## 2018年2月14日水曜日

### 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(逆数、平方根、3次までのテイラー多項式)

1. $\begin{array}{}f\left(x,y\right)={\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{1}{2}}\\ {D}_{1}f\left(x,y\right)=-\frac{1}{2}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{3}{2}}\left(1+y\right)\\ {D}_{2}f\left(x,y\right)=-\frac{1}{2}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{3}{2}}\left(1+x\right)\\ {D}_{1}^{2}f\left(x,y\right)=\frac{3}{4}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{5}{2}}{\left(1+y\right)}^{2}\\ {D}_{1}{D}_{2}f\left(x,y\right)=\frac{3}{4}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{5}{2}}\left(1+x\right)\left(1+y\right)-\frac{1}{2}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{3}{2}}\\ {D}_{2}^{2}f\left(x,y\right)=\frac{3}{4}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{5}{2}}{\left(1+x\right)}^{2}\\ {D}_{1}^{3}f\left(x,y\right)=-\frac{15}{8}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{7}{2}}{\left(1+y\right)}^{3}\\ {D}_{1}^{2}{D}_{2}f\left(x,y\right)=-\frac{15}{8}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{7}{2}}\left(1+x\right){\left(1+y\right)}^{2}+\frac{3}{4}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{5}{2}}\left(1+y\right)2\\ {D}_{1}{D}_{2}^{2}f\left(x,y\right)=-\frac{15}{8}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{7}{2}}{\left(1+y\right)}^{2}\left(1+y\right)+\frac{3}{4}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{5}{2}}\left(1+x\right)2\\ {D}_{2}^{3}f\left(x,y\right)=-\frac{15}{8}{\left(\left(1+x\right)\left(1+y\right)\right)}^{-\frac{7}{2}}{\left(1+x\right)}^{3}\end{array}$

よって、 求める3次のテイラー多項式は、

$\begin{array}{}f\left(0,0\right)+\left({D}_{1}f\left(0,0\right)x+{D}_{2}f\left(0,0\right)y\right)\\ +\frac{1}{2!}\left({D}_{1}^{2}f\left(0,0\right){x}^{2}+2{D}_{1}{D}_{2}f\left(0,0\right)xy+{D}_{2}^{2}f\left(0,0\right){y}^{2}\right)\\ +\frac{1}{3!}\left({D}_{1}^{3}f\left(0,0\right){x}^{3}+3{D}_{1}^{2}{D}_{2}f\left(0,0\right){x}^{2}y+3{D}_{1}{D}_{2}^{2}f\left(0,0\right)x{y}^{2}+{D}_{2}^{3}f\left(0,0\right){y}^{3}\right)\\ =1+\left(-\frac{1}{2}x-\frac{1}{2}y\right)\\ +\frac{1}{2}\left(\frac{3}{4}{x}^{2}+2\left(\frac{3}{4}-\frac{1}{2}\right)xy+\frac{3}{4}{y}^{2}\right)\\ +\frac{1}{6}\left(-\frac{15}{8}{x}^{3}+3\left(-\frac{15}{8}+\frac{3}{2}\right){x}^{2}y+3\left(-\frac{15}{8}+\frac{3}{2}\right)x{y}^{2}-\frac{15}{8}{y}^{3}\right)\\ =1-\frac{1}{2}\left(x+y\right)+\frac{1}{8}\left(3{x}^{2}+2xy+3{y}^{2}\right)-\frac{1}{16}\left(5{x}^{3}+3{x}^{2}y+3x{y}^{2}+5{y}^{3}\right)\end{array}$

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, sqrt, Derivative, factorial

a, b = symbols('a, b', nonzero=True)
x, y = symbols('x, y')
f = 1 / sqrt((1 + x) * (1 + y))
d = {x: 0, y: 0}
Dx = Derivative(f, x, 1)
Dy = Derivative(f, y, 1)
Dxx = Derivative(f, x, 2)
Dyy = Derivative(f, y, 2)
Dxy = Derivative(Dx, y, 1)
Dxxx = Derivative(f, x, 3)
Dyyy = Derivative(f, y, 3)
Dxxy = Derivative(Dxx, y, 1)
Dxyy = Derivative(Dyy, x, 1)
expr = f.subs(d) + (Dx.subs(d) * x + Dy.subs(d) * y) + 1 / factorial(2) * (Dxx.subs(d) * x ** 2 + 2 * Dxy.subs(d) * x * y + Dyy.subs(d) * y ** 2) + \
1 / factorial(3) * (Dxxx.subs(d) * x ** 3 + 3 * Dxxy.subs(d) * x ** 2 *
y + 3 * Dxyy.subs(d) * x * y ** 2 + Dyyy.subs(d) * y ** 3)

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


$./sample10.py 1 ─────────────────── _________________ ╲╱ (x + 1)⋅(y + 1) ⎛ 3 ⎞│ ⎛ 2 ⎞│ 3 ⎜ d ⎛ 1 ⎞⎟│ 2 ⎜ d ⎛ 1 ⎞⎟│ x ⋅⎜───⎜─────────⎟⎟│ x ⋅⎜───⎜─────────⎟⎟│ ⎜ 3⎜ _______⎟⎟│ 2 ⎜ 2⎜ _______⎟⎟│ 2 ⎝dx ⎝╲╱ x + 1 ⎠⎠│x=0 3⋅x ⋅y ⎝dx ⎝╲╱ x + 1 ⎠⎠│x=0 3⋅x⋅y x⋅y ─────────────────────── - ────── + ─────────────────────── - ────── + ─── + x⋅ 6 16 2 16 4 ⎛ 3 ⎞│ ⎛ 2 ⎞│ 3 ⎜ d ⎛ 1 ⎞⎟│ 2 ⎜ d ⎛ 1 ⎞⎟│ y ⋅⎜───⎜─────────⎟⎟│ y ⋅⎜───⎜─────────⎟⎟│ ⎜ 3⎜ _______⎟⎟│ ⎜ 2⎜ _______⎟⎟│ ⎛d ⎛ 1 ⎞⎞│ ⎝dy ⎝╲╱ y + 1 ⎠⎠│y=0 ⎝dy ⎝╲╱ y + 1 ⎠⎠│y=0 ⎛d ⎜──⎜─────────⎟⎟│ + ─────────────────────── + ─────────────────────── + y⋅⎜─ ⎜dx⎜ _______⎟⎟│ 6 2 ⎜d ⎝ ⎝╲╱ x + 1 ⎠⎠│x=0 ⎝ ⎛ 1 ⎞⎞│ ─⎜─────────⎟⎟│ + 1 y⎜ _______⎟⎟│ ⎝╲╱ y + 1 ⎠⎠│y=0 3 2 2 2 3 2 5⋅x 3⋅x ⋅y 3⋅x 3⋅x⋅y x⋅y x 5⋅y 3⋅y y - ──── - ────── + ──── - ────── + ─── - ─ - ──── + ──── - ─ + 1 16 16 8 16 4 2 16 8 2$


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">
<br>
<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="sample10.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_y0 = document.querySelector('#y0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
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) => 1 / Math.sqrt((1 + x) * (1 + y)),
g = (x, y) => 1 - (x + y) / 2 + (3 * x ** 2 + 2 * x * y + 3 * y ** 2) / 8 -
(5 * x ** 3 + 3 * x ** 2 * y + 3 * x * y ** 2 + 5 * y ** 3) / 16;

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),
y0 = parseFloat(input_y0.value);

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

let points = [],
lines = [],
fns = [[(x) => f(x, y0), 'red'],
[(x) => g(x, y0), '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])

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();