Author: Luigi Morino Publisher: Springer Nature ISBN: 3662632071 Category : Technology & Engineering Languages : en Pages : 1019
Book Description
Mathematics plays an important role in mechanics and other human endeavours. Validating examples in this first volume include, for instance: the connection between the golden ratio (the “divine proportion" used by Phidias and many other artists and enshrined in Leonardo's Vitruvian Man, shown on the front cover), and the Fibonacci spiral (observable in botany, e.g., in the placement of sunflower seeds); is the coast of Tuscany infinitely long?; the equal-time free fall of a feather and a lead ball in a vacuum; a simple diagnostic for changing your car's shocks; the Kepler laws of the planets; the dynamics of the Sun-Earth-Moon system; the tides' mechanism; the laws of friction and a wheel rolling down a partially icy slope; and many more. The style is colloquial. The emphasis is on intuition - lengthy but intuitive proofs are preferred to simple non-intuitive ones. The mathematical/mechanical sophistication gradually increases, making the volume widely accessible. Intuition is not at the expense of rigor. Except for grammar-school material, every statement that is later used is rigorously proven. Guidelines that facilitate the reading of the book are presented. The interplay between mathematics and mechanics is presented within a historical context, to show that often mechanics stimulated mathematical developments - Newton comes to mind. Sometimes mathematics was introduced independently of its mechanics applications, such as the absolute calculus for Einstein's general theory of relativity. Bio-sketches of all the scientists encountered are included and show that many of them dealt with both mathematics and mechanics.
Author: Saunders MacLane Publisher: Springer Science & Business Media ISBN: 1461248728 Category : Mathematics Languages : en Pages : 486
Book Description
This book records my efforts over the past four years to capture in words a description of the form and function of Mathematics, as a background for the Philosophy of Mathematics. My efforts have been encouraged by lec tures that I have given at Heidelberg under the auspices of the Alexander von Humboldt Stiftung, at the University of Chicago, and at the University of Minnesota, the latter under the auspices of the Institute for Mathematics and Its Applications. Jean Benabou has carefully read the entire manuscript and has offered incisive comments. George Glauberman, Car los Kenig, Christopher Mulvey, R. Narasimhan, and Dieter Puppe have provided similar comments on chosen chapters. Fred Linton has pointed out places requiring a more exact choice of wording. Many conversations with George Mackey have given me important insights on the nature of Mathematics. I have had similar help from Alfred Aeppli, John Gray, Jay Goldman, Peter Johnstone, Bill Lawvere, and Roger Lyndon. Over the years, I have profited from discussions of general issues with my colleagues Felix Browder and Melvin Rothenberg. Ideas from Tammo Tom Dieck, Albrecht Dold, Richard Lashof, and Ib Madsen have assisted in my study of geometry. Jerry Bona and B.L. Foster have helped with my examina tion of mechanics. My observations about logic have been subject to con structive scrutiny by Gert Miiller, Marian Boykan Pour-El, Ted Slaman, R. Voreadou, Volker Weispfennig, and Hugh Woodin.
Author: Apostolos Doxiadis Publisher: Princeton University Press ISBN: 1400842689 Category : Mathematics Languages : en Pages : 593
Book Description
Why narrative is essential to mathematics Circles Disturbed brings together important thinkers in mathematics, history, and philosophy to explore the relationship between mathematics and narrative. The book's title recalls the last words of the great Greek mathematician Archimedes before he was slain by a Roman soldier—"Don't disturb my circles"—words that seem to refer to two radically different concerns: that of the practical person living in the concrete world of reality, and that of the theoretician lost in a world of abstraction. Stories and theorems are, in a sense, the natural languages of these two worlds—stories representing the way we act and interact, and theorems giving us pure thought, distilled from the hustle and bustle of reality. Yet, though the voices of stories and theorems seem totally different, they share profound connections and similarities. A book unlike any other, Circles Disturbed delves into topics such as the way in which historical and biographical narratives shape our understanding of mathematics and mathematicians, the development of "myths of origins" in mathematics, the structure and importance of mathematical dreams, the role of storytelling in the formation of mathematical intuitions, the ways mathematics helps us organize the way we think about narrative structure, and much more. In addition to the editors, the contributors are Amir Alexander, David Corfield, Peter Galison, Timothy Gowers, Michael Harris, David Herman, Federica La Nave, G.E.R. Lloyd, Uri Margolin, Colin McLarty, Jan Christoph Meister, Arkady Plotnitsky, and Bernard Teissier.
Author: Mark Levi Publisher: Princeton University Press ISBN: 1400830478 Category : Mathematics Languages : en Pages : 197
Book Description
Everybody knows that mathematics is indispensable to physics--imagine where we'd be today if Einstein and Newton didn't have the math to back up their ideas. But how many people realize that physics can be used to produce many astonishing and strikingly elegant solutions in mathematics? Mark Levi shows how in this delightful book, treating readers to a host of entertaining problems and mind-bending puzzlers that will amuse and inspire their inner physicist. Levi turns math and physics upside down, revealing how physics can simplify proofs and lead to quicker solutions and new theorems, and how physical solutions can illustrate why results are true in ways lengthy mathematical calculations never can. Did you know it's possible to derive the Pythagorean theorem by spinning a fish tank filled with water? Or that soap film holds the key to determining the cheapest container for a given volume? Or that the line of best fit for a data set can be found using a mechanical contraption made from a rod and springs? Levi demonstrates how to use physical intuition to solve these and other fascinating math problems. More than half the problems can be tackled by anyone with precalculus and basic geometry, while the more challenging problems require some calculus. This one-of-a-kind book explains physics and math concepts where needed, and includes an informative appendix of physical principles. The Mathematical Mechanic will appeal to anyone interested in the little-known connections between mathematics and physics and how both endeavors relate to the world around us.
Author: Dr. Gérard G. Emch Publisher: Courier Corporation ISBN: 0486151719 Category : Science Languages : en Pages : 336
Book Description
This systematic algebraic approach offers a careful formulation of the problems' physical motivations as well as self-contained descriptions of the mathematical methods for arriving at solutions. 1972 edition.
Author: David M. Bressoud Publisher: Springer Science & Business Media ISBN: 1461209595 Category : Mathematics Languages : en Pages : 399
Book Description
Second Year Calculus: From Celestial Mechanics to Special Relativity covers multi-variable and vector calculus, emphasizing the historical physical problems which gave rise to the concepts of calculus. The book guides us from the birth of the mechanized view of the world in Isaac Newton's Mathematical Principles of Natural Philosophy in which mathematics becomes the ultimate tool for modelling physical reality, to the dawn of a radically new and often counter-intuitive age in Albert Einstein's Special Theory of Relativity in which it is the mathematical model which suggests new aspects of that reality. The development of this process is discussed from the modern viewpoint of differential forms. Using this concept, the student learns to compute orbits and rocket trajectories, model flows and force fields, and derive the laws of electricity and magnetism. These exercises and observations of mathematical symmetry enable the student to better understand the interaction of physics and mathematics.
Author: S. L. Sobolev Publisher: Courier Corporation ISBN: 9780486659640 Category : Science Languages : en Pages : 452
Book Description
This volume presents an unusually accessible introduction to equations fundamental to the investigation of waves, heat conduction, hydrodynamics, and other physical problems. Topics include derivation of fundamental equations, Riemann method, equation of heat conduction, theory of integral equations, Green's function, and much more. The only prerequisite is a familiarity with elementary analysis. 1964 edition.
Author: Avi Wigderson Publisher: Princeton University Press ISBN: 0691189137 Category : Computers Languages : en Pages : 434
Book Description
From the winner of the Turing Award and the Abel Prize, an introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography
Author: Anatole Katok Publisher: Cambridge University Press ISBN: 9780521575577 Category : Mathematics Languages : en Pages : 828
Book Description
This book provided the first self-contained comprehensive exposition of the theory of dynamical systems as a core mathematical discipline closely intertwined with most of the main areas of mathematics. The authors introduce and rigorously develop the theory while providing researchers interested in applications with fundamental tools and paradigms. The book begins with a discussion of several elementary but fundamental examples. These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. Over 400 systematic exercises are included in the text. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate up.
Author: A. A. Kirillov Publisher: Springer Science & Business Media ISBN: 1461381533 Category : Mathematics Languages : en Pages : 351
Book Description
Even the simplest mathematical abstraction of the phenomena of reality the real line-can be regarded from different points of view by different mathematical disciplines. For example, the algebraic approach to the study of the real line involves describing its properties as a set to whose elements we can apply" operations," and obtaining an algebraic model of it on the basis of these properties, without regard for the topological properties. On the other hand, we can focus on the topology of the real line and construct a formal model of it by singling out its" continuity" as a basis for the model. Analysis regards the line, and the functions on it, in the unity of the whole system of their algebraic and topological properties, with the fundamental deductions about them obtained by using the interplay between the algebraic and topological structures. The same picture is observed at higher stages of abstraction. Algebra studies linear spaces, groups, rings, modules, and so on. Topology studies structures of a different kind on arbitrary sets, structures that give mathe matical meaning to the concepts of a limit, continuity, a neighborhood, and so on. Functional analysis takes up topological linear spaces, topological groups, normed rings, modules of representations of topological groups in topological linear spaces, and so on. Thus, the basic object of study in functional analysis consists of objects equipped with compatible algebraic and topological structures.
Author: Luigi Morino
Author: Saunders MacLane
Author: Apostolos Doxiadis
Author: Mark Levi
Author: Dr. Gérard G. Emch
Author: David M. Bressoud
Author: S. L. Sobolev
Author: Avi Wigderson
Author: Anatole Katok
Author: A. A. Kirillov