPDWLFVLVPRUHZLWKWKHVWXGHQWVRI¿UVW\HDURIDQ8QGHUJUDGXDWH&RXUVH A beginner, therefore, needs a customized book that is at the same time easy to comprehend and covers plenty of applications along with the needed part of theory. This book is especially designed to cater to the QHHGVRIVXFKVWXGHQWVZKRVWXG\0DWKHPDWLFDLQWKH¿UVWVHPHVWHUDQGVHFRQGVHPHVWHURI%6F+RQV  0DWKHPDWLFVLQWKH8QLYHUVLW\RI'HOKLDQGRWKHU&HQWUDO8QLYHUVLWLHVZKHUHWKH&%&6FXUULFXOXPLVEHLQJRIIHUHG The text introduces the fundamentals of Practicals in Mathematica to the reader in the easiest form and is supplemented with solved examples. This Book will be very useful for students in preparing their core practical SDSHUVRI6HPHVWHU,DVZHOODV6HPHVWHU,,RQ&DOFXOXV$QDO\VLVDQG'LIIHUHQWLDO(TXDWLRQVRIIHUHGE\WKH 8QLYHUVLW\RI'HOKLXQGHU&%&6V\VWHP7KHDXWKRUVKDYHXWLOL]HGWKHLUZLGHH[SHULHQFHRIWHDFKLQJWKLVSDSHUWR 8QGHUJUDGXDWHFRXUVHVLQLQFRUSRUDWLQJPLQXWHVWRIWKHGHWDLOVZKLFKDEHJLQQHUPLJKW¿QGGLI¿FXOWWRXQGHUVWDQG 7KHOXFLGSUHVHQWDWLRQRIWKHWKHRU\LVZHOOFRPSOHPHQWHGE\SOHQW\RIVROYHGH[DPSOHVDQGXQVROYHGH[HUFLVHV,Q presence of so many books on Mathematica around, this title, for the targeted readers, will turn out to be the best in its class. The content of book is divided into two parts. The ¿UVWSDUWintroduces the students to Calculus which contains different types of functions, Conics and Concoids , Parametric curve and polar curves and their plotting etc.. Becoming familiar with the trigonometric functions, a student will thus connect with the book through this chapter very easily. The ¿UVWFKDSWHUintroduces the students to the Mathematica software, its launching and few basic commands. The second chapter, Operating in mathematica provides knowledge about texting features of mathematica and how to create a powerful presentation in it . The Third chapter is about plotting of function and their graphs. The fourth chapter introduces to the students the hyperbolic trigonometric functions and their sketching . The ¿IWK FKDSWHU provides quick introduction to the methods of calculus for vector valued functions and gives a foundation for study of multivariate calculus which they will study in the next semester. Sixth Chapter is on sketching of parametric and polar curve which may be studied independently. Chapter Seven and Eight together offer a good introduction to the methods of tracing of conics and are further applied to the study of Concoid section in Chapter Nine. Chapter Ten and Eleven are introduction to matrix operations and Plotting of real numbers. The Chapter Twelve of Part-1 is on Complex numbers , their operations and the graphical representation . The Second part of this book is on Differential equations and Analysis . The Chapter One is about Basics and Algebraic Calculations. Chapter Two provides a quick introduction to second and third order differential equation and plotting of its solutions. Chapter Three is about modeling in differential equations which covers all the existing models. These Modeling methods utilizes almost everything the reader has studied in theory, hence offering a panoramic view. Chapter Four is on Sequences and Chapter Five LVRQ,Q¿QLWH6HULHVZKLFKWHOOVDERXW SORWWLQJRIVHTXHQFHVFRQYHUJHQFHRIOLPLWV%RO]DQR:HLHUVWUDVVWKHRUHP&DXFK\URRWWHVWDQG'¶$OHPEHUW5DWLR test.

viii | Practical Mathematica

The authors feel highly grateful and would like to thank the founder and CEO of Wolfram Research for developing a great software which is extremely beneficial for Physics, Mathematics and Computer Science students. We are highly thankful to our families, friends and colleagues without whose good wishes this work would be difficult to complete. We are especially thankful to our colleagues in other colleges who shared their practical paper knowledge with us. We would like to express our thanks to our beloved student Riya Jain for helping us during the content development. We are also very thankful to ANE Books Pvt. Ltd. (Our Publisher) who not only showed faith in us but also kept us motivating to bring out the book in the present form. We sincerely KRSHWKDWWKHVWXGHQWVZLOO¿QGWKHWLWOHFRPSUHKHQVLYHDQGIULHQGO\DQGWKHERRNZLOOEHDEOHWRVHUYHLWVGHVLUHG purpose. Any kind of feedback from the readers in terms of suggestions, comments or criticisms is most welcome and it will help us to improve the book further. 10 th April, 2019

Authors [email protected], [email protected]

CONTENTS OF THE BOOK

Syllabus of Practical Paper

v

Preface

vii

UNIT 1 - CALCULUS ( SEMESTER 1)

CHAPTER 1 GETTING STARTED WITH MATHEMATICA

1-10

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1.3 Basic arithmetic computation

3

1.4 Writing commands in Mathematica

5

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1.7 Saving work and quitting Mathematica

9

1.8 Getting help with Mathematica

9

CHAPTER 2 OPERATING IN MATHEMATICA

11-16

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2.2

Writing text in notebook

11

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2.4

Writing assistant palette

12

2.5

Editing in Mathematica notebook

13

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2.7

Creating Slideshows

14

2.8

Printing the document

14

2.9

Loading packages

15

CHAPTER 3 PLOTTING OF FUNCTIONS

16-41

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3.2

17

Plotting of functions 3.2.1 Plotting of function By using Plot command

17

3.2.2 Plotting of function By Manipulate command

24

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3.4

Limits of a Function

33

3.5

Derivatives of a function

34

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3.7

37

Plotting of graphs of polynomials of degree 4 and 5, their derivative graph and comparing them

CHAPTER 4 HYPERBOLIC FUNCTION

42-53

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4.7 Plotting of function by Manipulate/ Animate command

48

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CHAPTER 5 VECTOR VALUED FUNCTIONS

54-62

5.1

Vectors

54

5.2

Vector Valued Function

57

5.3

Limit of a Vector Valued Function

57

5.4

Algebra of Limits

58

5.4.1

Addition and Subtraction of Limits

58

5.4.2

Limit of a Scalar Multiple

58

5.4.3

Limit of a Dot and Cross product of a Vector Valued Functions

58

5.5

Differentiation of a vector valued function

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CHAPTER 6 SKETCHING OF PARAMETRIC AND POLAR CURVES

59 

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CHAPTER 7 TRACING OF CONICS 7.1

7.2

Meaning of Conics

  

81-101 81

7.1.1 Circle

81

7.1.2 Parabola

83

7.1.3 Ellipse

87

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Tracing of Conics by Manipulation or Animation

CHAPTER 8 SURFACE OF REVOLUTION OF A CURVE

95

102-118

8.1 Surface of Revolution

102

8.2 Obtaining surface of revolution about x - axis , y- axis and z-axis

102

8.3 Obtaining surface of revolution about x-y axis , y-z axis and z-x axis

110

8.4 Miscellaneous examples

114

Contents | xi

CHAPTER 9 SKETCHING OF CONCOIDS

119-136

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9.2

119

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9.5

128

Paraboloid 9.5.1 Elliptical Paraboloid

128

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CHAPTER 10 MATRICES

137-152

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10.2

139

Matrix operations

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10.4

144

System of linear equations +RPRJHQHRXV6\VWHPRIOLQHDUHTXDWLRQV



10.4.2 Nonhomogeneous system of linear equations

148

CHAPTER 11

PLOTTING OF FINITE AND INFINITE SUBSETS OF R

153-159

11.1 Finding a number between two real numbers

153

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CHAPTER 12 COMPLEX NUMBERS AND THEIR REPRESENTATION

160-170

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UNIT 2 -DIFFERENTIAL EQUATIONS AND REAL ANALYSIS (SEMESTER 2)

CHAPTER 1 BASICS AND ALGEBRAIC CALCULATIONS

171-176

1.1 Algebraic calculations in Mathematica

171

1.2 Factoring and Expanding Polynomials

172

1.3 Some Logical Operators

174

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CHAPTER 2 PLOTTING OF SOLUTIONS OF FIRST, SECOND AND THIRD ORDER DIFFERENTIAL EQUATIONS

177-194

2.1 Differential Equations

177

2.1 First order differential equations

178

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2.4 Second order ordinary differential equations

180

2.5 Plotting of solutions of family of second order differential equations

180

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CHAPTER 3 MODELLING IN MATHEMATICA

195-257

3.1 Growth model

195

3.2 Decay model

199

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3.4 Case of single cold pill and a course of cold pill

210

3.5 Limited growth of population model(with and without harvesting)

214

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3.7 Predator prey model

218

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3.9 Battle model

232

CHAPTER 4 SEQUENCES

235-257

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4.7 Limit theorems

237

4.8 Limit point of a sequence

237

4.9 Computation of limits in Mathematica

237

4.10 Study of the convergence of sequence by plotting its terms

237

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4.12 Bolzano Weierstrass theorem

244

4.13 Cauchy Sequence

248

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4.15 Plotting of recursive sequences

253

CHAPTER 5 INFINITE SERIES

258-284

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5.2 Sequence of Partial Sums

258

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SOLUTIONS TO THE CHAPTER EXERCISE BIBLIOGRAPHY

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