Advanced Calculus - Fundamentals of Mathematics

by

Carlos Polanco

DOI: 10.2174/97898114150811190101
eISBN: 978-981-14-1508-1, 2019
ISBN: 978-981-14-1507-4



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Vector calculus is an essential mathematical tool for performing mathematical analysis of physical and natural phenomena. It is employ...[view complete introduction]
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Table of Contents

Foreword I

- Pp. i
Vladimir N. Uversky
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Foreword II

- Pp. ii
Thomas Buhse
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Preface

- Pp. iii-iv (2)
Carlos Polanco
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Acknowledgements

- Pp. v
Carlos Polanco
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List of Credits

- Pp. vi-viii (3)
Carlos Polanco
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List of Symbols

- Pp. ix-xiii (5)
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Vector Algebra

- Pp. 1-20 (20)
Carlos Polanco

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Real–Valued Functions

- Pp. 21-45 (25)
Carlos Polanco

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Critical Points

- Pp. 46-63 (18)
Carlos Polanco

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Vector-Valued Functions

- Pp. 64-74 (11)
Carlos Polanco

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Integration Over Bounded Regions

- Pp. 75-94 (20)
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Integration Over Unbounded Regions

- Pp. 95-102 (8)
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Integration Over Curves and Surfaces

- Pp. 103-112 (10)
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Theorems of Vector Calculus

- Pp. 113-123 (11)
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Geometric Algebra

- Pp. 124-138 (15)
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Theorems of Differential Forms

- Pp. 139-148 (10)
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Solutions for Chapters

- Pp. 149-176 (28)
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References

- Pp. 177-179 (3)
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Subject Index

- Pp. 180-183 (4)
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Foreword

FOREWORD I

In his book, Dr. Carlos Polanco elegantly describes fundamentals of Vector Calculus and its applications. The author was able to overcome the usual gap between mathematicians and users, describing the main topics related to Vector Calculus Theory in an easily comprehensible way utilising multiple useful examples and providing exercises. This book can be used as an auxiliary book for students interested in this field as well as a reference book for the seasoned Researcher.

Vladimir N. Uversky
Russian Academy of Sciences
Pushchino, Moscow region, Russia

FOREWORD II

With Advanced Calculus, Carlos Polanco introduces an experienced view on Vector Calculus that presents a valuable reference for students as well as for their teachers. This book is very well structured, nicely written and provides a comprehensive insight into the complexity of this field. Especially, the presentation of many examples and case studies will help the readers to deepen their acquired knowledge and to relate the theory to practice. It will certainly also help researchers in related fields to refresh their knowledge and to serve as a solid and clear source on Vector Calculus. Rounding up, Carlos Polanco’s book should become part of many bookshelves.

Thomas Buhse
Universidad Autonoma del Estado de Morelos
Cuernavaca Morelos, Mexico


Preface

Vector calculus is an essential mathematical tool for physical and natural phenomena analysis; it is a very important subject for students in the Faculties of Science and Engineering. This ebook has been designed to cover two academic semesters for second-year students, as it contains the fundamentals related to Vector Calculus. Here, the reader will find a concise and clear study of this mathematical field, it provides many examples, exercises, and a case study in each chapter. The solutions to the exercises are also included at the end of the ebook. The reader will be able to understand it even without a rigorous mathematical knowledge and will be able to immediately practise the concepts. Its main purpose is to enable students to put into practise what they are learning about the subject

This ebook on Vector Calculus differs from others in the intensive use of affine transformations (or mapping), to modify the integration region of real-valued functions or vector-valued functions.

The first part of the ebook introduces the Vector Algebra over the real coordinate space Rn, but cases and examples are displayed on the plane and space for a better understanding of the concepts. In this part is reviewed the dot and cross product operators, the orthogonal projection, various representations and conversions of the Cartesian and Polar coordinate systems on a plane, and the Cartesian, cylindrical, and spherical coordinate systems in a space are reviewed.

The second part is dedicated to real-valued functions, level surfaces, domain, image, graphic representations, properties of limits, continuity, differentiability, gradient, and polynomial approximation as well as the identification of critical points, the use of the Hessian matrix and Bordered Hessian matrix to characterise critical points, the implicit Function Theorem, and the inverse Function Theorem. The final chapter of this part introduces the vector-valued functions, graphic representations, and the rotational and divergence operators.

The third part presents the evaluation of integrals from mapping on a plane and in space. It separately reviews integrals of real-valued functions over bounded and unbounded regions, and curve and surface integrals of vector-valued functions over vector fields. The concept of mapping is carefully introduced, emphasising how adequate mapping simplifies integrals.

The fourth part reviews the Stokes, Gauss, and Green theorems applying mapping. It also includes examples with the complete resolution of both sides of the equalities that these theorems state.

The fifth part is a survey of exterior product and it is based on the books of Hermann Grassmann, Jerrold Marsden, and David Hestenes. We greatly appreciate the recommendation of Prof. Hestenes to include this material.

The last part covers the differential forms and the exterior product focused on line integral, surface integral, the Stokes, Gauss, and Green theorems.

The author hopes the reader interested in studying the fundamentals of calculus on several variables, finds useful the material presented here and that those who start studying this field find this information motivating. The author would like to acknowledge the Faculty of Sciences at Universidad Nacional Autónoma de México for support.

Consent for Publication

Not applicable.

Conflict of Interest

The authors confirm that this article content has no conflict of interest.



Carlos Polanco
Department of Mathematics
Faculty of Sciences
Universidad Nacional Autónoma de México
México City
México

List of Contributors

Author(s):
Carlos Polanco




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