数学 - Python - JavaScript - 解析学 - 多変数の関数 - 微分可能性と勾配ベクトル(第n偏微分係数、連続性、極限)

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〈3〉(松坂和夫(著)、岩波書店)の第14章(多変数の関数)、14.1(微分可能性と勾配ベクトル)、問題7.を取り組んでみる。

$\begin{array}{l} (x, y) \neq 0 \\ (D_{1} f) (x, y) \\ = \frac{y (x^{2} + y^{2}) - x y \cdot 2 x}{{(x^{2} + y^{2})}^{2}} \\ = \frac{y^{3} - x^{2} y}{{(x^{2} + y^{2})}^{2}} \\ (D_{2} f) (x, y) \\ = \frac{x^{3} - x y^{2}}{{(x^{2} + y^{2})}^{2}} \end{array}$

$\begin{array}{l} (x, y) = (0, 0) \\ (D_{1} f) (x, y) \\ = \lim_{h \to 0} \frac{f (0 + h, 0) - f (0, 0)}{h} \\ = \lim_{h \to 0} \frac{f (h, 0)}{h} \\ = \lim_{h \to 0} \frac{0}{h} \\ = 0 \\ (D_{2} f) (x, y) = 0 \end{array}$

よって、すべての点において第1偏微係数、第2偏微係数が存在する。

原点における連続性について。

$\begin{array}{l} a \neq 0 \\ y = a x \end{array}$

という直線を考える。

この直線上の点が原点に近づくとき、

$\begin{array}{l} \lim_{(x, y) \to (0, 0)} f (x, y) \\ = \lim_{x \to 0} \frac{x \cdot a x}{x^{2} + a^{2} x^{2}} \\ = \frac{a}{1 + a^{2}} \\ \neq 0 \end{array}$

よって f は原点において連続ではない。

コード(Emacs)

Python 3

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

x, y = symbols('x, y')
f = x * y / (x ** 2 + y ** 2)

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

入出力結果(Terminal, Jupyter(IPython))

$ ./sample7.py
∂ ⎛  x⋅y  ⎞
──⎜───────⎟
∂x⎜ 2    2⎟
  ⎝x  + y ⎠

-y⋅(x - y)⋅(x + y) 
───────────────────
              2    
     ⎛ 2    2⎞     
     ⎝x  + y ⎠     


∂ ⎛  x⋅y  ⎞
──⎜───────⎟
∂y⎜ 2    2⎟
  ⎝x  + y ⎠

x⋅(x - y)⋅(x + y)
─────────────────
             2   
    ⎛ 2    2⎞    
    ⎝x  + y ⎠    


$

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="0.001">
<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="1">

<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="sample7.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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_a0],
    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);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [],
        f = (x) => x === 0 ? 0 : x * (a0 * x) / (x ** 2 + (a0 * x) ** 2),
        fns = [[f, '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 =
a =

Kamimura's blog

2018年1月14日日曜日

数学 - Python - JavaScript - 解析学 - 多変数の関数 - 微分可能性と勾配ベクトル(第n偏微分係数、連続性、極限)

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