Overview

This book, Differential and Integral Calculus - Real Functions of One or Several Real Variables, presents the fundamental concepts of differential and integral calculus. It is intended for advanced mathematics students as well as professionals requiring a solid command of mathematical analysis tools for technical or scientific tasks. The book is divided into six major parts: Differential Calculus: Chapters covering the fundamentals of real functions of one variable, normed vector spaces, and differential operators. Each section is accompanied by solved exercises to reinforce understanding. Integral Calculus: In-depth discussions on integrals of multivariable functions, line integrals, and surface integrals. Practical exercises to illustrate physical and engineering applications. Calculus of Variations and Differential Equations: Exploration of the principles of calculus of variations, existence and uniqueness theorems, and dynamical systems. Application of Fourier analysis to evolution equations, with solved exercises to strengthen comprehension. Analysis on Differential Manifolds: Introduction to differential manifolds, tensor calculus, and Morse theory, with applications in general relativity and geometry. Each chapter is followed by solved exercises, allowing mastery of advanced concepts. Numerical Methods and Integration Schemes: Presentation of discretization methods, integration schemes, and advanced numerical methods such as finite elements and spectral methods. Practical exercises for solving problems in fluid dynamics and structural mechanics. Stochastic Calculus and Applications: Introduction to stochastic processes and stochastic differential equations, with applications in finance, biology, and physics. Exercises to apply stochastic calculus to random models and control processes. Conclusion and Appendices: The book concludes with a chapter dedicated to multivariable integration theorems, including Green's, Stokes', and Gauss' theorems, and their extensions to higher dimensions. The appendices provide a review of fundamental theorems in functional analysis, such as the best approximation theorem, Riesz's lemma, and the Arzelà-Ascoli theorem. This book is thus a comprehensive and structured guide for anyone seeking to master differential and integral calculus, with particular attention to practical applications in various scientific and technical fields.

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