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El Producto Vectorial EL PRODUCTO VECTORIAL Juan Francisco Gonz´alezHern´andez *** Resumen Se desarrolla una exposici´onhist´orica del origen de la operaci´onque llamamos producto vectorial, destac´andoselos problemas y conceptos ma- tem´aticos que llevaron a su introducci´onen el Algebra.´ Se incluye una descripci´onde las m´asdestacadas y representativas aplicaciones de este producto en F´ısica y Matem´aticas, as´ı como una visi´ongeneral de las diferentes alternativas para generalizar este producto a espacios de m´as dimensiones. Finalmente, se discute acerca de la mejor manera de intro- ducir este concepto a nivel te´orico y pr´actico en los ´ultimos cursos de la E.S.O. y en Bachillerato, destac´andosealgunos puntos pedag´ogicos y did´acticos que pueden ser de inter´es pr´actico. ´Indice 1. Introducci´on 2 2. Historia y desarrollo del producto vectorial 3 2.1. W. R.Hamilton: complejos, cuaternios y vectores . 4 2.2. Grassmann y el c´alculo de extensiones . 8 2.3. Gibbs, Heaviside y el an´alisis vectorial . 10 2.4. William K. Clifford . 12 2.5. Una s´ıntesis moderna del producto vectorial . 13 2.5.1. Definici´on. 14 2.5.2. Ejemplos . 14 2.5.3. Propiedades interesantes . 14 2.5.4. Matrices, SO(3) y producto vectorial . 14 2.5.5. Notaci´ontensorial y producto vectorial . 15 2.5.6. Vectores axiales y quiralidad . 15 3. Aplicaciones del producto vectorial 16 3.1. Aplicaciones en Matem´aticas . 16 3.1.1. C´alculode ´areasy vol´umenes . 16 3.1.2. Teorema de Stokes . 16 3.1.3. Algebras´ de Lie y producto vectorial . 17 3.1.4. M´etodo de Cayley-Dickson, semioctavas, octavas y Matri- ces de Zorn . 18 3.1.5. Otras utilidades matem´aticas . 19 *k: jfgh.teorfi[email protected] **H: 655 246 827 1 3.2. Aplicaciones en la F´ısica . 19 3.2.1. Momento angular y torque . 19 3.2.2. Vector de Laplace-Runge-Lenz . 20 3.2.3. Sistemas de referencia no inerciales . 21 3.2.4. Redes de Bravais . 21 3.2.5. Expresi´onde la fuerza de Lorentz y vector de Poynting . 22 3.2.6. Ley de Biot-Savart . 23 3.2.7. Ecuaciones de Maxwell . 23 3.2.8. Rotacional y vorticidad . 23 4. Generalizaciones 24 4.1. Algebras´ normadas en D=1, 2, 4, 8 . 24 4.1.1. Las ´algebras con divisi´on K = R, C, H, O . 24 4.1.2. A × B s´oloexiste en D = 0, 1, 3, 7 . 25 4.2. Producto exterior y Algebra´ de Grassmann . 29 4.2.1. Algebra´ de Grassmann para principiantes . 29 4.2.2. Algebras´ de Grassmann en serio. 31 4.3. Forma de volumen y dual de Hodge . 32 4.4. Algebras´ de Clifford (Algebra´ geom´etrica) . 33 5. Pedagog´ıay Did´actica 35 5.1. La regla del tornillo y del sacacorchos . 35 5.2. La regla de la mano derecha . 36 5.3. La t´ecnica e intuici´onalgebraicas . 36 5.4. Problemas comunes de tipo did´actico y recursos pedag´ogicos . 37 6. Conclusiones y discusi´on 38 A. Cronolog´ıa algebraica 39 B. Perfiles biogr´aficos 45 B.1. W. R. Hamilton . 45 B.2. Hermann G¨unther Grassmann . 48 B.3. Oliver Heaviside . 49 B.4. Josiah W. Gibbs . 52 B.5. Arthur Cayley . 52 B.6. W. K. Clifford . 53 B.7. Nabla . 54 1. Introducci´on La tercera operaci´onaritm´etica b´asica, el producto, es en Matem´aticas y en F´ısicauna de las fuentes de mayores sorpresas. Para n´umeros convencionales, tales como los n´umeros reales, R, la definici´ondel producto no presenta ning´un problema, ya que simplemente es cuesti´onde una extensi´on trivial de las reglas para el producto que tenemos en los n´umeros enteros, Z, y los racionales, Q. Sin embargo, todos nos hemos enfrentado alguna vez a la dificultad intr´ınse- ca que plantea el producto y su operaci´oninversa, la divisi´on, en cuerpos o 2 estructuras algebraicas m´asgenerales. Los ejemplos t´ıpicos que hoy d´ıase en- se˜nan en la E.S.O. y Bachillerato comprenden dos casos importantes: el ´algebra vectorial y el ´algebramatricial. El concepto de producto de dos vectores, o de dos matrices, es algo inherentemente complicado y sorprendente cuando se ve por primera vez. Esta ha sido las principales motivaciones, junto a la vertiente hist´orica, hoy totalmente arrinconada por desgracia, la que nos ha llevado a la realizaci´onde este trabajo. El conocimiento de los pasos l´ogicos que hist´orica- mente llevaron a sentar las bases del c´alculo vectorial y tensorial por un lado, y por otro del Algebra,´ tienen gran importancia para profundizar en el concepto sobre el cual, de una u otra forma, descansan la mayor parte de aplicaciones ´utiles: la noci´onde operaci´on. La organizaci´ondel presente trabajo es como sigue: en la secci´on1, estu- diamos c´omoapareci´oel concepto de vector y las distintas clases de producto durante la g´enesis de los Cuaternios en el siglo XIX, donde Hamilton introdujo la noci´onde vector y m´astarde Gibbs y Heaviside desarrollaron tanto la no- taci´oncomo la maquinaria matem´aticaconocida hoy no ya s´olopor f´ısicos y matem´aticos,sino por una gran parte de los estudiantes preuniversitarios, al mismo tiempo que se exponen cu´ales fueron los posteriores desarrollos hist´ori- cos. En la secci´on2, hacemos una extensa revisi´onde las aplicaciones que tiene la operaci´onde producto vectorial en las disciplinas de F´ısica y Matem´aticas. En la secci´on3, realizamos una s´ıntesis autocontenida de las distintas formas de generalizar, en un sentido concreto matem´atico, la noci´onde producto vectorial a espacios de m´asdimensiones usando dos enfoques diferentes. Finalmente, en la secci´on4, presentamos una discusi´onque se espera sea ´util a instructores y profesores, y donde cobran particular importancia tres aspectos: la visi´onespa- cial, la regla del tornillo o sacacorchos y la t´ecnica o intuici´onalgebraica, ´este ´ultimo normalmente poco destacado y valorado en libros y manuales. 2. Historia y desarrollo del producto vectorial La historia de la operaci´onproducto vectorial se encuentra ´ıntimamente uni- da al origen del Algebra´ y la propia evoluci´onnatural de la teor´ıade n´umeros[1]. En concreto, para ser m´asprecisos, su creaci´oncomo entidad propia se encuen- tra, como veremos, en el mismo per´ıodo de tiempo que llev´oa la introducci´on del an´alisis vectorial [2, 3, 4]. Entre los siglos XVI y XVIII tuvo lugar una aut´entica revoluci´onmatem´atica,consistente en la puesta en com´unde la Geo- metr´ıaCl´asica eucl´ıdea y el Algebra´ aparecida en el Renacimiento. Mediante los m´etodos de la Geometr´ıa Anal´ıtica,y del reci´eninventado C´alculo infinitesi- mal, se produjo una explosi´onde nuevos y poderosos resultados pero tambi´enal planteamiento de nuevas dificultades. As´ı,a finales del siglo XVIII y principios del siglo XIX, quedaban ya pocos matem´aticosatados a los m´etodos exclusiva- mente geom´etricos, usando preferentemente tambi´enya los nuevos desarrollos algebraicos procedentes de la teor´ıade ecuaciones. Sin embargo, conviene des- tacar que esta curiosa operaci´on, el producto vectorial, no apareci´oen primer lugar con los vectores, sino que fue m´asbien al rev´es, los vectores y las distin- tas operaciones entre ellos se crearon a partir de la soluci´onalgebraica de un problema concreto, que no es otro que la de la generalizaci´onde la noci´onde n´umero complejo. 3 2.1. W. R.Hamilton: complejos, cuaternios y vectores Sir William Rowan Hamilton fue sin lugar a dudas una persona superdo- tada en muchos aspectos: a los 5 a˜nosle´ıagriego, lat´ın, hebreo y dominaba a los diez a˜nosmedia docena de lenguas orientales. Estudi´oen el Trinity Colle- ge de Dubl´ın, y a los 22 a˜nosera ya astr´onomoreal de Irlanda, director del Observatorio de Dunsink y profesor de Astronom´ıa. Public´opronto un trabajo sobre rayos en Optica,´ y estudi´ootros interesantes fen´omenos naturales cuya explicaci´ony confirmaci´onexperimental le dieron enorme prestigio, tanto como para ser elevado a la nobleza a los 30 a˜nos.Adem´as, fue la primera persona que present´oun trabajo original sobre el Algebra´ de n´umeros complejos ( aunque dicen las rumorolog´ıasmatem´aticasque Gauss ya conoc´ıaese sistema) siendo el primero en formalizarlo en 1833. As´ı, en ese a˜no,el art´ıculoque present´oa la Irish Academy, introdujo el concepto de n´umero complejo como un ´algebra 1formal de pares de n´umeros reales con operaciones suma y producto definidas como en la actualidad. De esta forma: (a, b) + (c, d) = (a + c, b + d) λ (a, b) = (λa, λb) (a, b)(c, d) = (ac − bd, ad + bc) El producto vectorial y el producto escalar surgieron a posteriori de la teor´ıa de los cuaternios de Hamilton, que son un conjunto de “n´umeros”de la forma siguiente: Q = a + bi + cj + dk (1) con a, b, c, d, ∈ R y los elementos base hipercomplejos cumpliendo las relaciones fundamentales siguientes: i 2 = j 2 = k 2 = ijk = −1 (2) ¿C´omo lleg´oHamilton a semejantes relaciones? Bien, la cuesti´ones ma- tem´aticamente no trivial, pero el problema es una “simple” generalizaci´on.Por analog´ıacon los n´umeros complejos, Hamilton se pregunt´osi ser´ıaposible hacer lo mismo para tripletes de n´umeros lo que hab´ıahecho para pares o dobletes de n´umeros reales, lo que hab´ıahecho para construir de una manera novel los n´umeros complejos.
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