2018年3月29日木曜日

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10章(積分の性質)、4(広義の積分)、練習問題10.を取り組んでみる。


  1. 0 0 1 1 tan x dx = 0 1 cos x sin x dx 0 1 1 sin x dx

    よって前問9により、 積分

    0 1 1 sin x dx

    は存在するので、定理らにより、積分

    0 1 1 tan x dx

    は存在する。

コード(Emacs)

Python 3

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

x = symbols('x')
f = 1 / sqrt(sin(x))
c = Rational(1, 10)
g = 1 / sqrt(c * x)
h = 1 / sqrt(tan(x))
I1 = Integral(f, (x, 0, 1))
I2 = Integral(g, (x, 0, 1))
I3 = Integral(h, (x, 0, 1))
for I in [I1, I2, I3]:
    for t in [I, I.doit()]:
        pprint(t)
        print()
    print()

p = plot(f, g, h, (x, -0.1, 1.1), ylim=(0, 10), legend=True, show=False)

for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color
p.save('sample10.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample10.py
1              
⌠              
⎮     1        
⎮ ────────── dx
⎮   ________   
⎮ ╲╱ sin(x)    
⌡              
0              

1              
⌠              
⎮     1        
⎮ ────────── dx
⎮   ________   
⎮ ╲╱ sin(x)    
⌡              
0              


1       
⌠       
⎮ √10   
⎮ ─── dx
⎮  √x   
⌡       
0       

2⋅√10


1              
⌠              
⎮     1        
⎮ ────────── dx
⎮   ________   
⎮ ╲╱ tan(x)    
⌡              
0              

1              
⌠              
⎮     1        
⎮ ────────── dx
⎮   ________   
⎮ ╲╱ tan(x)    
⌡              
0              


$

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">
<br>
<label for="c0">c = </label>
<input id="c0" type="number" value="10">

<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="sample10.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_c0 = document.querySelector('#c0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              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 f = (x) => 1 / Math.sqrt(Math.sin(x)),
    g = (x) => 1 / Math.sqrt(Math.tan(x)),
    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),
        c0 = parseFloat(input_c0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [[0, y1, 0, y2, 'brown'],
                 [1, y1, 1, y2, 'orange']],
        h = (x) => 1 / Math.sqrt(Math.sqrt(1 / c0 * x)),
        fns1 = [],
        fns2 = [];

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

                points.push([x, y, color]);
            }
        });

    fns1
        .forEach((o) => {
            let [f, color] = o;
            
            lines.push([x1, f(x1), x2, f(x2), color]);
        });
    
    fns2
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx0) {
                let g = f(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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