数学 - Python - JavaScript - 解析学 - 距離空間の位相 - n次元実数空間における曲線(対数関数、積分、置換積分法、曲線の長さ、パラメーター表示に変換)

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12章(距離空間の位相)、12.4(n次元実数空間における曲線)、問題5.を取り組んでみる。

5. 
   

   曲線のパラメーター表示を

   $\left(t,\log t\right)$

   と考える。

   $\begin{array}{}\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}\left(t\right)=\left(t,\log t\right)\\ \mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}\text{'}\left(t\right)=\left(1,\frac{1}{t}\right)\\ \left|\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}\text{'}\left(t\right)\right|=\sqrt{1+\frac{1}{t^2}}\end{array}$

   もとめる曲線の長さ。

   $\begin{array}{}{\int \; <!--\; integral\; -->}_1^3\left|\mathrm{\gamma \; <!--\; greek\; small\; letter\; gamma\; -->}\text{'}\left(t\right)\right|\mathrm{dt}\\ ={\int \; <!--\; integral\; -->}_1^3\sqrt{1+\frac{1}{t^2}}\mathrm{dt}\\ ={\int \; <!--\; integral\; -->}_1^3\sqrt{\frac{t^2+1}{t^2}}\mathrm{dt}\\ ={\int \; <!--\; integral\; -->}_1^3\frac{\sqrt{t^2+1}}{t}\mathrm{dt}\end{array}$
   $\begin{array}{}u=\sqrt{t^2+1}\\ \frac{du}{\mathrm{dt}}=\frac{2t}{2\sqrt{t^2+1}}\\ \frac{du}{\mathrm{dt}}=\frac{t}{\sqrt{t^2+1}}\\ \mathrm{dt}=\frac{\sqrt{t^2+1}}{t}du\\ 1\le \; <!--\; less-than\; or\; equal\; to\; -->t\le \; <!--\; less-than\; or\; equal\; to\; -->3\\ \sqrt{2}\le \; <!--\; less-than\; or\; equal\; to\; -->u\le \; <!--\; less-than\; or\; equal\; to\; -->\sqrt{10}\end{array}$
   $\begin{array}{}L\left(t\right)=\underset{\sqrt{2}}{\overset{\sqrt{10}}{\int \; <!--\; integral\; -->}}\frac{\sqrt{t^2+1}}{t}·\; <!--\; middle\; dot\; -->\frac{\sqrt{t^2+1}}{t}du\\ =\underset{\sqrt{2}}{\overset{\sqrt{10}}{\int \; <!--\; integral\; -->}}\frac{t^2+1}{t^2}du\\ ={\int \; <!--\; integral\; -->}_2^{\sqrt{10}}\left(1+\frac{1}{t^2}\right)du\\ =\underset{\sqrt{2}}{\overset{\sqrt{10}}{\int \; <!--\; integral\; -->}}1du+{\int \; <!--\; integral\; -->}_{\sqrt{2}}^{\sqrt{10}}\frac{1}{t^2}du\end{array}$
   $\begin{array}{}{u}^2=t^2+1\\ t^2=u^2-1\end{array}$
   $\begin{array}{}L\left(t\right)=\sqrt{10}-\sqrt{2}+\underset{\sqrt{2}}{\overset{\sqrt{10}}{\int \; <!--\; integral\; -->}}\frac{1}{u^2-1}du\\ \frac{1}{u^2-1}=\frac{1}{u+1}·\; <!--\; middle\; dot\; -->\frac{1}{u-1}\\ \frac{A}{u+1}+\frac{B}{u-1}=\frac{\left(A+B\right)u-A+B}{u^2-1}\\ A+B=0\\ -A+B=1\\ B=\frac{1}{2},A=-\frac{1}{2}\\ \frac{1}{u^2-1}=-\frac{1}{2\left(u+1\right)}+\frac{1}{2\left(u-1\right)}\end{array}$
   $\begin{array}{}L\left(t\right)\\ =\sqrt{10}-\sqrt{2}+\frac{1}{2}\underset{\sqrt{2}}{\overset{\sqrt{10}}{\int \; <!--\; integral\; -->}}\left(-\frac{1}{u+1}+\frac{1}{u-1}\right)du\\ =\sqrt{10}-\sqrt{2}+\frac{1}{2}{\left[-\log \left(u+1\right)+\log \left(u-1\right)\right]}_{\sqrt{2}}^{\sqrt{10}}\\ =\sqrt{10}-\sqrt{2}+\frac{1}{2}\left(\log \frac{\sqrt{10}-1}{\sqrt{10}+1}-\log \frac{\sqrt{2}-1}{\sqrt{2}+1}\right)\\ =\sqrt{10}-\sqrt{2}+\frac{1}{2}\log \frac{\left(\sqrt{10}-1\right)\left(\sqrt{2}+1\right)}{\left(\sqrt{10}+1\right)\left(\sqrt{2}-1\right)}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, Integral

t = symbols('t')
f = sqrt(1 + 1 / t ** 2)
I = Integral(f, (t, 1, 3))
for o in [I, I.doit()]:
    pprint(o)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample5.py
3                 
⌠                 
⎮      ________   
⎮     ╱     1     
⎮    ╱  1 + ──  dt
⎮   ╱        2    
⎮ ╲╱        t     
⌡                 
1                 

-√2 - asinh(1/3) + log(1 + √2) + √10

$

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="0.001">
<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">

<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="sample5.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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) => Math.log(x);

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);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        lines = [[1, y1, 1, y2, 'red'],
                 [3, y1, 3, y2, 'blue']],
        fns = [[f, 'green']];

    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 =