Papers by Nestor Eduardo Gutierrez camayo
Hello! I'm not the book Author, my interest i'ts only share this book and the others in this webs... more Hello! I'm not the book Author, my interest i'ts only share this book and the others in this website to the entire people!
Los elementos fundamentales del análisis vectorial son los vectores y los escalares. Usaremos la ... more Los elementos fundamentales del análisis vectorial son los vectores y los escalares. Usaremos la notación R para denotar la recta numérica que se asocia con el conjunto de números reales, R 2 para denotar el plano cartesiano y R 3 para el espacio ordinario en tres dimensiones. Leyes del álgebra vectorial El teorema siguiente es válido: teorema 1.1: Suponga que A, B y C son vectores y m y n son escalares. Entonces se cumplen las siguientes leyes: [A 1 ] (A 1 B) 1 C 5 (A 1 B) 1 C Ley asociativa para la suma [A 2 ] Existe un vector cero, 0, tal que para todo vector A, A 1 0 5 0 1 A 5 A Existencia del elemento cero [A 3 ] Para todo vector A, existe un vector 2A tal que A 1 (2A) 5 (2A) 1 A 5 0 Existencia de los negativos [A 4 ] A 1 B 5 B 1 A Ley conmutativa para la suma [M 1 ] m(A 1 B) 5 mA 1 mB Ley distributiva [M 2 ] (m 1 n)A 5 mA 1 nA Ley distributiva [M 3 ] m(nA) 5 (mn)A Ley asociativa [M 4 ] 1(A) 5 A Multiplicación por la unidad Las ocho leyes anteriores son los axiomas que dei nen una estructura abstracta llamada espacio vectorial. Dichas leyes se agrupan en dos conjuntos, indicados por sus leyendas. Las primeras cuatro leyes se rei eren a la suma de vectores. Con ellas es posible demostrar las propiedades siguientes de la suma de vectores. a) Cualquier suma A 1 1 A 2 1 … 1 A n de vectores no requiere de paréntesis y no depende del orden de los sumandos. b) El vector cero, 0, es único y el negativo, 2A, de un vector A es único. c) (Ley de cancelación). Si A 1 C 5 B 1 C, entonces A 5 B. Las cuatro leyes restantes se rei eren a la multiplicación por un escalar. Con su empleo se demuestran las propiedades siguientes. proposición 1.2: a) Para cualquier escalar m y el vector cero, 0, se cumple que m0 5 0. b) Para cualquier vector A y el escalar 0, se cumple que 0A 5 0. c) Si mA 5 0, entonces m 5 0 o A 5 0. d) Para cualquier vector A y un escalar m, se cumple que (2m)A 5 m(2A) 5 2(mA). 1.3 VECTORES UNITARIOS Los vectores unitarios son aquellos que tienen una longitud igual a uno. Suponga que A es un vector cualquiera con longitud)A) > 0. Entonces A/)A) es un vector unitario, denotado por a, que tiene la misma dirección que A. Asimismo, cualquier vector A puede representarse con un vector unitario a en la dirección de A multiplicado por la magnitud de A. Es decir, A 5)A)a. EJEMPLO 1.1 Suponga que)A) 5 3. Entonces a 5)A)/3 es un vector unitario en la dirección de A. También se cumple que A 5 3a. 1.4 LOS VECTORES UNITARIOS RECTANGULARES: i, j, k Un conjunto importante de vectores unitarios, denotados por i, j y k, son aquellos que tienen las direcciones de los ejes x, y y z, respectivamente, de un sistema de coordenadas rectangulares de tres dimensiones. [Vea la i gura 1-4a).] El sistema de coordenadas que se ilustra en la i gura 1-4a), que será el que usemos a menos que se especii que otro caso, se denomina sistema de coordenadas de la mano derecha. El sistema se caracteriza por la siguiente propiedad: si doblamos los dedos de la mano derecha con un giro de 90° a partir del eje positivo de las x y hacia el eje y positivo, entonces el pulgar apuntará en la dirección del eje positivo de las z.
Small Projects-These are optional activities that may take up to a few weeks to complete. Large P... more Small Projects-These are optional activities that may take up to a few weeks to complete. Large Projects-These could be a semester-long project, a capstone project, or an honors thesis. Additional Exercises-Additional exercises may include any of the previous activities. They appear at the end of chapters and are optional. xi Chapter 1: Complex Dynamics This chapter investigates chaos and fractals as they relate to dynamical systems that come from iterating complex-valued functions, i.e., given an initial value z 0 we consider the values z 1 D f .z 0 /; z 2 D f .z 1 / D f .f .z 0 //, z 3 D f .z 2 / D f .f .f .z 0 ///; : : : , and how the sequence fz n g behaves. This kind of iteration arises in Newton's method for approximating roots, and so our chapter begins by asking: Which initial values will work for Newton's method (i.e., converge to a root)? If the initial value z 0 is changed slightly, will similar or drastically different behavior result? The questions are considered computationally, visually, and experimentally with the aid of computer applets. We then consider the iteration of any complex analytic map, which leads to the mathematics behind the Mandelbrot set, and much more. Chapter2: Soap Films, Differential Geometry, and Minimal Surfaces Dipping a wire frame into soapy water produces an iridescent soap film that clings to the frame. Such soap films are related to minimal surfaces, which are beautiful geometric objects that minimize surface area locally. Visually, minimal surfaces can be thought of as saddle surfaces that bend upward in one direction in the same amount that they bend downward in the perpendicular direction. In this chapter, we present the necessary background from differential geometry, a field of mathematics in which the ideas and techniques of calculus are applied to geometric shapes, to give an introduction to minimal surfaces. Then we use ideas from complex analysis to present a nice way to describe minimal surfaces and to relate the geometry of the surface with this description. This allows us to begin investigating some of Chapter 6: Circle Packing Circle packings are configurations of circles with prescribed patterns of tangency. They exist in quite amazing and often visually stunning variety, but what are they doing in a book on complex analysis? Complex analysis is, at its heart, a Introduction xv chapters and gave me lots of advice from the students' perspective. These students include
* 'First move, then teach. ' In modern languages, the mentioned type of moving (the heart) is cal... more * 'First move, then teach. ' In modern languages, the mentioned type of moving (the heart) is called motivating; both terms go back to the same Latin root.
Using this bo ok Marginal notes refer to bibliographic references, to other pages or to challenge... more Using this bo ok Marginal notes refer to bibliographic references, to other pages or to challenge solutions. In the colour edition, such notes and also the pointers to footnotes and to other websites are typeset in green. Over time, links on the internet tend to disappear. Most links can be recovered via www.archive.org, which keeps a copy of old internet pages. In the free pdf edition of this book, available at www.motionmountain.net, all green pointers and links are clickable. The pdf edition also contains all films; they can be watched directly in Adobe Reader. Solutions and hints for challenges are given in the appendix. Challenges are classified as easy (e), standard student level (s), difficult (d) and research level (r). Challenges for which no solution has yet been included in the book are marked (ny).
* 'First move, then teach.' In modern languages, the mentioned type of moving (the heart) is call... more * 'First move, then teach.' In modern languages, the mentioned type of moving (the heart) is called motivating; both terms go back to the same Latin root.
* 'First move, then teach.' In modern languages, the mentioned type of moving (the heart) is call... more * 'First move, then teach.' In modern languages, the mentioned type of moving (the heart) is called motivating; both terms go back to the same Latin root.
* 'First move, then teach.' In modern languages, the mentioned type of moving (the heart) is call... more * 'First move, then teach.' In modern languages, the mentioned type of moving (the heart) is called motivating; both terms go back to the same Latin root.
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Papers by Nestor Eduardo Gutierrez camayo