## 2018年2月13日火曜日

### 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(三角関数(余弦)、指数関数、3次までのテイラー多項式)

1. $\begin{array}{}{D}_{1}f\left(x,y\right)=a{e}^{ax}\mathrm{cos}by\\ {D}_{2}f\left(x,y\right)=-b{e}^{ax}\mathrm{sin}by\\ {D}_{1}^{2}f\left(x,y\right)={a}^{2}{e}^{ax}\mathrm{cos}by\\ {D}_{1}{D}_{2}f\left(x,y\right)=-ab{e}^{ax}\mathrm{sin}by\\ {D}_{2}^{2}f\left(x,y\right)=-{b}^{2}{e}^{ax}\mathrm{cos}by\\ {D}_{1}^{3}f\left(x,y\right)={a}^{3}{e}^{ax}\mathrm{cos}by\\ {D}_{1}^{2}{D}_{2}f\left(x,y\right)=-{a}^{2}b{e}^{ax}\mathrm{sin}by\\ {D}_{1}{D}_{2}^{2}f\left(x,y\right)=-a{b}^{2}{e}^{ax}\mathrm{cos}by\\ {D}_{2}^{3}f\left(x,y\right)={b}^{3}{e}^{ax}\mathrm{sin}by\end{array}$

よって、 求める3次のテイラー多項式は、

$\begin{array}{}f\left(0,0\right)+\left({D}_{1}f\left(0,0\right)x+{D}_{2}f\left(0,0\right)y\right)\\ +\frac{1}{2!}\left({D}_{1}^{2}f\left(0,0\right){x}^{2}+2{D}_{1}{D}_{2}f\left(0,0\right)xy+{D}_{2}^{2}f\left(0,0\right){y}^{2}\right)\\ +\frac{1}{3!}\left({D}_{1}^{3}f\left(0,0\right){x}^{3}+3{D}_{1}^{2}{D}_{2}f\left(0,0\right){x}^{2}y+3{D}_{1}{D}_{2}^{2}f\left(0,0\right)x{y}^{2}+{D}_{2}^{3}f\left(0,0\right){y}^{3}\right)\\ =1+\left(ax\right)+\frac{1}{2}\left({a}^{2}{x}^{2}-{b}^{2}{y}^{2}\right)+\frac{1}{6}\left({a}^{3}{x}^{3}-3a{b}^{2}x{y}^{2}\right)\end{array}$

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, exp, cos, Derivative, factorial

a, b = symbols('a, b', nonzero=True)
x, y = symbols('x, y')
f = exp(a * x) * cos(b * 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()

$./sample10.py a⋅x ℯ ⋅cos(b⋅y) ⎛ 3 ⎞│ ⎛ 2 ⎞│ 3 ⎜ ∂ ⎛ a⋅x⎞⎟│ 2 ⎜ ∂ ⎛ a⋅x⎞⎟│ 3 x ⋅⎜───⎝ℯ ⎠⎟│ x ⋅⎜───⎝ℯ ⎠⎟│ y ⋅ 2 2 ⎜ 3 ⎟│ ⎜ 2 ⎟│ a⋅b ⋅x⋅y ⎝∂x ⎠│x=0 ⎝∂x ⎠│x=0 ⎛∂ ⎛ a⋅x⎞⎞│ - ───────── + ────────────────── + ────────────────── + x⋅⎜──⎝ℯ ⎠⎟│ + ─── 2 6 2 ⎝∂x ⎠│x=0 ⎛ 3 ⎞│ ⎛ 2 ⎞│ ⎜ ∂ ⎟│ 2 ⎜ ∂ ⎟│ ⎜───(cos(b⋅y))⎟│ y ⋅⎜───(cos(b⋅y))⎟│ ⎜ 3 ⎟│ ⎜ 2 ⎟│ ⎝∂y ⎠│y=0 ⎝∂y ⎠│y=0 ⎛∂ ⎞│ ─────────────────── + ────────────────────── + y⋅⎜──(cos(b⋅y))⎟│ + 1 6 2 ⎝∂y ⎠│y=0 3 3 2 2 2 2 2 2 a ⋅x a ⋅x a⋅b ⋅x⋅y b ⋅y ───── + ───── - ───────── + a⋅x - ───── + 1 6 2 2 2$

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="a0">a = </label>
<input id="a0" type="number" value="1">
<label for="b0">b = </label>
<input id="b0" type="number" value="1">
<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="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_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 = [],
lines = [],
f = (x, y) => Math.exp(a0 * x) * Math.cos(b0 * y),
g = (x, y) => 1 +
a0 * x +
1 / 2 * (a0 ** 2 * x ** 2 - b0 ** 2 * y ** 2) +
1 / 6 * (a0 ** 3 * x ** 3 - 3 * a0 * b0 ** 2 * x * y ** 2),
fns = [[(x) => f(x, y0), 'red'],
[(x) => f(x0, x), 'green'],
[(x) => g(x, y0), 'blue'],
[(x) => g(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();