Elements of Mathematical Analysis

ISBN-13: 9783031458538 / Angielski / Miękka / 2023

Costas J. Papachristou

Opis

This book provides a comprehensive yet informal introduction to differentiating and integrating real functions with one variable. It also covers basic first-order differential equations and introduces higher-dimensional differentiation and integration.
 The focus is on significant theoretical proofs, accompanied by illustrative examples for clarity. A comprehensive bibliography aids deeper understanding. The concept of a function's differential is a central theme, relating to the "differential" within integrals. The discussion of indefinite integrals (collections of antiderivatives) precedes definite integrals, naturally connecting the two. The Appendix offers essential math formulas, exercise properties, and an in-depth exploration of continuity and differentiability. Select exercise solutions are provided.  This book suits short introductory math courses for novice physics/engineering students. It equips them with vital differential and integral calculus tools for real-world applications. It is also useful for first-year undergraduates, reinforcing advanced calculus foundations for better Physics comprehension.

Spis treści

1. FUNCTIONS 

1.1  Real Numbers   

1.2  Functions   

1.3  Domain of Definition of a Function  

1.4  Implicit and Multiple-Valued Functions 

1.5  Exponential and Logarithmic Functions   

1.6  Linear Function   

1.7  Quadratic Function   

1.8  Even and Odd Functions  

1.9  Periodic Functions    

1.10  Inverse Function    

1.11  Monotonicity of a Function    

2. DERIVATIVE AND DIFFERENTIAL

2.1  Definition   

2.2  Differentiation Rules    

2.3  Derivatives of Trigonometric Functions   

2.4  Table of Derivatives of Elementary Functions    

2.5  Derivatives of Composite Functions    

2.6  Derivatives of Functions of the Form  y=[f(x)]φ(x)    

2.7  Differential of a Function    

2.8  Differential Operators    

2.9  Derivative of a Composite Function by Using the Differential    

2.10  Geometrical Significance of the Derivative and the Differential    

2.11  Higher-Order Derivatives    

2.12  Derivatives of Implicit Functions    

3. SOME APPLICATIONS OF DERIVATIVES

3.1  Tangent and Normal Lines on Curves    

3.2  Angle of Intersection of Two Curves    

3.3  Maximum and Minimum Values of a Function    

3.4  Indeterminate Forms and L’Hospital’s Rule    

4. INDEFINITE INTEGRAL

4.1  Antiderivatives of a Function    

4.2  The Indefinite Integral    

4.3  Basic Integration Rules    

4.4  Integration by Substitution (Change of Variable)    

4.5  Integration by Parts (Partial Integration)   

4.6  Integration of Rational Functions    

5. DEFINITE INTEGRAL

5.1  Definition and Properties    

5.2  Integration by Substitution    

5.3  Integration of Even, Odd and  Periodic Functions    

5.4  Integrals with Variable Limits   

5.5  Improper Integrals: Infinite Limits   

5.6  Improper Integrals: Unbounded Integrand    

5.7  The Definite Integral as a Plane Area   

6. SERIES

6.1  Series of Constants    

6.2  Positive Series    

6.3  Absolutely Convergent Series   

6.4  Functional Series   

6.5  Expansion of Functions into Power Series    

7. AN ELEMENTARY INTRODUCTION TO DIFFERENTIAL EQUATIONS

7.1  Two Basic Theorems  

7.2  First-Order Differential Equations   

7.3  Some Special Cases   

7.4  Examples  

8. INTRODUCTION TO DIFFERENTIATION IN HIGHER DIMENSIONS

8.1  Partial Derivatives and Total Differential    

8.2  Exact Differential Equations   

8.3  Integrating Factor    

8.4  Line Integrals on the Plane   

9. COMPLEX NUMBERS       

9.1  The Notion of a Complex Number    

9.2  Polar Form of a Complex Number    

9.3  Exponential Form of a Complex Number  

9.4  Powers and Roots of Complex Numbers  

10. INTRODUCTION TO COMPLEX ANALYSIS   

10.1  Analytic Functions and the Cauchy-Riemann Relations    

10.2  Integrals of Complex Functions   

10.3  The Cauchy-Goursat Theorem    

10.4  Indefinite Integral of an Analytic Function  

APPENDIX

ANSWERS TO SELECTED EXERCISES

SELECTED BIBLIOGRAPHY

INDEX

O autorze

Dr. Costas J. Papachristou is a senior faculty member at the Hellenic Naval Academy (The Naval Academy of Greece). He received a Ph.D. in Theoretical Physics from Brigham Young University, USA, and spent another year at BYU as a research faculty in Physics. At the HNA, he teaches freshman- and sophomore-level Physics. His research interests are mainly in Theoretical and Mathematical Physics (in particular, symmetry and integrability properties of classical field equations, classical electromagnetism), but he is also active in Physics Education. He is an amateur musician and a columnist for Greek newspapers and magazines.