## 2018年1月13日土曜日

### 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(方程式、楕円、接戦、局面、接平面、グラディエント(gradient)、直交、内積(スカラー積、ドット積))

1. $\begin{array}{}f\left(x,y\right)=\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}\\ n=gradf\left({x}_{0},{y}_{0}\right)=\left(\frac{2{x}_{0}}{{a}^{2}},\frac{2{y}_{0}}{{b}^{2}}\right)\\ n·\left(x,y\right)=n·\left({x}_{0},{y}_{0}\right)\\ \frac{2{x}_{0}x}{{a}^{2}}+\frac{2{y}_{0}y}{{b}^{2}}=\frac{2{x}_{0}^{2}}{{a}^{2}}+\frac{2{y}_{0}^{2}}{{b}^{2}}\\ \frac{2{x}_{0}x}{{a}^{2}}+\frac{2{y}_{0}y}{{b}^{2}}=2\\ \frac{{x}_{0}x}{{a}^{2}}+\frac{{y}_{0}y}{{b}^{2}}=1\end{array}$

2. $\begin{array}{}f\left(x,y\right)={x}^{2}+{y}^{2}\\ u=\left(\left(x,y\right),z\right)\\ g\left(u\right)=f\left(x,y\right)-z\\ grad\left(g\left(1,-2,5\right)\right)·\left(\left(\left(x,y\right),z\right)-\left(\left(1,-2\right),5\right)\right)=0\\ gradf\left(1,-2\right)·\left(x-1,y+2\right)-\left(z-5\right)=0\\ \left(2,-4\right)·\left(x-1,y+2\right)-\left(z-5\right)=0\\ 2x-2-4y-8-z+5=0\\ 2x-4y-z=5\end{array}$

3. $\begin{array}{}f\left(x,y,z\right)=xyz\\ n=gradf\left({x}_{0},{y}_{0},{z}_{0}\right)=\left({y}_{0}{z}_{0},{z}_{0}{x}_{0},{x}_{0}{y}_{0}\right)\\ n·\left(x,y,z\right)=n·\left({x}_{0},{y}_{0},{z}_{0}\right)\\ x{y}_{0}{z}_{0}+{x}_{0}y{z}_{0}+{x}_{0}{y}_{0}z=3{x}_{0}{y}_{0}{z}_{0}\\ x{y}_{0}{z}_{0}+{x}_{0}y{z}_{0}+{x}_{0}{y}_{0}z=3\end{array}$

4. $\begin{array}{}f\left(x,y\right)=xy\\ u=\left(\left(x,y\right),z\right)\\ g\left(u\right)=f\left(x,y\right)-{z}^{2}\\ grad\left(g\left(\left(1,4\right),2\right)\right)=\left(\left(\left(x,y\right),z\right)-\left(\left(1,4\right),2\right)\right)=0\\ gradf\left(1,4\right)·\left(x-1,y-4\right)-4\left(z-2\right)=0\\ 4\left(x-1\right)+\left(y-4\right)-4\left(z-2\right)=0\\ 4x+y-4z=0\end{array}$

コード(Emacs)

Python 3

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

x, y, x0, y0, a, b = symbols('x, y, x0, y0, a, b')
eq1 = x ** 2 / a ** 2 + y ** 2 / b ** 2 - 1
eq2 = x0 * x / a ** 2 + y0 * y / b ** 2 - 1
for eq in [eq1, eq2]:
pprint(eq)
print()
for t in solve(eq, y):
pprint(t)
print()
print()


$./sample6.py 2 2 y x -1 + ── + ── 2 2 b a _________________ -b⋅╲╱ (a - x)⋅(a + x) ─────────────────────── a _________________ b⋅╲╱ (a - x)⋅(a + x) ───────────────────── a y⋅y₀ x⋅x₀ -1 + ──── + ──── 2 2 b a 2 ⎛ 2 ⎞ b ⋅⎝a - x⋅x₀⎠ ────────────── 2 a ⋅y₀$


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">
<label for="b0">b = </label>
<input id="b0" type="number" value="1">

<label for="x0">x0 = </label>
<input id="x0" type="number" value="0">
<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="sample6.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_a0 = document.querySelector('#a0'),
input_b0 = document.querySelector('#b0'),
input_x0 = document.querySelector('#x0'),
input_y0 = document.querySelector('#y0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_a0, input_b0, 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 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),
b0 = parseFloat(input_b0.value),
x0 = parseFloat(input_x0.value),
y0 = parseFloat(input_y0.value);

if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}
if (x0 ** 2 / a0 ** 2 + y0 ** 2 / b0 ** 2 !== 1) {
p(点(${x0},${y0})は楕円上の点ではない);
}

let points = [],
lines = [],
f1 = (x) => -b0 * Math.sqrt(a0 ** 2 - x ** 2) / a0,
f2 = (x) => -f1(x),
g = (x) => b0 ** 2 * (a0 ** 2 - x * x0) / (a0 ** 2 * y0),
fns = [[f1, 'red'],
[f2, 'red'],
[g, '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();