Anthony Tromba Livres

This advanced-level study approaches mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is directed to mathematicians, engineers and physicists who wish to see this classical subject in a modern setting with examples of newer mathematical contributions. Prerequisites include a solid background in advanced calculus and the basics of geometry and functional analysis.The first two chapters cover the background geometry ― developed as needed ― and use this discussion to obtain the basic results on kinematics and dynamics of continuous media. Subsequent chapters deal with elastic materials, linearization, variational principles, the use of functional analysis in elasticity, and bifurcation theory. Carefully selected problems are interspersed throughout, while a large bibliography rounds out the text.

The goal of this text is to help students learn to use calculus intelligently for solving a wide variety of mathematical and physical problems. Examples and Exercises The exercise sets have been carefully constructed to be of maximum use to the students.

This volume is the first in a three-part treatise on minimal surfaces, focusing on boundary value problems. It serves as a revised and expanded version of earlier monographs. The book opens with fundamental concepts of surface theory in three-dimensional Euclidean space, introducing minimal surfaces as stationary points of area or surfaces with zero mean curvature. A minimal surface is defined as a nonconstant harmonic mapping that is conformally parametrized and may have branch points. The classical theory of minimal surfaces is explored, featuring numerous examples, Björling’s initial value problem, reflection principles, and important theorems by Bernstein, Heinz, Osserman, and Fujimoto. The second part addresses Plateau’s problem and its modifications, presenting a new elementary proof that the area and Dirichlet integral share the same infimum for admissible surfaces spanning a prescribed contour. This leads to a simplified solution for minimizing both area and Dirichlet integral, along with new proofs of Riemann and Korn-Lichtenstein's mapping theorems, and a solution to the simultaneous Douglas problem for contours with multiple components. The volume also covers stable minimal surfaces, deriving curvature estimates and presenting uniqueness and finiteness results. Additionally, it develops a theory of unstable solutions to Plateau’s problems based on Courant’s mountain pass lemma and solves Dirichlet’s problem for non

Focusing on minimal surfaces with free boundaries, the book explores their boundary behavior and presents key results, including asymptotic expansions and Gauss-Bonnet formulas. It tackles the challenges of deriving regularity proofs for non-minimizers through indirect reasoning and monotonicity formulas. Geometric properties, enclosure theorems, and isoperimetric inequalities are examined, alongside discussions of obstacle problems and Plateau’s problem in Riemannian manifolds. The final chapter introduces a novel approach to the absence of interior branch points in area-minimizing solutions.

The exploration of minimal surfaces is deepened through a focus on existence, regularity, and uniqueness theorems for surfaces with partially free boundaries, highlighting the concept of "edge-crawling." A priori estimates for higher-dimensional minimal surfaces and singular integral minimizers are also discussed, leading to significant Bernstein theorems. Additionally, the book presents a comprehensive global theory, addressing the Douglas problem with Teichmüller theory, deriving index theorems, and introducing a topological perspective through Fredholm vector fields, all reflecting Smale's vision.

The MznLnx Exam Prep series is designed to help you pass your exams. Editors at MznLnx review your textbooks and then prepare these practice exams to help you master the textbook material. Unlike study guides, workbooks, and practice tests provided by the texbook publisher and textbook authors, MznLnx gives you all of the material in each chapter in exam form, not just samples, so you can be sure to nail your exam.

One of the most elementary questions in mathematics is whether an area minimizing surface spanning a contour in three space is immersed or not; i.e. does its derivative have maximal rank everywhere. The purpose of this monograph is to present an elementary proof of this very fundamental and beautiful mathematical result. The exposition follows the original line of attack initiated by Jesse Douglas in his Fields medal work in 1931, namely use Dirichlet's energy as opposed to area. Remarkably, the author shows how to calculate arbitrarily high orders of derivatives of Dirichlet's energy defined on the infinite dimensional manifold of all surfaces spanning a contour, breaking new ground in the Calculus of Variations, where normally only the second derivative or variation is calculated. The monograph begins with easy examples leading to a proof in a large number of cases that can be presented in a graduate course in either manifolds or complex analysis. Thus this monograph requires only the most basic knowledge of analysis, complex analysis and topology and can therefore be read by almost anyone with a basic graduate education.