数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 曲線の長さ - 媒介変数で表された曲線の長さ(指数関数、三角関数(正弦と余弦))

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28.を取り組んでみる。

$\begin{array}{l} \int_{0}^{1} \sqrt{{(\frac{d}{d t} (e^{t} \cos t))}^{2} + {(\frac{d}{d t} (e^{t} \sin t))}^{2}} d t \\ = \int_{0}^{1} \sqrt{{(e^{t} \cos t - e^{t} \sin t)}^{2} + {(e^{t} \sin t + e^{t} \cos t)}^{2}} d t \\ = \int_{0}^{1} \sqrt{e^{2 t} ({(\cos t - \sin t)}^{2} + {(\sin t + \cos t)}^{2})} d t \\ = \int_{0}^{1} e^{t} \sqrt{\cos^{2} t - 2 \sin t \cos t + \sin^{2} t + \sin^{2} t + 2 \sin t \cos t + \cos^{2} t} d t \\ = \int_{0}^{1} e^{t} \sqrt{\cos^{2} t + \sin^{2} t + \sin^{2} t + \cos^{2} t} d t \\ = \int_{0}^{1} e^{t} \sqrt{2 (\sin^{2} t + \cos^{2} t)} d t \\ = \int_{0}^{1} e^{t} \sqrt{2} d t \\ = \sqrt{2} \int_{0}^{1} e^{t} d t \\ = \sqrt{2} {[e^{t}]}_{0}^{1} \\ = \sqrt{2} (e - 1) \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Integral, Derivative, sin, cos, sqrt, exp

print('28.')

t = symbols('t')
x = exp(t) * cos(t)
y = exp(t) * sin(t)
f = sqrt(Derivative(x, t, 1) ** 2 + Derivative(y, t, 1) ** 2)
I = Integral(f, (t, 0, 1))
for o in [I, I.doit().factor()]:
    pprint(o)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample28.py
28.
1                                              
⌠                                              
⎮      _____________________________________   
⎮     ╱                2                  2    
⎮    ╱  ⎛d ⎛ t       ⎞⎞    ⎛d ⎛ t       ⎞⎞     
⎮   ╱   ⎜──⎝ℯ ⋅sin(t)⎠⎟  + ⎜──⎝ℯ ⋅cos(t)⎠⎟   dt
⎮ ╲╱    ⎝dt           ⎠    ⎝dt           ⎠     
⌡                                              
0                                              

   ⎛            ___________________⎞
   ⎜           ╱    2         2    ⎟
√2⋅⎝-1 + 4⋅ℯ⋅╲╱  cos (1) + sin (1) ⎠
────────────────────────────────────
                 6                  

$

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="-100">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="100">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-100">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="100">
<br>
<label for="t1">t1 = </label>
<input id="t1" type="number" value="-5">
<label for="t2">t2 = </label>
<input id="t2" type="number" value="5">
<label for="dt">dt = </label>
<input id="dt" type="number" value="0.001">

<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="sample28.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_t1 = document.querySelector('#t1'),
    input_t2 = document.querySelector('#t2'),
    input_dt = document.querySelector('#dt'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_t1, input_t2, input_dt],
    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 fx = (t) => Math.exp(t) * Math.cos(t),
    fy = (t) => Math.exp(t) * Math.sin(t);

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),
        t1 = parseFloat(input_t1.value),
        t2 = parseFloat(input_t2.value),
        dt = parseFloat(input_dt.value);


    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [],
        fns = [],
        fns1 = [],
        fns2 = [];

    for (let t = t1; t <= t2; t += dt) {
        points.push([fx(t), fy(t), 'green']);
    }
    console.log(points);
    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();

r = dx =
x1 = x2 =
y1 = y2 =
t1 = t2 = dt =

Kamimura's blog

2017年10月13日金曜日

数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 曲線の長さ - 媒介変数で表された曲線の長さ(指数関数、三角関数(正弦と余弦))

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