Advanced Calculus Explored with Applications in Physics, Chemistry, and Beyond Hamza E. Alsamraee pdf

 

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Book name: Advanced Calculus Explored with Applications in Physics, Chemistry, and Beyond Hamza E. Alsamraee

Book Format: pdf

Author: by Hamza E. Alsamraee

Number of pages: 444

Book size: 7.35 MB

Publisher: by Hamza E. Alsamraee

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Description

Hi! My name is Hamza Alsamraee, and I am a senior (12th grade) at Centreville High School, Virginia. I have always had an affinity for mathematics, and from a very young age was motivated to pursue my curiosity. When I entered a new school in 7th grade after moving, I encountered some new mathematics I was unequipped for. Namely, I did not know what a linear equation even was! I was rather low-spirited, as I was stuck in an ever-lasting loop of confusion in class.

My mother and father soon began teaching me to the best of their ability. Fortunately, their efforts were effective, and I got a B on my first linear equations test! It was a huge improvement from being totally lost, but I wanted to know more. I did not care much about the grade, but I did care that I did not completely master the material.

I soon entered in a period of rapid learning, delving into curriculums significantly beyond my coursework simply for the sake of mastering higher mathematics. It seemed to me that the more I explored the field, the more beautiful the results were.

 

 

Table of Contents

About the Author 15

Preface 23

I Introductory Chapters 27

1 Differential Calculus 29

1.1 The Limit . .  . . 30

1.1.1 L’Hopital’s Rule   36

1.1.2 More Advanced Limits  . 40

1.2 The Derivative  . 45

1.2.1 Product Rule .  46

1.2.2 Quotient Rule   . . 48

1.2.3 Chain Rule  .  . . 49

1.3 Exercise Problems   . . . 56

78 CONTENTS

2 Basic Integration 59

2.1 Riemann Integral  60

2.2 Lebesgue Integral  63

2.3 The u-substitution   . . . 65

2.4 Other Problems  . 82

2.5 Exercise Problems   . . . 99

3 Feynman’s Trick 101

3.1 Introduction  . . . 102

3.2 Direct Approach  103

3.3 Indirect Approach   . . . 128

3.4 Exercise Problems   . . . 131

4 Sums of Simple Series 135

4.1 Introduction  . . . 136

4.2 Arithmetic and Geometric Series    136

4.3 Arithmetic-Geometric Series  . . 141

4.4 Summation by Parts   . . 146

4.5 Telescoping Series  152

4.6 Trigonometric Series   . . 159

4.7 Exercise Problems   . . . 163CONTENTS 9

II Series and Calculus 165

5 Prerequisites 167

5.1 Introduction  . . . 168

5.2 Ways to Prove Convergence  . . 172

5.2.1 The Comparison Test  . 172

5.2.2 The Ratio Test   . 173

5.2.3 The Integral Test   176

5.2.4 The Root Test   . 181

5.2.5 Dirichlet’s Test   . 184

5.3 Interchanging Summation and Integration  . . . 185

6 Evaluating Series 191

6.1 Introduction  . . . 192

6.2 Some Problems  . 193

6.2.1 Harmonic Numbers  . . . 204

6.3 Exercise Problems   . . . 214

7 Series and Integrals 215

7.1 Introduction  . . . 216

7.2 Some Problems  . 216

7.3 Exercise Problems   . . . 23010 CONTENTS

8 Fractional Part Integrals 233

8.1 Introduction  . . . 234

8.2 Some Problems  . 235

8.3 Open Problems  . 255

8.4 Exercise Problems   . . . 255

III A Study in the Special Functions 257

9 Gamma Function 261

9.1 Definition   262

9.2 Special Values  . . 262

9.3 Properties and Representations  264

9.4 Some Problems  . 271

9.5 Exercise Problems   . . . 275

10 Polygamma Functions 277

10.1 Definition   278

10.2 Special Values  . . 279

10.3 Properties and Representations  280

10.4 Some Problems  . 282

10.5 Exercise Problems   . . . 294CONTENTS 11

11 Beta Function 295

11.1 Definition   296

11.2 Special Values  . . 297

11.3 Properties and Representations  297

11.4 Some Problems  . 302

11.5 Exercise Problems   . . . 309

12 Zeta Function 311

12.1 Definition   312

12.2 Special Values  . . 312

12.3 Properties and Representations  317

12.4 Some Problems  . 326

12.5 Exercise Problems   . . . 335

IV Applications in the Mathematical Sciences and Beyond 339

13 The Big Picture 341

13.1 Introduction  . . . 342

13.2 Goal of the Part  343

14 Classical Mechanics 345

14.1 Introduction  . . . 34612 CONTENTS

14.1.1 The Lagrange Equations  346

14.2 The Falling Chain   . . . 347

14.3 The Pendulum  . 353

14.4 Point Mass in a Force Field  . . 357

15 Physical Chemistry 363

15.1 Introduction  . . . 364

15.2 Sodium Chloride’s Madelung Constant   . 370

15.3 The Riemann Series Theorem in Action   371

15.4 Pharmaceutical Connections  . . 377

15.5 The Debye Model  378

16 Statistical Mechanics 381

16.1 Introduction  . . . 382

16.2 Equations of State   . . . 383

16.3 Virial Expansion  385

16.3.1 Lennard-Jones Potential  386

16.4 Blackbody Radiation   . . 388

16.5 Fermi-Dirac (F-D) Statistics  . . 393

17 Miscellaneous 401

17.1 Volume of a Hypersphere of Dimension N  . . . 402CONTENTS 13

17.1.1 Spherical Coordinates  . 402

17.1.2 Calculation   . . . 404

17.1.3 Discussion  407

17.1.4 Applications   . . . 409

17.1.5 Mathematical Connections    410

V Appendices 413

Appendix A 415

Appendix B 421

Acknowledgements 425

Answers 427

Integral Table 435

Trigonometric Identities 439

Alphabetical Index 443