2018年2月13日火曜日

学習環境

解析入門〈3〉(松坂 和夫(著)、岩波書店)の第14章(多変数の関数)、14.2(高次偏導関数、テイラーの定理)、問題10-(b).を取り組んでみる。


    1. D 1 f x , y = a e a x cos b y D 2 f x , y = - b e a x sin b y D 1 2 f x , y = a 2 e a x cos b y D 1 D 2 f x , y = - a b e a x sin b y D 2 2 f x , y = - b 2 e a x cos b y D 1 3 f x , y = a 3 e a x cos b y D 1 2 D 2 f x , y = - a 2 b e a x sin b y D 1 D 2 2 f x , y = - a b 2 e a x cos b y D 2 3 f x , y = b 3 e a x sin b y

      よって、 求める3次のテイラー多項式は、

      f 0 , 0 + D 1 f 0 , 0 x + D 2 f 0 , 0 y + 1 2 ! D 1 2 f 0 , 0 x 2 + 2 D 1 D 2 f 0 , 0 x y + D 2 2 f 0 , 0 y 2 + 1 3 ! D 1 3 f 0 , 0 x 3 + 3 D 1 2 D 2 f 0 , 0 x 2 y + 3 D 1 D 2 2 f 0 , 0 x y 2 + D 2 3 f 0 , 0 y 3 = 1 + a x + 1 2 a 2 x 2 - b 2 y 2 + 1 6 a 3 x 3 - 3 a b 2 x y 2

macOS High Sierraの標準搭載されているグラフ作成ソフト、Grapher で作成。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, exp, cos, Derivative, factorial

a, b = symbols('a, b', nonzero=True)
x, y = symbols('x, y')
f = exp(a * x) * cos(b * y)
d = {x: 0, y: 0}
Dx = Derivative(f, x, 1)
Dy = Derivative(f, y, 1)
Dxx = Derivative(f, x, 2)
Dyy = Derivative(f, y, 2)
Dxy = Derivative(Dx, y, 1)
Dxxx = Derivative(f, x, 3)
Dyyy = Derivative(f, y, 3)
Dxxy = Derivative(Dxx, y, 1)
Dxyy = Derivative(Dyy, x, 1)
expr = f.subs(d) + (Dx.subs(d) * x + Dy.subs(d) * y) + 1 / factorial(2) * (Dxx.subs(d) * x ** 2 + 2 * Dxy.subs(d) * x * y + Dyy.subs(d) * y ** 2) + \
    1 / factorial(3) * (Dxxx.subs(d) * x ** 3 + 3 * Dxxy.subs(d) * x ** 2 *
                        y + 3 * Dxyy.subs(d) * x * y ** 2 + Dyyy.subs(d) * y ** 3)


for t in [f, expr, expr.doit()]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample10.py
 a⋅x         
ℯ   ⋅cos(b⋅y)

                 ⎛  3      ⎞│         ⎛  2      ⎞│                            
               3 ⎜ ∂ ⎛ a⋅x⎞⎟│       2 ⎜ ∂ ⎛ a⋅x⎞⎟│                          3 
              x ⋅⎜───⎝ℯ   ⎠⎟│      x ⋅⎜───⎝ℯ   ⎠⎟│                         y ⋅
     2    2      ⎜  3      ⎟│         ⎜  2      ⎟│                            
  a⋅b ⋅x⋅y       ⎝∂x       ⎠│x=0      ⎝∂x       ⎠│x=0     ⎛∂ ⎛ a⋅x⎞⎞│         
- ───────── + ────────────────── + ────────────────── + x⋅⎜──⎝ℯ   ⎠⎟│    + ───
      2               6                    2              ⎝∂x      ⎠│x=0      

⎛  3          ⎞│         ⎛  2          ⎞│                              
⎜ ∂           ⎟│       2 ⎜ ∂           ⎟│                              
⎜───(cos(b⋅y))⎟│      y ⋅⎜───(cos(b⋅y))⎟│                              
⎜  3          ⎟│         ⎜  2          ⎟│                              
⎝∂y           ⎠│y=0      ⎝∂y           ⎠│y=0     ⎛∂           ⎞│       
─────────────────── + ────────────────────── + y⋅⎜──(cos(b⋅y))⎟│    + 1
       6                        2                ⎝∂y          ⎠│y=0    

 3  3    2  2      2    2          2  2    
a ⋅x    a ⋅x    a⋅b ⋅x⋅y          b ⋅y     
───── + ───── - ───────── + a⋅x - ───── + 1
  6       2         2               2      

$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="a0">a = </label>
<input id="a0" type="number" value="1">
<label for="b0">b = </label>
<input id="b0" type="number" value="1">
<label for="x0">x0 = </label>
<input id="x0" type="number" value="1">
<label for="y0">y0 = </label>
<input id="y0" type="number" value="1">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample10.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    input_x0 = document.querySelector('#x0'),
    input_y0 = document.querySelector('#y0'),    
    input_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_x0, input_y0, input_a0, input_b0],
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        x0 = parseFloat(input_x0.value),
        y0 = parseFloat(input_y0.value),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value);
        
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [],
        f = (x, y) => Math.exp(a0 * x) * Math.cos(b0 * y),
        g = (x, y) => 1 +
        a0 * x +
        1 / 2 * (a0 ** 2 * x ** 2 - b0 ** 2 * y ** 2) +
        1 / 6 * (a0 ** 3 * x ** 3 - 3 * a0 * b0 ** 2 * x * y ** 2),
        fns = [[(x) => f(x, y0), 'red'],
               [(x) => f(x0, x), 'green'],
               [(x) => g(x, y0), 'blue'],
               [(x) => g(x0, x), 'orange']];

    fns
        .forEach((o) => {
            let [fn, color] = o;
            
            for (let x = x1; x <= x2; x += dx) {
                let y = fn(x);
                
                if (Math.abs(y) < Infinity) {
                    points.push([x, y, color]);
                }
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);

    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    p(fns.join('\n'));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();








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