MATERIAL GEOMETRY: GROUPOIDS IN CONTINUUM MECHANICS

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This monograph is the first in which the theory of groupoids and algebroids is applied to the study of the properties of uniformity and homogeneity of continuous media. It is a further step in the application of differential geometry to the mechanics of continua, initiated years ago with the introduction of the theory of G-structures, in which the group G denotes the group of material symmetries, to study smoothly uniform materials. The new approach presented in this book goes much further by being much more general. It is not a generalization per se, but rather a natural way of considering the algebraic-geometric structure induced by the so-called material isomorphisms. This approach has allowed us to encompass non-uniform materials and discover new properties of uniformity and homogeneity that certain material bodies can possess, thus opening a new area in the discipline. Sample Chapter(s) Preface Chapter 2: Continuum Mechanics: Elastic Simple Bodies Contents: Preface About the Authors Introduction Fundamentals: Continuum Mechanics: Elastic Simple Bodies Groupoids Algebroids Material Groupoid: Material Algebroid Characteristic Distributions and Material Bodies Appendices: Foliations and Distributions Covariant Derivatives Principal Bundles and Connections Bibliography Index Readership: Graduate and postgraduate students interested in Continuum Mechanics, Mathematical Physics and Differential Geometry. Researchers in Elasticity, Applied Mathematics and Differential Geometry. And those taking a master or doctorate course that seeks the interaction between mathematics and mechanical engineering.

JOURNEY THROUGH THE WONDERS OF PLANE GEOMETRY, A

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Geometry is one of the most beautiful aspects of mathematics. This beauty is because you can "see" geometry at work. Most people are exposed to the very basic elements of geometry throughout their schooling with the most concentrated study in the secondary school curriculum. High schools in the United States offer one year of concentrated study of geometry that shows students how a mathematician functions, since everything that is accepted beyond the basic axioms must be proved. Unfortunately, as the course is only one year long, there is still very much in geometry left unexplored for the general audience. That is the challenge of this book, in which we will present a plethora of amazing geometric relationships. We begin with the special relationships of the Golden Ratio, before considering unexpected concurrencies and collinearities. Next, we present some surprising results that arise when squares and similar triangles are placed on triangle sides, followed by a discussion of concyclic points and the relationships between circles and various linear figures. Moving on to more advanced aspects of linear geometry, we consider the geometric wonders of polygons. Finally, we address geometric surprises and fallacies, before concluding with a chapter on the useful concept of homothety, which is not included in the American year-long course in geometry. Request Inspection Copy Sample Chapter(s) Introduction Chapter 9: Geometric Surprises Contents: About the Authors Introduction The Golden Ratio in Geometry Unexpected Concurrencies Unexpected Collinearities Squares on Triangle Sides Similar Triangles on Triangle Sides Discovering Concyclic Points Circle Wonders Polygons and Polygrams Geometric Surprises Geometric Fallacies Homothety, Similarity, and Applications Index Readership: This book is suitable for secondary/high school level students and teachers, as well as a general readership, particularly those interested in bridging the gaps in their knowledge of geometry.

Solving Problems in Point Geometry (1版)

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【簡介】 For over two millennia, the complexities of elementary geometry have challenged learners, burdened by the intricacies of auxiliary graphics and cumbersome calculations. Inspired by Leibniz's query, this book introduces a groundbreaking method: point geometry. By operating directly on points, it integrates the strengths of coordinate, vector, and mass point methods, simplifying operations and problem-solving. Central to this method is the identity approach, which streamlines complex problems into concise equations, unlocking multiple propositions with ease. Through meticulously crafted examples, readers are invited to explore the joy of mathematical thinking. Beyond mathematics, point geometry holds promise for artificial intelligence, offering a simple yet rich knowledge representation and reasoning method. With most solutions generated by computer programs, the potential for simplifying reasoning methods is immense, paving the way for a brighter future in both education and AI advancement. In this ambitious endeavor, the authors seek to simplify knowledge representation and reasoning, reduce the burden of learning, and accelerate the progress of artificial intelligence. This book is not just a guide to geometry; it's a catalyst for transformative thinking and discovery. Request Inspection Copy 【目錄】