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

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

## Information

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

## 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.

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