数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数の積分(平方根、逆数、正弦、余弦、逆三角関数(逆正弦関数))

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)
Nebo(Windows アプリ)
iPad Pro + Apple Pencil
MyScript Nebo(iPad アプリ)
参考書籍

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)、3(三角関数の積分)、練習問題14、15、16、17.を取り組んでみる。

$\begin{array}{l} x = 3 \sin t \\ \frac{dx}{dt} = 3 \cos t \\ \int \frac{1}{\sqrt{9 - 9 \sin^{2} t}} \cdot 3 \cos t dt \\ = \int 1 dt \\ = t \\ = \arcsin \frac{x}{3} \end{array}$
$\begin{array}{l} x = \sqrt{3} \sin t \\ \frac{dx}{dt} = \sqrt{3} \cos t \\ \int \frac{1}{\sqrt{3 - 3 \sin^{2} t}} \sqrt{3} \cos t dt \\ = \int A dt \\ = t \\ = \arcsin \frac{x}{\sqrt{3}} \end{array}$
$\begin{array}{l} x = \frac{1}{\sqrt{2}} \sin t \\ \frac{dx}{dt} = \frac{1}{\sqrt{2}} \cos t \\ \int \frac{1}{\sqrt{2 - 2 \sin^{2} t}} \cdot \frac{1}{\sqrt{2}} \cos t dt \\ = \frac{1}{2} \int 1 dt \\ = \frac{1}{2} t \\ = \frac{1}{2} \arcsin (\sqrt{2} x) \end{array}$
$\begin{array}{l} x = \frac{a}{b} \sin t \\ \frac{dx}{dt} = \frac{a}{b} \cos t \\ \int \frac{1}{a \sqrt{1 - \sin^{2} t}} \frac{a}{b} \cos t dt \\ = \frac{1}{b} \int 1 dt \\ = \frac{1}{b} t \\ = \frac{1}{b} \arcsin (\frac{b}{a} x) \end{array}$

コード(Emacs)

Python 3

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

x, a, b = symbols('x, a, b')
fs = [1 / sqrt(9 - x ** 2),
      1 / sqrt(3 - x ** 2),
      1 / sqrt(2 - 4 * x ** 2),
      1 / sqrt(a ** 2 - b ** 2 * x ** 2)]

for i, f in enumerate(fs, 14):
    print(f'{i}.')
    I = Integral(f, x)
    for t in [I, I.doit()]:
        pprint(t)
        print()

p = plot(*fs[:3], legend=True, show=False)
for i, color in enumerate(['red', 'green', 'blue']):
    p[i].line_color = color
p.save('sample14.svg')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample14.py
14.
⌠                 
⎮       1         
⎮ ───────────── dx
⎮    __________   
⎮   ╱    2        
⎮ ╲╱  - x  + 9    
⌡                 

    ⎛x⎞
asin⎜─⎟
    ⎝3⎠

15.
⌠                 
⎮       1         
⎮ ───────────── dx
⎮    __________   
⎮   ╱    2        
⎮ ╲╱  - x  + 3    
⌡                 

    ⎛√3⋅x⎞
asin⎜────⎟
    ⎝ 3  ⎠

16.
⌠                   
⎮        1          
⎮ ─────────────── dx
⎮    ____________   
⎮   ╱      2        
⎮ ╲╱  - 4⋅x  + 2    
⌡                   

⎧-ⅈ⋅acosh(√2⋅x)         │ 2│    
⎪───────────────  for 2⋅│x │ > 1
⎪       2                       
⎨                               
⎪  asin(√2⋅x)                   
⎪  ──────────       otherwise   
⎩      2                        

17.
⌠                   
⎮        1          
⎮ ─────────────── dx
⎮    ____________   
⎮   ╱  2    2  2    
⎮ ╲╱  a  - b ⋅x     
⌡                   

⎧        ⎛b⋅x⎞                  
⎪-ⅈ⋅acosh⎜───⎟       │ 2  2│    
⎪        ⎝ a ⎠       │b ⋅x │    
⎪──────────────  for │─────│ > 1
⎪      b             │   2 │    
⎪                    │  a  │    
⎨                               
⎪      ⎛b⋅x⎞                    
⎪  asin⎜───⎟                    
⎪      ⎝ a ⎠                    
⎪  ─────────        otherwise   
⎪      b                        
⎩                               

$

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="a0">a = </label>
<input id="a0" type="number" value="2">
<label for="b0">b = </label>
<input id="b0" type="number" value="3">

<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="sample14.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_a0 = document.querySelector('#a0'),
    input_b0 = document.querySelector('#b0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0, input_b0],
    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),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    
    
    let points = [],
        lines = [[-Math.PI, y1, -Math.PI, y2, 'orange'],
                 [Math.PI, y1, Math.PI, y2, 'brown']],
        fns = [[(x) => 1 / Math.sqrt(9 - x ** 2), 'red'],
               [(x) => 1 / Math.sqrt(3 - x ** 2), 'green'],
               [(x) => 1 / Math.sqrt(2 - 4 * x ** 2), 'blue'],
               [(x) => 1 / Math.sqrt(a0 ** 2 - b0 ** 2 * x ** 2), 'purple']], 
        fns1 = [],
        fns2 = [];

    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();

r = dx =
x1 = x2 =
y1 = y2 =
a = b =

Kamimura's blog

2018年6月7日木曜日

数学 - Python - JavaScript - 解析学 - 積分 - 積分の計算 - 三角関数の積分(平方根、逆数、正弦、余弦、逆三角関数(逆正弦関数))

0 コメント:

コメントを投稿