## 2018年1月27日土曜日

### 数学 - Python - JavaScript - 解析学 - 多変数の関数 - 高次偏導関数、テイラーの定理(反復偏微分、連続性、nニュートン商、極限)

1. $\begin{array}{}\left({D}_{1}{D}_{2}f\right)\left(x,y\right)\\ =\frac{\left(5{x}^{4}-12{x}^{2}{y}^{2}-{y}^{4}\right){\left({x}^{2}+{y}^{2}\right)}^{2}-\left({x}^{5}-4{x}^{3}{y}^{2}-x{y}^{4}\right)2\left({x}^{2}+{y}^{2}\right)2x}{{\left({x}^{2}+{y}^{2}\right)}^{4}}\end{array}$
$\begin{array}{}\left({D}_{2}{D}_{1}f\right)\left(x,y\right)\\ =\frac{\left({x}^{4}+12{x}^{2}{y}^{2}-5{y}^{4}\right){\left({x}^{2}+{y}^{2}\right)}^{2}-\left({x}^{4}y+4{x}^{2}{y}^{3}-{y}^{5}\right)2\left({x}^{2}+{y}^{2}\right).2y}{{\left({x}^{2}+{y}^{2}\right)}^{4}}\end{array}$

2. $\begin{array}{}\left({D}_{1}{D}_{2}f\right)\left(0,0\right)\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{D}_{2}f\left(0+h,0\right)-{D}_{2}f\left(0,0\right)}{h}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{D}_{2}f\left(h,0\right)-0}{h}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{h}^{5}}{{h}^{4}}·\frac{1}{h}\\ =\underset{h\to 0}{\mathrm{lim}}1\\ =1\end{array}$
$\begin{array}{}\left({D}_{2}{D}_{1}f\right)\left(0,0\right)\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{D}_{1}f\left(0,h\right)-{D}_{1}f\left(0,0\right)}{h}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{-{h}^{5}}{{h}^{4}}·\frac{1}{h}\\ =\underset{h\to 0}{\mathrm{lim}}-1\\ =-1\end{array}$

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, Limit, Derivative

x, y = symbols('x, y')

f = x * y * (x ** 2 - y ** 2) / (x ** 2 + y ** 2)
D1 = Derivative(f, x, 1)
D2 = Derivative(f, y, 1)
D12 = Derivative(D2, x, 1)
D21 = Derivative(D1, y, 1)

for D in [D12, D21]:
pprint(D.factor())
print()

h = symbols('h', nonzero=True)

for dir in ['+', '-']:
l1 = Limit(D2.subs({x: h, y: 0}) / h, h, 0, dir=dir)
l2 = Limit(D1.subs({x: 0, y: h}) / h, h, 0, dir=dir)
for l in [l1, l2]:
for t in [l, l.doit()]:
pprint(t)
print()
print()
print()

$./sample1_2.py ⎛ 4 2 2 4⎞ (x - y)⋅(x + y)⋅⎝x + 10⋅x ⋅y + y ⎠ ──────────────────────────────────── 3 ⎛ 2 2⎞ ⎝x + y ⎠ ⎛ 4 2 2 4⎞ (x - y)⋅(x + y)⋅⎝x + 10⋅x ⋅y + y ⎠ ──────────────────────────────────── 3 ⎛ 2 2⎞ ⎝x + y ⎠ ⎛⎛ ⎛ ⎛ 2 2⎞⎞⎞│ ⎞ ⎜⎜∂ ⎜h⋅y⋅⎝h - y ⎠⎟⎟│ ⎟ ⎜⎜──⎜─────────────⎟⎟│ ⎟ ⎜⎜∂y⎜ 2 2 ⎟⎟│ ⎟ ⎜⎝ ⎝ h + y ⎠⎠│y=0⎟ lim ⎜───────────────────────⎟ h─→0⁺⎝ h ⎠ 1 ⎛⎛ ⎛ ⎛ 2 2⎞⎞⎞│ ⎞ ⎜⎜∂ ⎜h⋅x⋅⎝- h + x ⎠⎟⎟│ ⎟ ⎜⎜──⎜───────────────⎟⎟│ ⎟ ⎜⎜∂x⎜ 2 2 ⎟⎟│ ⎟ ⎜⎝ ⎝ h + x ⎠⎠│x=0⎟ lim ⎜─────────────────────────⎟ h─→0⁺⎝ h ⎠ -1 ⎛⎛ ⎛ ⎛ 2 2⎞⎞⎞│ ⎞ ⎜⎜∂ ⎜h⋅y⋅⎝h - y ⎠⎟⎟│ ⎟ ⎜⎜──⎜─────────────⎟⎟│ ⎟ ⎜⎜∂y⎜ 2 2 ⎟⎟│ ⎟ ⎜⎝ ⎝ h + y ⎠⎠│y=0⎟ lim ⎜───────────────────────⎟ h─→0⁻⎝ h ⎠ 1 ⎛⎛ ⎛ ⎛ 2 2⎞⎞⎞│ ⎞ ⎜⎜∂ ⎜h⋅x⋅⎝- h + x ⎠⎟⎟│ ⎟ ⎜⎜──⎜───────────────⎟⎟│ ⎟ ⎜⎜∂x⎜ 2 2 ⎟⎟│ ⎟ ⎜⎝ ⎝ h + x ⎠⎠│x=0⎟ lim ⎜─────────────────────────⎟ h─→0⁻⎝ h ⎠ -1$

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="x0">x0 = </label>
<input id="x0" type="number" value="1">
<label for="y0">y0 = </label>
<input id="y0" 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="sample1_2.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
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'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_x0, input_y0],
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 f = (x, y) => x * y * (x ** 2 - y ** 2) / (x ** 2 + y ** 2);

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);

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

let points = [],
lines = [],
fns = [[(x) => f(x, y0), 'green'],
[(x) => f(x0, x), 'orange']];

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])

let yscale = d3.scaleLinear()
.domain([y1, y2])

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();