2018年12月9日日曜日

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)

1. 半径2の球体の体積.

$\begin{array}{}\frac{4}{3}\pi {2}^{3}\\ =\frac{32}{3}\pi \end{array}$

半径1の円柱状の穴について、円柱座標を考える。

$\begin{array}{}0\le r\le 1\\ 0\le \theta \le 2\pi \\ -\sqrt{{2}^{2}-{r}^{2}}\le z\le \sqrt{{2}^{2}-{r}^{2}}\end{array}$

よって円柱状の穴の体積は、

$\begin{array}{}2\underset{0}{\overset{1}{\int }}\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\sqrt{4-{r}^{2}}}{\int }}r\mathrm{dz}d\theta dr\\ =2\underset{0}{\overset{2\pi }{\int }}d\theta \underset{0}{\overset{1}{\int }}r\sqrt{4-{r}^{2}}dr\\ =4\pi {\left[-\frac{1}{3}{\left(4-{r}^{2}\right)}^{\frac{3}{2}}\right]}_{0}^{1}\\ =-\frac{4}{3}\pi \left({\left(4-1\right)}^{\frac{3}{2}}-{4}^{\frac{3}{2}}\right)\\ =-\frac{4}{3}\pi \left({3}^{\frac{3}{2}}-8\right)\\ =-4\pi \sqrt{3}+\frac{32}{3}\pi \end{array}$

よって、求める穴の空いた求体の体積は、

$\begin{array}{}\frac{32}{3}k-\left(-4\pi \sqrt{3}+\frac{32}{3}\pi \right)\\ =4\pi \sqrt{3}\end{array}$

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()


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


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,
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])

let yscale = d3.scaleLinear()
.domain([y1, y2])

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();