数学 - Python - JavaScript - 解析学 - 多変数の関数 - 高次偏導関数、テイラーの定理(ラプラス演算子(ラプラシアン)、調和関数)

解析入門〈3〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

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2-(a).を取り組んでみる。

2. 1. 
      
      $\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->x}=\frac{x^2+y^2-2x^2}{{\left(x^2+y^2\right)}^2}=\frac{y^2-x^2}{{\left(x^2+y^2\right)}^2}$
      $\begin{array}{}\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->x^2}=\frac{-2x{\left(x^2+y^2\right)}^2+\left(x^2-y^2\right)2\left(x^2+y^2\right)2x}{{\left(x^2+y^2\right)}^4}\\ =\frac{-2x\left(x^2+y^2\right)+4x\left(x^2-y^2\right)}{{\left(x^2+y^2\right)}^3}\end{array}$
      $\frac{\partial \; <!--\; partial\; differential\; -->f}{\partial \; <!--\; partial\; differential\; -->y}=\frac{-x·\; <!--\; middle\; dot\; -->2y}{{\left(x^2+y^2\right)}^2}=\frac{-2xy}{{\left(x^2+y^2\right)}^2}$
      $\begin{array}{}\frac{{\partial \; <!--\; partial\; differential\; -->}^{2}f}{\partial \; <!--\; partial\; differential\; -->y^2}=\frac{-2x{\left(x^2+y^2\right)}^2+2xy·\; <!--\; middle\; dot\; -->2\left(x^2+y^2\right)·\; <!--\; middle\; dot\; -->2y}{{\left(x^2+y^2\right)}^4}\\ =\frac{-2x\left(x^2+y^2\right)+8x{y}^2}{{\left(x^2+y^2\right)}^3}\end{array}$
      $\begin{array}{}-4x\left(x^2+y^2\right)+4x\left(x^2-y^2\right)+8x{y}^2\\ =-4x{y}^2-4x{y}^2+8x{y}^2\\ =0\end{array}$

      よって、

      $\mathrm{\Delta \; <!--\; greek\; capital\; letter\; delta\; -->}f=0$

      なので、問題の関数は調和関数である。

      （証明終）

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Limit, Derivative

x, y = symbols('x, y')

f = x / (x ** 2 + y ** 2)
Ds = [Derivative(f, t, 2) for t in [x, y]]

for t in Ds + [sum(Ds).doit().factor()]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample2.py
  2         
 ∂ ⎛   x   ⎞
───⎜───────⎟
  2⎜ 2    2⎟
∂x ⎝x  + y ⎠

  2         
 ∂ ⎛   x   ⎞
───⎜───────⎟
  2⎜ 2    2⎟
∂y ⎝x  + y ⎠

0

$

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="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="sample2.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'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_x0, input_y0],
    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 f = (x, y) => x / (x ** 2 + y ** 2);

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);
        
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [],
        fns = [[(x) => f(x, y0), 'green'],
               [(x) => f(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();

r = dx =
x1 = x2 =
y1 = y2 =
x0 = y0 =