2017年8月23日水曜日

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2部(微分と基本的な関数)、第6章(曲線をえがくこと)、4(極座標)、練習問題10、11、12.を取り組んでみる。


  1. cos3θ =cos( 2θ+θ ) =cos2θcosθsin2θsinθ =( cos 2 θ sin 2 θ )cosθ2sinθcosθsinθ =( 2 cos 2 θ1 )cosθ2( 1 cos 2 θ )cosθ =2 cos 3 θcosθ2cosθ+2 cos 3 θ =4 cos 3 θ3cosθ x=rcosθ cosθ= x x 2 + y 2 x 2 + y 2 = 4 x 3 ( x 2 + y 2 ) x 2 + y 2 3x x 2 + y 2 ( x 2 + y 2 ) 2 =4 x 3 3x( x 2 + y 2 ) x 4 +4 x 2 y 2 + y 4 =4 x 3 3 x 3 3x y 2 x 4 x 3 +4 x 2 y 2 +3x y 2 + y 4 =0

  2. cos3θ =cos( 2θ+θ ) =cos2θcosθsin2θsinθ =( cos 2 θ sin 2 θ )cosθ2sinθcosθsinθ =( 2 cos 2 θ1 )cosθ2( 1 cos 2 θ )cosθ =2 cos 3 θcosθ2cosθ+2 cos 3 θ =4 cos 3 θ3cosθ x=rcosθ cosθ= x x 2 + y 2 x 2 + y 2 =| 4 x 3 ( x 2 + y 2 ) x 2 + y 2 3x x 2 + y 2 | ( x 2 + y 2 ) 2 =| 4 x 3 3x( x 2 + y 2 ) |

  3. x=rcosθ cosθ= x r cosθ= x x 2 + y 2 x 2 + y 2 = 6x x 2 + y 2 x 2 + y 2 =6x ( x3 ) 2 + y 2 = 3 2

    中心(3, 0)、半径3の円。

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, solve, sqrt, plot

x, y = symbols('x y', real=True)
eqs = [x ** 4 - x ** 3 + 4 * x ** 2 * y ** 2 + 3 * x * y ** 2 + y ** 4,
       (x ** 2 + y ** 2) ** 2 - abs(4 * x ** 3 - 3 * x * (x ** 2 + y ** 2)),
       (x - 3) ** 2 + y ** 2 - 3 ** 2]

for i, eq in enumerate(eqs, 10):
    print(f'{i}.')
    s = solve(eq, y)
    pprint(s)
    try:
        p = plot(*s, show=False, legend=True)
        p.save(f'sample{i}.svg')
    except Exception as err:
        print(type(err), err)
    print()

入出力結果(Terminal, IPython)

$ ./sample10.py
10.
⎡       ________________________________________        ______________________
⎢      ╱               __________________              ╱               _______
⎢     ╱               ╱     2                         ╱               ╱     2 
⎢    ╱        2   x⋅╲╱  12⋅x  + 28⋅x + 9    3⋅x      ╱        2   x⋅╲╱  12⋅x  
⎢-  ╱    - 2⋅x  - ─────────────────────── - ─── ,   ╱    - 2⋅x  - ────────────
⎣ ╲╱                         2               2    ╲╱                         2

__________________         ________________________________________        ___
___________               ╱               __________________              ╱   
                         ╱               ╱     2                         ╱    
+ 28⋅x + 9    3⋅x       ╱        2   x⋅╲╱  12⋅x  + 28⋅x + 9    3⋅x      ╱     
─────────── - ─── , -  ╱    - 2⋅x  + ─────────────────────── - ─── ,   ╱    - 
               2     ╲╱                         2               2    ╲╱       

_____________________________________⎤
            __________________       ⎥
           ╱     2                   ⎥
   2   x⋅╲╱  12⋅x  + 28⋅x + 9    3⋅x ⎥
2⋅x  + ─────────────────────── - ─── ⎥
                  2               2  ⎦

11.
⎡      ______________________________         ______________________________  
⎢     ╱   ⎛         ___________    ⎞         ╱   ⎛         ___________    ⎞   
⎢√2⋅╲╱  x⋅⎝-2⋅x - ╲╱ -16⋅x + 9  + 3⎠   -√2⋅╲╱  x⋅⎝-2⋅x + ╲╱ -16⋅x + 9  + 3⎠   
⎢────────────────────────────────────, ──────────────────────────────────────,
⎢                 2                                      2                    
⎢                                                                             
⎣                                                                             

       ______________________________         ______________________________  
      ╱   ⎛         ___________    ⎞         ╱    ⎛        ___________    ⎞   
 √2⋅╲╱  x⋅⎝-2⋅x + ╲╱ -16⋅x + 9  + 3⎠   -√2⋅╲╱  -x⋅⎝2⋅x + ╲╱ -16⋅x + 9  - 3⎠   
 ────────────────────────────────────, ──────────────────────────────────────,
                  2                                      2                    
                                                                              
                                                                              

 ⎧      _____________________________                                         
 ⎪     ╱   ⎛         __________    ⎞        2 ⎛          __________    ⎞      
 ⎪√2⋅╲╱  x⋅⎝-2⋅x - ╲╱ 16⋅x + 9  - 3⎠       x ⋅⎝8⋅x + 3⋅╲╱ 16⋅x + 9  + 9⎠      
 ⎨───────────────────────────────────  for ───────────────────────────── ≥ 0, 
 ⎪                 2                                     2                    
 ⎪                                                                            
 ⎩                nan                                otherwise                

⎧      _____________________________                                         ⎧
⎪     ╱   ⎛         __________    ⎞        2 ⎛          __________    ⎞      ⎪
⎪√2⋅╲╱  x⋅⎝-2⋅x + ╲╱ 16⋅x + 9  - 3⎠       x ⋅⎝8⋅x - 3⋅╲╱ 16⋅x + 9  + 9⎠      ⎪
⎨───────────────────────────────────  for ───────────────────────────── ≥ 0, ⎨
⎪                 2                                     2                    ⎪
⎪                                                                            ⎪
⎩                nan                                otherwise                ⎩

       _____________________________                                          
      ╱    ⎛        __________    ⎞         2 ⎛          __________    ⎞      
-√2⋅╲╱  -x⋅⎝2⋅x - ╲╱ 16⋅x + 9  + 3⎠        x ⋅⎝8⋅x - 3⋅╲╱ 16⋅x + 9  + 9⎠      
─────────────────────────────────────  for ───────────────────────────── ≥ 0, 
                  2                                      2                    
                                                                              
                 nan                                 otherwise                

⎧       _____________________________                                        ⎤
⎪      ╱    ⎛        __________    ⎞         2 ⎛          __________    ⎞    ⎥
⎪-√2⋅╲╱  -x⋅⎝2⋅x + ╲╱ 16⋅x + 9  + 3⎠        x ⋅⎝8⋅x + 3⋅╲╱ 16⋅x + 9  + 9⎠    ⎥
⎨─────────────────────────────────────  for ───────────────────────────── ≥ 0⎥
⎪                  2                                      2                  ⎥
⎪                                                                            ⎥
⎩                 nan                                 otherwise              ⎦
<class 'TypeError'> Invalid comparison of complex -3550.0 + 1843.23085911668*I

12.
⎡  ____________     ____________⎤
⎣╲╱ x⋅(-x + 6) , -╲╱ -x⋅(x - 6) ⎦

$

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="dΘ">dΘ = </label>
<input id="dΘ" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" 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_dΘ = document.querySelector('#dΘ'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dΘ, 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 f10 = (Θ) => Math.cos(3 * Θ),
    f11 = (Θ) => Math.abs(Math.cos(3 * Θ)),
    f12 = (Θ) => 6 * Math.cos(Θ);
        
let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),= parseFloat(input_dΘ.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

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

    let points = [],
        lines = [],
        fns = [[f10, 'rgba(255, 0, 0, 0.1)'],
               [f11, 'rgba(0, 255, 0, 0.1)'],
               [f12, 'rgba(0, 0, 255, 0.1)']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let Θ = 0; Θ <= 2 * Math.PI; Θ +=) {
                let r = f(Θ),
                    x = r * Math.cos(Θ),
                    y = r * Math.sin(Θ);

                points.push([x, y, 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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