2018年3月20日火曜日

学習環境

解析入門〈3〉(松坂 和夫(著)、岩波書店)の第14章(多変数の関数)、14.3(極値問題)、問題1-(b).を取り組んでみる。



    1. D 1 f x , y = 2 x a 2 D 2 f x , y = - 2 y b 2
      D 1 f x , y = 0 D 2 f x , y = 0 x = 0 , y = 0

      よって、臨界点は

      0 , 0

      第2次導関数の計算。

      D 1 2 f x , y = 2 a 2 D 2 2 f x , y = - 2 b 2 D 1 D 2 f x , y = 0

      よって、

      Δ 0 , 0 = D 1 D 2 f 0 , 0 - D 1 2 f 0 , 0 2 2 f 0 , 0 = 0 - 2 a 2 · - 2 b 2 = 4 a 2 b 2 > 0

      ゆえに極値点ではない。

macOS High Sierraの標準搭載されているグラフ作成ソフト、Grapher で作成。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Derivative, Matrix, solve

x, y, a, b = symbols('x, y, a, b')
f = x ** 2 / a ** 2 - y ** 2 / b ** 2

for t in [x, y]:
    for n in range(1, 3):
        D = Derivative(f, t, n)
        for s in [D, D.doit()]:
            pprint(s)
            print()
Dxy = Derivative(Derivative(f, x, 1).doit(), y, 1).doit()
pprint(Dxy)

delta = Dxy - Derivative(f, x, 2).doit() * Derivative(f, y, 2).doit()

critical_point = solve(
    (Derivative(f, x, 1).doit(), Derivative(f, y, 1).doit()), dict=True)

for t in [delta, critical_point]:
    pprint(t)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample1.py
  ⎛   2    2⎞
∂ ⎜  y    x ⎟
──⎜- ── + ──⎟
∂x⎜   2    2⎟
  ⎝  b    a ⎠

2⋅x
───
  2
 a 

  2⎛   2    2⎞
 ∂ ⎜  y    x ⎟
───⎜- ── + ──⎟
  2⎜   2    2⎟
∂x ⎝  b    a ⎠

2 
──
 2
a 

  ⎛   2    2⎞
∂ ⎜  y    x ⎟
──⎜- ── + ──⎟
∂y⎜   2    2⎟
  ⎝  b    a ⎠

-2⋅y 
─────
   2 
  b  

  2⎛   2    2⎞
 ∂ ⎜  y    x ⎟
───⎜- ── + ──⎟
  2⎜   2    2⎟
∂y ⎝  b    a ⎠

-2 
───
  2
 b 

0
  4  
─────
 2  2
a ⋅b 

[{x: 0, y: 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="0.0001" 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">
<br>
<label for="x0">x0 = </label>
<input id="x0" type="number" value="0">
<label for="y0">y0 = </label>
<input id="y0" type="number" value="0">
<br>
<label for="a0">a0 = </label>
<input id="a0" type="number" value="2">
<label for="b0">b0 = </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="sample1.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_x0 = document.querySelector('#x0'),
    input_y0 = document.querySelector('#y0'),            
    input_a0 = document.querySelector('#a0'),        
    input_b0 = document.querySelector('#b0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_x0, input_y0, 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),
        x0 = parseFloat(input_x0.value),
        y0 = parseFloat(input_y0.value),
        a0 = parseFloat(input_a0.value),
        b0 = parseFloat(input_b0.value);
        
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        f = (x, y) => x ** 2 / a0 ** 2 - y ** 2 / b0 ** 2,
        g = (x) => f(x, y0),
        h = (y) => f(x0, y),
        lines = [],
        fns = [[g, 'red'],
               [h, '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();









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