2018年10月2日火曜日

学習環境

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第3部(積分)、第11章(積分の計算)、補充問題(三角関数に関する積分)23.を取り組んでみる。


  1. x 2 a 2 - x 2 3 2 dx = x 2 a 2 - x 2 - 3 2 dx = f a 2 - x 2 - 1 2 - 1 a 2 - x 2 dx = x a 2 - x 2 - arcsin x a

コード(Emacs)

Python 3

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

print('23.')

a, x = symbols('a, x')
f = x ** 2 / (a ** 2 - x ** 2) ** Rational(3, 2)
I = Integral(f, x)
for t in [I, I.doit().simplify()]:
    pprint(t)
    print()

a0 = 2
p = plot(f.subs({a: a0}), I.doit().simplify().subs(
    {a: a0}), legend=True, show=False)
colors = ['red', 'green']
for i, c in enumerate(colors):
    p[i].line_color = c
p.save('sample23.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample23.py
23.
⌠                
⎮       2        
⎮      x         
⎮ ──────────── dx
⎮          3/2   
⎮ ⎛ 2    2⎞      
⎮ ⎝a  - x ⎠      
⌡                

⎧                                    │ 2│    
⎪       ⎛x⎞          ⅈ⋅x             │x │    
⎪ⅈ⋅acosh⎜─⎟ - ─────────────────  for │──│ > 1
⎪       ⎝a⎠           _________      │ 2│    
⎪                    ╱       2       │a │    
⎪                   ╱       x                
⎪             a⋅   ╱   -1 + ──               
⎪                 ╱          2               
⎪               ╲╱          a                
⎨                                            
⎪       ⎛x⎞          x                       
⎪ - asin⎜─⎟ + ────────────────    otherwise  
⎪       ⎝a⎠           ________               
⎪                    ╱      2                
⎪                   ╱      x                 
⎪             a⋅   ╱   1 - ──                
⎪                 ╱         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="0.001" 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">

<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="sample23.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_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => x ** 2 / Math.sqrt((2 ** 2 - x ** 2) ** 3),
    g = (x) => x / Math.sqrt(2 ** 2 - x ** 2) - Math.asin(x / 2),
    fns = [[f, 'red'],
           [g, 'green']];

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 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, 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].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();







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