数学 - Python - JavaScript - 解析学 - 重積分の変数変換 - 変数変換定理(穴の開いた球体の体積、円柱座標)

解析入門(下)
楽天ブックス(Kobo) Yahoo!

学習環境

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

解析入門(下) (松坂和夫 数学入門シリーズ 6) (松坂 和夫(著)、岩波書店)の第24章(重積分の変数変換)、24.2(変数変換定理)、問題4.を取り組んでみる。

4. 
   

   半径2の球体の体積.

   $\begin{array}{}\frac{4}{3}\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}{2}^3\\ =\frac{32}{3}\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\end{array}$

   半径1の円柱状の穴について、円柱座標を考える。

   $\begin{array}{}0\le \; <!--\; less-than\; or\; equal\; to\; -->r\le \; <!--\; less-than\; or\; equal\; to\; -->1\\ 0\le \; <!--\; less-than\; or\; equal\; to\; -->\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\le \; <!--\; less-than\; or\; equal\; to\; -->2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\\ -\sqrt{2^2-r^2}\le \; <!--\; less-than\; or\; equal\; to\; -->z\le \; <!--\; less-than\; or\; equal\; to\; -->\sqrt{2^2-r^2}\end{array}$

   よって円柱状の穴の体積は、

   $\begin{array}{}2\underset{0}{\overset{1}{\int \; <!--\; integral\; -->}}\underset{0}{\overset{2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{\int \; <!--\; integral\; -->}}\underset{0}{\overset{\sqrt{4-r^2}}{\int \; <!--\; integral\; -->}}r\mathrm{dz}d\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}dr\\ =2\underset{0}{\overset{2\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}{\int \; <!--\; integral\; -->}}d\mathrm{\theta \; <!--\; greek\; small\; letter\; theta\; -->}\underset{0}{\overset{1}{\int \; <!--\; integral\; -->}}r\sqrt{4-r^2}dr\\ =4\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}{\left[-\frac{1}{3}{\left(4-r^2\right)}^{\frac{3}{2}}\right]}_0^1\\ =-\frac{4}{3}\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\left({\left(4-1\right)}^{\frac{3}{2}}-4^{\frac{3}{2}}\right)\\ =-\frac{4}{3}\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\left(3^{\frac{3}{2}}-8\right)\\ =-4\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\sqrt{3}+\frac{32}{3}\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\end{array}$

   よって、求める穴の空いた求体の体積は、

   $\begin{array}{}\frac{32}{3}k-\left(-4\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\sqrt{3}+\frac{32}{3}\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\right)\\ =4\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\sqrt{3}\end{array}$

コード(Emacs)

Python 3

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

print('4.')

r, theta, z = symbols('r, θ, z')

I = Rational(4, 3) * pi * 2 ** 3 - 2 * \
    Integral(
        Integral(
            Integral(r, (z, 0, sqrt(4 - r ** 2))),
            (theta, 0, 2 * pi)),
        (r, 0, 1))


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

入出力結果(Terminal, Jupyter(IPython))

$ ./sample4.py
4.
             __________                  
            ╱    2                       
    1 2⋅π ╲╱  - r  + 4                   
    ⌠  ⌠        ⌠                    32⋅π
- 2⋅⎮  ⎮        ⎮       r dz dθ dr + ────
    ⌡  ⌡        ⌡                     3  
    0  0        0                        

4⋅√3⋅π

$

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">

<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="sample4.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 fns = [[x => Math.sqrt(4 - x ** 2), 'red'],
           [x => -Math.sqrt(4 - x ** 2), 'green']];

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, 'blue'],
                 [1, y1, 1, y2, 'brown']];
    
    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 =