Celso Melchiades Doria - Differentiability in Banach Spaces, Differential Forms and Applications

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Chapters flowchart

There are also three appendices to make the textbook as self-contained as possible, and to support the reader in gaining an understanding of the basic terminology, the notations, the concepts and some basic theorems used in the text.

The appendices are not supposed to be read; instead they should serve as a guiding tool to recall the basics according to the needs of the readers. In Appendix A, we fix some notations, enunciate some theorems largely used throughout the other chapters, and further develop some elementary content according to our needs. In this way, Appendix A is recommended as a reference source for prerequisites and references. Appendix B introduces very basic concepts from the differentiable manifold and Lie groups. Appendix C deals with tensor algebra, which is a basic concept to have a full understanding of the content in Chap..

At the turn of the century, at the end of 1999, there were many speculations about results of greater importance that were reached in the 1st Millenium. One day, while waiting for a medical appointment, there was a magazine listing some results that were considered to be of great significance among those obtained for the development of human knowledge. To my surprise and joy, one of the noted results was the Fundamental Theorem of Calculus. I had not thought of this possibility, but I immediately agreed to its inclusion, not only because I am a mathematician but also because it is a fact that all of Classical Mechanics, Thermodynamics and Electromagnetism were developed using calculus. Consequently, technological advancements achieved in the exact sciences and social sciences depended largely on the development of calculus.

In general, when we refer to calculus, we are discussing the techniques of differentiation and integration. In most textbooks, the study of the derivative precedes that of the integral, but not historically. The method of exhaustion developed by Archimedes was an infinite sum process, analogous to that used nowadays to define the integral of a function.

The concept of derivative of a function appeared in the sixteenth century after the emergence of analytical geometry to calculate the relative rate of change of a quantity. In the period noted, many ideas in physics were evolving rapidly due to the scientific method. In this period, the development of mechanics was latent. The Pioneers of Calculus as we know it today were Sir Isaac Newton (16421727), who developed the Method of Flows, and Gottfried Wilhelm von Leibniz (16461716) who developed Calculus, as he named it, and also gave a good part of the notation used to this day. Newton discovered the basic Laws of Classical Mechanics, then applied them together with the Method of Flows to demonstrate Keplers Laws.

Newtons 2nd Law states that a Force on a body of mass generates a relative rate of change of velocity in relation to time. More precisely, in our current mathematical language, the second Law states that the force acting on a body of mass is given by