## 2018年12月12日水曜日

### 数学 - Python - JavaScript - 解析学 - 重積分の変数変換 - 変数変換定理(曲線カルジオイド(cardioid)の内側、円の外側、面積、極座標、積分、三角関数(余弦、正弦)、累乗(べき乗))

1. 曲線がジオイドの内側の面積。（必要な範囲のみ）

$\begin{array}{}2\underset{0}{\overset{\frac{\pi }{2}}{\int }}\underset{0}{\overset{a\left(1+\mathrm{cos}\theta \right)}{\int }}rdrd\theta \\ =\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\left[{r}^{2}\right]}_{0}^{a\left(1+\mathrm{cos}\theta \right)}d\theta \\ ={a}^{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\left(1+\mathrm{cos}\theta \right)}^{2}d\theta \\ ={a}^{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(1+2\mathrm{cos}\theta +{\mathrm{cos}}^{2}\theta \right)d\theta \\ ={a}^{2}\left(\frac{\pi }{2}+2{\left[\mathrm{sin}\theta \right]}_{0}^{\frac{\pi }{2}}+{\left[\frac{1}{2}\mathrm{cos}\theta \mathrm{sin}\theta \right]}_{0}^{\frac{\pi }{2}}+\frac{1}{2}\underset{0}{\overset{\frac{\pi }{2}}{\int }}1d\theta \right)\\ ={a}^{2}\left(\frac{\pi }{2}+2+\frac{1}{2}\frac{\pi }{2}\right)\\ ={a}^{2}\left(2+\frac{3}{4}\pi \right)\end{array}$

曲線カルジオイドの内側の曲線

$r=a$

の内側の面積。

$\begin{array}{}2\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\int }_{0}^{a}rdrd\theta \\ =\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\left[{r}^{2}\right]}_{0}^{a}d\theta \\ =\underset{0}{\overset{\frac{\pi }{2}}{\int }}{a}^{2}d\theta \\ =\frac{\pi }{2}{a}^{2}\end{array}$

よって、求める面積は、

$\begin{array}{}{a}^{2}\left(2+\frac{3}{4}\pi \right)-\frac{\pi }{2}{a}^{2}\\ ={a}^{2}\left(2+\frac{\pi }{4}\right)\end{array}$

コード(Emacs)

Python 3

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

print('5-(c).')

r, theta, a = symbols('r, θ, a')

I = 2 * Integral(Integral(r, (r, a, a * (1 + cos(theta)))),
(theta, 0, pi / 2))

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


$./sample5.py 5-(c). π ─ 2 a⋅(cos(θ) + 1) ⌠ ⌠ 2⋅⎮ ⎮ r dr dθ ⌡ ⌡ 0 a 2 π⋅a 2 ⎛ 3⋅π⎞ - ──── + a ⋅⎜2 + ───⎟ 2 ⎝ 4 ⎠ 2 a ⋅(π + 8) ────────── 4$


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="sample5.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 a = 2,
fr = theta => a * (1 + Math.cos(theta)),
fr1 = theta => a;
fns = [];

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 = [];

for (let theta = 0; theta < 2 * Math.PI; theta += dx) {
let r0 = fr(theta),
r1 = fr1(theta);

points.push([r0 * Math.cos(theta), r0 * Math.sin(theta), 'red']);
points.push([r1 * Math.cos(theta), r1 * Math.sin(theta), '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])

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();