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)
- 数式入力ソフト(TeX, MathML): MathType
- MathML対応ブラウザ: Firefox、Safari
- MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
- 参考書籍
数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第20章(面積、体積、長さ - 積分法の応用)、20.3(曲線の長さ)、媒介変数で表された曲線の長さ、問27.を取り組んでみる。
コード(Emacs)
Python 3
#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, Derivative, sin, cos, sqrt, pi, solve, plot
print('27.')
a = symbols('a', positive=True)
t = symbols('t')
I = 3 * a / 2 * 4 * Integral(sin(2 * t), (t, 0, pi / 2))
for o in [I, I.doit()]:
pprint(o)
print()
x = symbols('x')
y = a * sin(t) ** 3
s = solve(x - a * cos(t) ** 3, t)
for t0 in s:
pprint(y.subs({t:t0}))
print()
入出力結果(Terminal, Jupyter(IPython))
$ ./sample27.py
27.
π
─
2
⌠
6⋅a⋅⎮ sin(2⋅t) dt
⌡
0
6⋅a
3/2
⎛ 2/3⎞
⎜ x ⎟
-a⋅⎜1 - ────⎟
⎜ 2/3⎟
⎝ a ⎠
3/2
⎛ 2 ⎞
⎜ ⎛ 3 ___ 3 ___⎞ ⎟
⎜ ⎜ ╲╱ x √3⋅ⅈ⋅╲╱ x ⎟ ⎟
-a⋅⎜- ⎜- ─────── - ──────────⎟ + 1⎟
⎜ ⎜ 3 ___ 3 ___ ⎟ ⎟
⎝ ⎝ 2⋅╲╱ a 2⋅╲╱ a ⎠ ⎠
3/2
⎛ 2 ⎞
⎜ ⎛ 3 ___ 3 ___⎞ ⎟
⎜ ⎜ ╲╱ x √3⋅ⅈ⋅╲╱ x ⎟ ⎟
-a⋅⎜- ⎜- ─────── + ──────────⎟ + 1⎟
⎜ ⎜ 3 ___ 3 ___ ⎟ ⎟
⎝ ⎝ 2⋅╲╱ a 2⋅╲╱ a ⎠ ⎠
3/2
⎛ 2/3⎞
⎜ x ⎟
a⋅⎜1 - ────⎟
⎜ 2/3⎟
⎝ a ⎠
3/2
⎛ 2 ⎞
⎜ ⎛ 3 ___ 3 ___⎞ ⎟
⎜ ⎜ ╲╱ x √3⋅ⅈ⋅╲╱ x ⎟ ⎟
a⋅⎜- ⎜- ─────── - ──────────⎟ + 1⎟
⎜ ⎜ 3 ___ 3 ___ ⎟ ⎟
⎝ ⎝ 2⋅╲╱ a 2⋅╲╱ a ⎠ ⎠
3/2
⎛ 2 ⎞
⎜ ⎛ 3 ___ 3 ___⎞ ⎟
⎜ ⎜ ╲╱ x √3⋅ⅈ⋅╲╱ x ⎟ ⎟
a⋅⎜- ⎜- ─────── + ──────────⎟ + 1⎟
⎜ ⎜ 3 ___ 3 ___ ⎟ ⎟
⎝ ⎝ 2⋅╲╱ a 2⋅╲╱ a ⎠ ⎠
$
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"> <br> <label for="a0">a0 = </label> <input id="a0" type="number" min="0" 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="sample27.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_a0 = document.querySelector('#a0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_a0],
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 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),
a0 = parseFloat(input_a0.value);
if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
let points = [],
lines = [],
f1 = (x) => -a0 * Math.sqrt((1 - ((x / a0) ** 2) ** (1 / 3)) ** 3),
f2 = (x) => -f1(x),
fns = [[f1, 'red'],
[f2, 'green']],
fns1 = [],
fns2 = [];
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]);
}
}
});
fns1.forEach((o) => {
let [fn, color] = o;
lines.push([x1, fn(x1), x2, fn(x2), color]);
});
fns2.forEach((o) => {
let [fn, color] = o;
for (let x = x1; x <= x2; x += dx0) {
let g = fn(x);
lines.push([x1, g(x1), x2, g(x2), 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('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.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.append('g')
.attr('transform', `translate(0, ${height - padding})`)
.call(xaxis);
svg.append('g')
.attr('transform', `translate(${padding}, 0)`)
.call(yaxis);
[fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};
inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();
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