数学 - Python - JavaScript - 解析学 - 重積分の変数変換 - 変数変換定理(カルジオイド(cardioid)、曲線、累乗、極座標、面積、積分、三角関数(余弦、正弦)、累乗(べき乗))

学習環境

解析入門(下) (松坂和夫数学入門シリーズ 6) (松坂和夫(著)、岩波書店)の第24章(重積分の変数変換)、24.2(変数変換定理)、問題5-(a).を取り組んでみる。

1. $\begin{array}{l} 2 \int_{0}^{π} \int_{0}^{a (1 + \cos θ)} r d r d θ \\ = \int_{0}^{π} a^{2} {(1 + \cos θ)}^{2} d θ \\ = a^{2} \int_{0}^{π} (1 + \cos^{2} θ + 2 \cos θ) d θ \\ = a^{2} ({[θ]}_{0}^{π} + 2 {[\sin θ]}_{0}^{π} + {[\frac{1}{2} \cos θ \sin θ]}_{0}^{π} + \frac{1}{2} \int_{0}^{π} 1 d θ) \\ = a^{2} (π + \frac{1}{2} π) \\ = \frac{3}{2} π a^{2} \end{array}$

コード(Emacs)

Python 3

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

print('5.')

r, theta, a = symbols('r, θ, a')

I = 2 * Integral(Integral(r, (r, 0, a * (1 + cos(theta)))),
                 (theta, 0, pi))


for t in [I, I.doit()]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample5.py
5.
  π a⋅(cos(θ) + 1)        
  ⌠       ⌠               
2⋅⎮       ⎮        r dr dθ
  ⌡       ⌡               
  0       0               

     2
3⋅π⋅a 
──────
  2   

$

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="0.005">
<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="sample5.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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 a = 2,
    fr = theta => a * (1 + Math.cos(theta)),
    fns = [];

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 = [];

    for (let theta = 0; theta < Math.PI; theta += dx) {
        let r0 = fr(theta);
        
        points.push([r0 * Math.cos(theta), r0 * Math.sin(theta), 'red']);
    }

    for (let theta = Math.PI; theta < 2 * Math.PI; theta += dx) {
        let r0 = fr(theta);
        
        points.push([r0 * Math.cos(theta), r0 * Math.sin(theta), 'green']);
    }
    
    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]);
                }
            }
        });
    
    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('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.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.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
    p(fns.join('\n'));
};

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

r = dx =
x1 = x2 =
y1 = y2 =

Kamimura's blog

2018年12月10日月曜日

数学 - Python - JavaScript - 解析学 - 重積分の変数変換 - 変数変換定理(カルジオイド(cardioid)、曲線、累乗、極座標、面積、積分、三角関数(余弦、正弦)、累乗(べき乗))

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