2017年10月31日火曜日

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45.を取り組んでみる。

    1. 球形の中心を通る断面を考える。

      s 2 + u 2 = R 2 u 2 = R 2 s 2 V= R xR u 2 πds = R xR ( R 2 s 2 )πds =π R xR ( R 2 s 2 )ds =π [ R 2 s 1 3 s 3 ] R xR =π( ( R 2 ( xR ) 1 3 ( xR ) 3 )( R 3 + 1 3 R 3 ) ) =π( R 2 x R 3 1 3 ( x 3 3 x 2 R+3x R 2 R 3 )+ R 3 1 3 R 3 ) =π( R 2 x 1 3 x 3 + x 2 Rx R 2 + 1 3 R 3 1 3 R 3 ) =π( 1 3 x 3 + x 2 R ) =π( R x 2 1 3 x 3 )

    2. dV dt =k x dV dx · dx dt =k x d dx ( π( R x 2 1 3 x 3 ) ) dx dt =k x π( 2Rx x 2 ) dx dt =k x dx dt = k x π( 2Rx x 2 ) dx dt = k x πx( 2Rx ) dx dt = k π x ( 2Rx )
    3. 微分方程式の一般解。

      dx dt = k π x ( 2Rx ) π x ( 2Rx )dx=kdt ( π x ( 2Rx ) )dx = kdt π ( 2R x x 3 2 )dx = kdt π( 2R 2 3 x 3 2 2 5 x 5 2 )=kt+C π x 3 2 ( 2R 2 3 2 5 x )=kt+C 2π x 3 2 ( 2 3 R 1 5 x )=kt+C

      0秒のとき、水深は2Rなので。初期条件t = 0、x=2RとしてCを求める。

      2π ( 2R ) 3 2 ( 2 3 R 1 5 ·2R )=k0+C C=2π ( 2R ) 3 2 ( 2 3 R 1 5 ·2R )

      よって、求める関係式。

      2π x 3 2 ( 2 3 R 1 5 x )=kt2π ( 2R ) 3 2 ( 2 3 R 1 5 ·2R )

    4. 前問の関係式で x = Rとする。

      2π R 3 2 ( 2 3 R 1 5 R )=kt2π ( 2R ) 3 2 ( 2 3 R 1 5 ·2R ) t= 2π ( 2R ) 3 2 ( 2 3 R 2 5 R )2π R 3 2 ( 2 3 R 1 5 R ) k = 2π R 3 2 ( 2 3 2 ( 2 3 R 2 5 R )( 2 3 R 1 5 R ) ) k = 2π R 3 2 ( 2 5 2 ( 1 3 R 1 5 R )( 2 3 R 1 5 R ) ) k

コード(Emacs)

Python 3

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

print('45.')

R, k = symbols('R k')
t = 2 * pi * R ** Rational(3, 2) * (2 ** Rational(5, 2)
                                    * (R / 3 - R / 5) - (2 * R / 3 - R / 5)) / k

for t0 in [t, t.expand(), t.factor()]:
    pprint(t0)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample45.py
45.
     3/2 ⎛  7⋅R   8⋅√2⋅R⎞
2⋅π⋅R   ⋅⎜- ─── + ──────⎟
         ⎝   15     15  ⎠
─────────────────────────
            k            

        5/2            5/2
  14⋅π⋅R      16⋅√2⋅π⋅R   
- ───────── + ────────────
     15⋅k         15⋅k    

     5/2            
2⋅π⋅R   ⋅(-7 + 8⋅√2)
────────────────────
        15⋅k        

$

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="r1">R = </label>
<input id="r1" type="number" min="0" value="2">

<label for="x0">x0 = </label>
<input id="x0" 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="sample45.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_r1 = document.querySelector('#r1'),
    input_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_r1, input_x0],
    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),
        r1 = parseFloat(input_r1.value),
        x0 = parseFloat(input_x0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [[x0, y1, x0, y2, 'blue']],
        f1 = (x) => Math.sqrt(r1 ** 2 - x ** 2),
        f2 = (x) => -f1(x),
        fns = [[f1, 'green'],
               [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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