Bøger af Alberto Guzmán

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Este libro va dirigido a padres, educadores, profesionales, cuidadores y cualquiera que esté interesado en conocer y afrontar la tarea de educar con solidez, potenciando las naturales capacidades de los niños y siempre con la vista puesta en su larga etapa de adultos.

This text is appropriate for a one-semester course in what is usually called ad- vanced calculus of several variables. The focus is on expanding the concept of continuity; specifically, we establish theorems related to extreme and intermediate values, generalizing the important results regarding continuous functions of one real variable. We begin by considering the function f(x, y, ... ) of multiple variables as a function of the single vector variable (x, y, ... ). It turns out that most of the n treatment does not need to be limited to the finite-dimensional spaces R , so we will often place ourselves in an arbitrary vector space equipped with the right tools of measurement. We then proceed much as one does with functions on R. First we give an algebraic and metric structure to the set of vectors. We then define limits, leading to the concept of continuity and to properties of continuous functions. Finally, we enlarge upon some topological concepts that surface along the way. A thorough understanding of single-variable calculus is a fundamental require- ment. The student should be familiar with the axioms of the real number system and be able to use them to develop elementary calculus, that is, to define continuous junction, derivative, and integral, and to prove their most important elementary properties. Familiarity with these properties is a must. To help the reader, we provide references for the needed theorems.

This text is appropriate for a one-semester course in what is usually called ad- vanced calculus of several variables. The approach taken here extends elementary results about derivatives and integrals of single-variable functions to functions in several-variable Euclidean space. The elementary material in the single- and several-variable case leads naturally to significant advanced theorems about func- tions of multiple variables. In the first three chapters, differentiability and derivatives are defined; prop- erties of derivatives reducible to the scalar, real-valued case are discussed; and two results from the vector case, important to the theoretical development of curves and surfaces, are presented. The next three chapters proceed analogously through the development of integration theory. Integrals and integrability are de- fined; properties of integrals of scalar functions are discussed; and results about scalar integrals of vector functions are presented. The development of these lat- ter theorems, the vector-field theorems, brings together a number of results from other chapters and emphasizes the physical applications of the theory.