2017年10月30日月曜日

学習環境

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)、20.4(簡単な微分方程式)、二三の応用、問44.を取り組んでみる。

  1. P(x, y)を平面上の任意の点とする。

    点Pを通る曲線を考える。

    y=k x 3

    この曲線の点Pにおける接線の傾きを考える。

    x0 k= y x 3 3k x 2 =3 y x 3 x 2 = 3y x

    求める曲線は、点Pにおいて問題の曲線の傾きと直交する接線をもつ曲線。

    求める曲線をy = f(x)とおく。

    dy dx · 3y x =1 3ydy=xdx 3ydy = xdx + C 1 3 2 y 2 = 1 2 x 2 + C 1 x 2 +3 y 2 = C 1 ( C 1 >0 ) 1 3 x 2 + y 2 =C( C>0 )

    x = 0のとき(y軸、直線)、原点において、問題の曲線の傾きは0なので、すべての問題の曲線と直交する。

    よって、求める曲線。

    1 3 x 2 + y 2 =C( C>0 ),x=0

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sqrt, plot, Rational, Derivative, solve

print('44.')

k, x, C = symbols('k x C', real=True)
f = k * x ** 3
g1 = sqrt(C - x ** 2 / 3)
g2 = -g1
gs = [g1.subs({C: C0}) for C0 in range(1, 6)] + \
    [g2.subs({C: C0}) for C0 in range(1, 6)]

for t in [f, g1, g2]:
    pprint(t)
    print()

k0 = Rational(1, 20)
C0 = 1
Df = Derivative(f, x, 1).doit().subs({k: k0})
Dg = Derivative(g1, x, 1).doit().subs({C: C0})
x0 = solve((f - g1).subs({k: k0, C: C0}), x)[0]
pprint(x0)
h1 = Df.subs({k: k0, x: x0, C: C0}) * (x - x0) + f.subs({k: k0, x: x0, C: C0})
h2 = Dg.subs({k: k0, x: x0, C: C0}) * (x - x0) + g1.subs({k: k0, x: x0, C: C0})

p = plot(f.subs({k: k0}), h1, h2, *gs, (x, -5, 5), ylim=(-5, 5), show=False)
p[0].line_color = 'red'
p[1].line_color = 'green'
p[2].line_color = 'orange'
p.save('sample44.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample44.py
44.
   3
k⋅x 

     ________
    ╱      2 
   ╱      x  
  ╱   C - ── 
╲╱        3  

      ________
     ╱      2 
    ╱      x  
-  ╱   C - ── 
 ╲╱        3  

$

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="k0">k = </label>
<input id="k0" type="number" value="1">

<label for="c0">C = </label>
<input id="c0" type="number" value="1">

<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="sample44.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_k0 = document.querySelector('#k0'),
    input_c0 = document.querySelector('#c0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_k0, input_c0],
    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),
        k0 = parseFloat(input_k0.value),
        c0 = parseFloat(input_c0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [],
        f = (x) => k0 * x ** 3,
        g1 = (x) => Math.sqrt(c0  - x ** 2 / 3),
        g2 = (x) => -g1(x),
        fns = [[f, 'red'],
               [g1, 'green'],
              [g2, '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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