Introduction to Mathematics with Maple

by Adams, Pamela W.; Smith, K.; Vyborny, Rudolf

ISBN13: 9789812389312

ISBN10: 9812389318

Format: Hardcover

Pub. Date: 2004-06-01

Publisher(s): WORLD SCIENTIFIC PUB CO INC

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Summary

New concepts are motivated before being introduced through rigorous definitions. All theorems are proved and great care is taken over the logical structure of the material presented. To facilitate understanding, a large number of diagrams are included. Most of the material is presented in the traditional way, but an innovative approach is taken with emphasis on the use of Maple and in presenting a modern theory of integration. To help readers with their own use of this software, a list of Maple commands employed in the book is provided. The book advocates the use of computers in mathematics in general, and in pure mathematics in particular. It makes the point that results need not be correct just because they come from the computer. A careful and critical approach to using computer algebra systems persists throughout the text.

Preface

vii

1. Introduction

(32)

1.1 Our aims

(3)

1.2 Introducing Maple

(5)

1.2.1 What is Maple?

(1)

1.2.2 Starting Maple

(1)

1.2.3 Worksheets in Maple

(1)

1.2.4 Entering commands into Maple

(2)

1.2.5 Stopping Maple

(1)

1.2.6 Using previous results

(1)

1.2.7 Summary

(1)

1.3 Help and error messages with Maple

(2)

1.3.1 Help

(1)

1.3.2 Error messages

(1)

1.4 Arithmetic in Maple

(7)

1.4.1 Basic mathematical operators

(1)

1.4.2 Special mathematical constants

(1)

1.4.3 Performing calculations

(1)

1.4.4 Exact versus floating point numbers

(5)

1.5 Algebra in Maple

(8)

1.5.1 Assigning variables and giving names

(2)

1.5.2 Useful inbuilt functions

(6)

1.6 Examples of the use of Maple

(7)

2. Sets

(30)

2.1 Sets

(12)

2.1.1 Union, intersection and difference of sets

(3)

2.1.2 Sets in Maple

(4)

2.1.3 Families of sets

(1)

2.1.4 Cartesian product of sets

(1)

2.1.5 Some common sets

(2)

2.2 Correct and incorrect reasoning

(2)

2.3 Propositions and their combinations

(4)

2.4 Indirect proof

(2)

2.5 Comments and supplements

(10)

2.5.1 Divisibility: An example of an axiomatic theory

(7)

3. Functions

(34)

3.1 Relations

(5)

3.2 Functions

(7)

3.3 Functions in Maple

(14)

3.3.1 Library of functions

(2)

3.3.2 Defining functions in Maple

(3)

3.3.3 Boolean functions

(1)

3.3.4 Graphs of functions in Maple

(8)

3.4 Composition of functions

(1)

3.5 Bijections

(1)

3.6 Inverse functions

(4)

3.7 Comments

(2)

4. Real Numbers

(32)

4.1 Fields

(3)

4.2 Order axioms

100

(5)

4.3 Absolute value

105

(3)

4.4 Using Maple for solving inequalities

108

(4)

4.5 Inductive sets

112

(2)

4.6 The least upper bound axiom

114

(6)

4.7 Operation with real valued functions

120

(1)

4.8 Supplement. Peano axioms. Dedekind cuts

121

(8)

5. Mathematical Induction

129

(38)

5.1 Inductive reasoning

129

(6)

5.2 Aim high!

135

(1)

5.3 Notation for sums and products

136

(9)

5.3.1 Sums in Maple

141

(2)

5.3.2 Products in Maple

143

(2)

5.4 Sequences

145

(1)

5.5 Inductive definitions

146

(3)

5.6 The binomial theorem

149

(3)

5.7 Roots and powers with rational exponents

152

(5)

5.8 Some important inequalities

157

(4)

5.9 Complete induction

161

(3)

5.10 Proof of the recursion theorem

164

(2)

5.11 Comments

166

(1)

6. Polynomials

167

(24)

6.1 Polynomial functions

167

(2)

6.2 Algebraic viewpoint

169

(6)

6.3 Long division algorithm

175

(3)

6.4 Roots of polynomials

178

(2)

6.5 The Taylor polynomial

180

(4)

6.6 Factorization

184

(7)

7. Complex Numbers

191

(16)

7.1 Field extensions

191

(3)

7.2 Complex numbers

194

(13)

7.2.1 Absolute value of a complex number

196

(1)

7.2.2 Square root of a complex number

197

(2)

7.2.3 Maple and complex numbers

199

(1)

7.2.4 Geometric representation of complex numbers. Trigonometric form of a complex number

199

(3)

7.2.5 The binomial equation

202

(5)

8. Solving Equations

207

(24)

8.1 General remarks

207

(2)

8.2 Maple commands solve and f solve

209

(4)

8.3 Algebraic equations

213

(15)

8.3.1 Equations of higher orders and f solve

225

(3)

8.4 Linear equations in several unknowns

228

(3)

9. Sets Revisited

231

(8)

9.1 Equivalent sets

231

(8)

10. Limits of Sequences

239

(40)

10.1 The concept of a limit

239

(10)

10.2 Basic theorems

249

(6)

10.3 Limits of sequences in Maple

255

(2)

10.4 Monotonic sequences

257

(6)

10.5 Infinite limits

263

(4)

10.6 Subsequences

267

(1)

10.7 Existence theorems

268

(7)

10.8 Comments and supplements

275

(4)

11. Series

279

(34)

11.1 Definition of convergence

279

(6)

11.2 Basic theorems

285

(4)

11.3 Maple and infinite series

289

(1)

11.4 Absolute and conditional convergence

290

(5)

11.5 Rearrangements

295

(2)

11.6 Convergence tests

297

(3)

11.7 Power series

300

(3)

11.8 Comments and supplements

303

(10)

11.8.1 More convergence tests

303

(3)

11.8.2 Rearrangements revisited

306

(1)

11.8.3 Multiplication of series

307

(2)

11.8.4 Concluding comments

309

(4)

12. Limits and Continuity of Functions

313

(44)

12.1 Limits

313

(9)

12.1.1 Limits of functions in Maple

319

(3)

12.2 The Cauchy definition

322

(7)

12.3 Infinite limits

329

(3)

12.4 Continuity at a point

332

(6)

12.5 Continuity of functions on closed bounded intervals

338

(15)

12.6 Comments and supplements

353

(4)

13. Derivatives

357

(50)

13.1 Introduction

357

(5)

13.2 Basic theorems on derivatives

362

(7)

13.3 Significance of the sign of derivative

369

(11)

13.4 Higher derivatives

380

(8)

13.4.1 Higher derivatives in Maple

381

(1)

13.4.2 Significance of the second derivative

382

(6)

13.5 Mean value theorems

388

(3)

13.6 The Bernoulli-l'Hospital rule

391

(3)

13.7 Taylor's formula

394

(4)

13.8 Differentiation of power series

398

(3)

13.9 Comments and supplements

401

(6)

14. Elementary Functions

407

(24)

14.1 Introduction

407

(1)

14.2 The exponential function

408

(3)

14.3 The logarithm

411

(4)

14.4 The general power

415

(3)

14.5 Trigonometric functions

418

(7)

14.6 Inverses to trigonometric functions

425

(5)

14.7 Hyperbolic functions

430

(1)

15 Integrals

431

(70)

15.1 Intuitive description of the integral

431

(7)

15.2 The definition of the integral

438

(7)

15.2.1 Integration in Maple

443

(2)

15.3 Basic theorems

445

(5)

15.4 Bolzano-Cauchy principle

450

(5)

15.5 Antiderivates and areas

455

(2)

15.6 Introduction to the fundamental theorem of calculus

457

(1)

15.7 The fundamental theorem of calculus

458

(11)

15.8 Consequences of the fundamental theorem

469

(8)

15.9 Remainder in the Taylor formula

477

(4)

15.10 The indefinite integral

481

(7)

15.11 Integrals over unbounded intervals

488

(4)

15.12 Interchange of limit and integration

492

(6)

15.13 Comments and supplements

498

(3)

Appendix A Maple Programming

501

(10)

A.1 Some Maple programs

501

(5)

A.1.1 Introduction

501

(1)

A.1.2 The conditional statement

502

(2)

A.1.3 The while statement

504

(2)

A.2 Examples

506

(5)

References

511

(2)

Index of Maple commands used in this book

513

(6)

Index

519

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