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))

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6.を取り組んでみる。

1. $\begin{array}{l} f (x, y) = \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} \\ n = g r a d f (x_{0}, y_{0}) = (\frac{2 x_{0}}{a^{2}}, \frac{2 y_{0}}{b^{2}}) \\ n \cdot (x, y) = n \cdot (x_{0}, y_{0}) \\ \frac{2 x_{0} x}{a^{2}} + \frac{2 y_{0} y}{b^{2}} = \frac{2 x_{0}^{2}}{a^{2}} + \frac{2 y_{0}^{2}}{b^{2}} \\ \frac{2 x_{0} x}{a^{2}} + \frac{2 y_{0} y}{b^{2}} = 2 \\ \frac{x_{0} x}{a^{2}} + \frac{y_{0} y}{b^{2}} = 1 \end{array}$
2. $\begin{array}{l} f (x, y) = x^{2} + y^{2} \\ u = ((x, y), z) \\ g (u) = f (x, y) - z \\ g r a d (g (1, - 2, 5)) \cdot (((x, y), z) - ((1, - 2), 5)) = 0 \\ g r a d f (1, - 2) \cdot (x - 1, y + 2) - (z - 5) = 0 \\ (2, - 4) \cdot (x - 1, y + 2) - (z - 5) = 0 \\ 2 x - 2 - 4 y - 8 - z + 5 = 0 \\ 2 x - 4 y - z = 5 \end{array}$
3. $\begin{array}{l} f (x, y, z) = x y z \\ n = g r a d f (x_{0}, y_{0}, z_{0}) = (y_{0} z_{0}, z_{0} x_{0}, x_{0} y_{0}) \\ n \cdot (x, y, z) = n \cdot (x_{0}, y_{0}, z_{0}) \\ x y_{0} z_{0} + x_{0} y z_{0} + x_{0} y_{0} z = 3 x_{0} y_{0} z_{0} \\ x y_{0} z_{0} + x_{0} y z_{0} + x_{0} y_{0} z = 3 \end{array}$
4. $\begin{array}{l} f (x, y) = x y \\ u = ((x, y), z) \\ g (u) = f (x, y) - z^{2} \\ g r a d (g ((1, 4), 2)) = (((x, y), z) - ((1, 4), 2)) = 0 \\ g r a d f (1, 4) \cdot (x - 1, y - 4) - 4 (z - 2) = 0 \\ 4 (x - 1) + (y - 4) - 4 (z - 2) = 0 \\ 4 x + y - 4 z = 0 \end{array}$

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()

入出力結果(Terminal, Jupyter(IPython))

$ ./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,
    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'),
    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])
        .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 = b = x0 = y0 =

Kamimura's blog

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))

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