Frase Célebre de John Cotton Dana

Quien se atreve a enseñar,
nunca debe dejar de aprender.

John Cotton Dana

Trazo de una Línea Punteada (Tabla Digitalizadora)

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Se traza la gráfica que determina un punto que se mueve sobre el plano. Convertir la ventana de Mathematica en una tabla digitalizadora.

Clear[s, list];

list = {}; unir = False; puntos = False;

Manipulate[If[puntos, AppendTo[list, s]]; 

 ListPlot[list, PlotStyle -> {Red, PointSize[0.01]}, 

  PlotRange -> {{-3, 3}, {-3, 3}}, AxesOrigin -> {0, 0}, 

  Joined -> unir], Text["Dejar Rastro"], 

 Button["Trazo", puntos = True], Button["No Trazo", puntos = False], 

 Text["Unir los puntos"], Button["Unir", unir = True], 

 Button["Puntear", unir = False], Text["Borrar"], 

 Button["Limpiar", list = {}], 

 Button["Último Trazo", list = Drop[list, -10]], {{s, {2, 2}}, 

  Locator}]

Para obtener los puntos del trazo

list

{ {-3., 0.27}, {-2.86,  0.36}, {-2.86, 0.36}, {-2.885, 0.43}, {-2.955, 0.44}, {-3., 0.41}, {-3., 0.29}, 

{-3., 0.22}, {-2.995, 0.18}, {-2.94, 0.17}, {-2.895, 0.19}, {-2.835, 0.32}, {-2.81, 0.44}, 

{-2.795, 0.71}, {-2.795, 0.89}, {-2.8, 1.12}, {-2.815, 1.29}, {-2.825, 1.44}, {-2.83, 1.53}, {-2.835, 1.58}, {-2.84, 1.6}, {-2.84, 1.59}, {-2.81, 1.52}, {-2.77, 1.38}, {-2.74, 1.24}, {-2.7, 1.11}, {-2.66, 1.03}, {-2.625, 0.93}, {-2.59, 0.84}, {-2.565, 0.76}, {-2.565, 0.67}, {-2.56, 0.6}, {-2.555, 0.58}, {-2.55,  0.65}, {-2.535, 0.83}, {-2.51, 1.05}, {-2.465, 1.24}, {-2.405, 1.42}, {-2.35, 1.52}, {-2.34, 1.56}, {-2.325, 1.59}, {-2.32, 1.61}, {-2.315, 1.62}, {-2.31, 1.62}, {-2.31, 1.61}, {-2.31, 1.6}, {-2.31, 1.55}, {-2.32, 1.38}, {-2.325, 1.26}, {-2.325, 1.1}, {-2.33, 0.97}, {-2.335, 0.84}, {-2.34, 0.72}, {-2.345, 0.56}, {-2.36, 0.46}, {-2.365, 0.37}, {-2.365, 0.32}, {-2.35, 0.26}, {-2.33, 0.24}, {-2.265, 0.31}, {-2.23, 0.43}, {-2.215,  0.6}, {-2.185, 0.76}, {-2.125, 0.88}, {-2.07, 0.92}, {-2.005, 0.9}, {-1.98, 0.79}, {-1.98, 0.71}, {-1.98, 0.68}, {-1.995, 0.76}, {-2.02, 0.85}, {-2.06, 0.94}, {-2.12, 0.96}, {-2.19, 0.87}, {-2.215, 0.71}, {-2.21, 0.46}, {-2.195, 0.36}, {-2.18, 0.27}, {-2.115, 0.27}, {-2.05, 0.32}, {-2.01, 0.48}, {-1.995, 0.68}, {-1.995, 0.8}, {-1.995, 0.79}, {-1.995, 0.69}, {-1.99, 0.51}, {-1.99, 0.4}, {-1.985, 0.3}, {-1.92, 0.22}, {-1.85, 0.2}, {-1.775, 0.33}, {-1.72, 0.63}, {-1.68, 0.98}, {-1.66, 1.4}, {-1.67, 1.69}, {-1.68, 1.89}, {-1.695, 1.95}, {-1.705, 1.9}, {-1.705, 1.63}, {-1.68, 1.3}, {-1.68, 1.07}, {-1.67, 0.87}, {-1.675, 0.72}, {-1.675, 0.57}, {-1.69, 0.46}, {-1.705, 0.38}, {-1.69, 0.36}, {-1.665, 0.33}, {-1.615, 0.36}, {-1.51, 0.48}, {-1.45, 0.67}, {-1.38, 0.96}, {-1.365, 1.33}, {-1.365, 1.68}, {-1.37, 1.91}, {-1.375, 1.92}, {-1.375, 1.68}, {-1.35, 1.36}, {-1.35, 1.15}, {-1.345, 0.91}, {-1.345, 0.71}, {-1.355, 0.56}, {-1.365, 0.45}, {-1.38, 0.4}, {-1.395, 0.39}, {-1.42, 0.48}, {-1.415, 0.64}, {-1.305, 0.88}, {-1.205, 0.96}, {-1.16, 0.92}, {-1.125, 0.86}, {-1.13, 0.75}, {-1.15, 0.61}, {-1.16, 0.47}, {-1.165, 0.4}, {-1.16, 0.34}, {-1.11, 0.33}, {-1.015, 0.36}, {-0.91, 0.47}, {-0.835, 0.67}, {-0.815, 0.89}, {-0.835, 1.01}, {-0.87, 1.04}, {-0.915, 1.01}, {-0.94, 0.89}, {-0.92, 0.67}, {-0.91, 0.51}, {-0.89, 0.39}, {-0.85, 0.32}, {-0.825, 0.29}, {-0.79, 0.29}, {-0.725, 0.41}, {-0.695, 0.53}, {-0.675, 0.72}, {-0.645, 0.88}, {-0.565, 0.99}, {-0.495, 1.01}, {-0.47, 0.89}, {-0.46, 0.75}, {-0.505,0.54}, {-0.515, 0.43}, {-0.52, 0.36}, {-0.52, 0.35}, {-0.485, 0.46}, {-0.44, 0.66}, {-0.375, 0.88}, {-0.275, 1.02}, {-0.215, 1.02}, {-0.19, 0.89}, {-0.175, 0.73}, {-0.195, 0.52}, {-0.21, 0.41}, {-0.22, 0.39}, {-0.22, 0.45}, {-0.175, 0.66}, {-0.11, 0.86}, {0.01, 1.01}, {0.115, 1.06}, {0.175, 1.04}, {0.155, 0.83}, {0.11, 0.62}, {0.09, 0.44}, {0.09, 0.36}, {0.175, 0.36}, {0.245, 0.41}, {0.305, 0.52}, {0.355, 0.67}, {0.405, 0.88}, {0.505, 1.1}, {0.62, 1.19}, {0.64, 1.1}, {0.65, 0.99}, {0.65, 0.96}, {0.68, 1.08}, {0.68,1.2}, {0.625, 1.24}, {0.535, 1.16}, {0.455, 0.96}, {0.425, 0.78}, {0.385, 0.59}, {0.375, 0.5}, {0.44, 0.48}, {0.455, 0.45}, {0.51, 0.46}, {0.6, 0.55}, {0.67, 0.74}, {0.72, 0.93}, {0.75,1.11}, {0.75, 1.12}, {0.745, 1.08}, {0.695,0.88}, {0.63, 0.68}, {0.6, 0.54}, {0.6, 0.44}, {0.59, 0.33}, {0.63, 0.27}, {0.7, 0.26}, {0.775, 0.3}, {0.835, 0.36}, {0.955, 0.53}, {1.04, 0.73}, {1.135, 1.01}, {1.215, 1.29}, {1.27, 1.6}, {1.3, 1.84}, {1.305,2.}, {1.31, 2.}, {1.31, 1.69}, {1.26, 1.4}, {1.21, 1.18}, {1.17, 1.01}, {1.135, 0.86}, {1.1, 0.7}, {1.11, 0.57}, {1.135, 0.52}, {1.21, 0.53}, {1.275, 0.61}, {1.345, 0.85}, {1.385, 0.99}, {1.425, 1.1}, {1.43, 1.15}, {1.41, 1.09}, {1.38, 0.92}, {1.32, 0.74}, {1.28, 0.56}, {1.29, 0.45}, {1.38, 0.43}, {1.505, 0.49}, {1.61, 0.67}, {1.68, 0.84}, {1.74, 0.98}, {1.835, 1.16}, {1.915, 1.22}, {1.98, 1.2}, {1.975, 1.1}, {1.915,0.97}, {1.93, 1.03}, {1.945, 1.19}, {1.93, 1.24}, {1.83, 1.16}, {1.755, 0.98}, {1.69, 0.72}, {1.65,0.53}, {1.655, 0.43}, {1.705, 0.36}, {1.775, 0.35}, {1.9, 0.43}, {2.015, 0.63}, {2.08, 0.78}, {2.165,0.95}, {2.24, 1.04}, {2.305, 1.05}, {2.325, 1.}, {2.285, 0.89}, {2.27, 0.85}, {2.3, 0.94}, {2.325,1.08}, {2.325, 1.17}, {2.27, 1.15}, {2.155, 0.94}, {2.07, 0.76}, {2.01, 0.6}, {1.995, 0.46}, {2.025,0.4}, {2.08, 0.39}, {2.15, 0.43}, {2.235, 0.51}, {2.32, 0.74}, {2.36, 0.88}, {2.395, 0.98}, {2.415,1.04}, {2.425, 1.04}, {2.39, 0.93}, {2.31, 0.75}, {2.28, 0.65}, {2.245, 0.53}, {2.205, 0.43}, {2.22,0.37}, {2.25, 0.33}, {2.325, 0.31}, {2.395, 0.32}, {2.465, 0.38}, {2.52, 0.42}, {2.58, 0.49}, {2.61,0.53}, {2.63, 0.56}, {2.63, 0.57}, {2.605, 0.57}, {-1.83, 1.44}, {-1.83, 1.42}, {-1.26, 1.53}, {-1.26,1.53}, {-1.28, 1.54}, {-1.315, 1.54}, {-1.885, 1.47}, {-1.875, 1.51}, {-1.865, 1.5}, {-1.835, 1.48}, {-1.81, 1.48},  {-1.73, 1.49}, {-1.71, 1.49}, {-1.71, 1.5}, {-1.69, 1.51}, {-1.68, 1.52}, {-1.665, 1.53}, {-1.655, 1.54}, {-1.645, 1.54}, {-1.63, 1.54}, {-1.61, 1.54}, {-1.605, 1.54}, {-1.59, 1.54}, {-1.58, 1.55}, {-1.55, 1.56}, {-1.55, 1.57}, {-1.54, 1.57}, {-1.54, 1.57}, {-1.535, 1.57}, {-1.53, 1.57},, {-1.525, 1.57}, {-1.525, 1.57}, {-1.555, 1.57}, {1.045, 1.51}, {1.05, 1.47}, {1.065, 1.48}, {1.18, 1.52}, {1.285, 1.57}, {1.41, 1.66}, {1.505, 1.71}, {1.57, 1.77}, {1.58, 1.77}, {1.545, 1.78}, {1.49, 1.36},  {1.51, 1.4}, {1.515,1.4}, {1.525, 1.4}}

Para aprender más sobre Mathematica ingrese aquí sitio de aprendizaje de Wolfram o en mi website ustamathematica.wixsite.com/basicas

Frase Célebre de Simeon Poisson

La vida es buena por dos cosas,
descubrir y enseñar matemáticas.

Simeon Poisson

Distancia media de dos puntos en un cuadrado unitario

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En el cuadrado unitario R = [0,1] × [0,1], deseamos determinar el promedio de las distancias de dos puntos tomados al azar. unitario.

El promedio de la longitud de todos los siguientes segmentos:

ss = 0; lin = {}; k = 0;

Manipulate[k++; If[k > 100, Break[]]; aa = RandomReal[1, {2, 2}]; 

 ss = ss + Apply[EuclideanDistance, aa]; AppendTo[lin, Line[aa]]; 

 Graphics[{Yellow, Rectangle[], Red, lin}, 

  PlotLabel -> Row[{"Promedio: ", (ss/k)}]], {n, 1, 10, Trigger}]

Calculamos el promedio de 100000 segmentos tomados de forma aleatoria en el cuadrado unitario., la respuesta tiende al número:

ss = 0; pro = {};

Do[ss = ss + Apply[EuclideanDistance, RandomReal[1, {2, 2}]]; 

  AppendTo[pro, ss/i], {i, 100000}];

Show[ListPlot[pro], 

 Plot[(Sqrt[2] + 2 + 5 Log[1 + Sqrt[2]])/15, {x, 0, 100000}, 

  PlotStyle -> Red]]

Veamos varias simulaciones, la línea roja es 0.286372 :

Para determinarlo de forma analítica, recordemos que el promedio de una función f del intervalo [a,b] en los reales es:

Así, la fórmula de la distancia es de los reales en cuarta dimensión y por tanto tomamos una integral cuádruple sobre un área de dimensión una unidad:   

Para aprender más sobre Mathematica ingrese aquí sitio de aprendizaje de Wolfram o en mi website ustamathematica.wixsite.com/basicas