数学 - 線型代数 - ベクトル空間 - 1次従属と1次独立(三角関数(正弦、余弦)、指数関数(微分、導関数))

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)
数式入力ソフト(TeX, MathML): MathType
MathML対応ブラウザ: Firefox、Safari
MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
参考書籍

線型代数入門(松坂和夫(著)、岩波書店)の第2章(ベクトル空間)、6(1次従属と1次独立)、問5.を取り組んでみる。

1. $\begin{array}{l} c_{1} \cos t + c_{2} \sin t = 0 \\ \sqrt{c_{1}^{2} + c_{2}^{2}} (\frac{c_{1}}{\sqrt{c_{1}^{2} + c_{2}^{2}}} \cos t + \frac{c_{2}}{\sqrt{c_{1}^{2} + c_{2}^{2}}} \sin t) = 0 \\ \sin θ = \frac{c_{1}}{\sqrt{c_{1}^{2} + c_{2}^{2}}} \\ \cos θ = \frac{c_{2}}{\sqrt{c_{1}^{2} + c_{2}^{2}}} \\ \sqrt{c_{1}^{2} + c_{2}^{2}} (\sin θ \cos t + \cos θ \sin t) = 0 \\ \sqrt{c_{1}^{2} + c_{2}^{2}} \sin (θ + t) = 0 \\ \sqrt{c_{1}^{2} + c_{2}^{2}} = 0 \\ c_{1} = c_{2} = 0 \end{array}$
2. $\begin{array}{l} c_{1} e^{t} + c_{2} t^{2} = 0 \\ c_{1} e^{t} + 2 c_{2} t = 0 \\ c_{1} e^{t} + 2 c_{2} = 0 \\ c_{1} e^{t} = 0 \\ c_{1} = 0 \\ c_{2} = 0 \end{array}$
3. $\begin{array}{l} c_{1} e^{t} + c_{2} e^{2 t} + c_{3} e^{3 t} = 0 \\ c_{1} e^{t} + 2 c_{2} e^{2 t} + 3 c_{3} e^{3 t} = 0 \\ c_{1} e^{t} + 4 c_{2} e^{2 t} + 9 c_{3} e^{3 t} = 0 \\ t = 1 \\ c_{1} + c_{2} + c_{3} = 0 \\ c_{1} + 2 c_{2} + 3 c_{3} = 0 \\ c_{1} + 4 c_{2} + 9 c_{3} = 0 \\ c_{3} = - c_{1} - c_{2} \\ c_{1} + 2 c_{2} - 3 c_{1} - 3 c_{2} = 0 \\ - 2 c_{1} - c_{2} = 0 \\ c_{2} = - 2 c_{1} \\ c_{3} = - c_{1} + 2 c_{1} = c_{1} \\ c_{1} - 8 c_{1} + 9 c_{1} = 0 \\ 2 c_{1} = 0 \\ c_{1} = 0 \\ c_{2} = 0 \\ c_{3} = 0 \end{array}$

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="c1">c1 = </label>
<input id="c1" type="number" value="1">
<label for="c2">c2 = </label>
<input id="c2" type="number" value="1">
<label for="c3">c3 = </label>
<input id="c3" 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="sample5.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_c1 = document.querySelector('#c1'),
    input_c2 = document.querySelector('#c2'),
    input_c3 = document.querySelector('#c3'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_c1, input_c2, input_c3],
    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),
        c1 = parseFloat(input_c1.value),
        c2 = parseFloat(input_c2.value),
        c3 = parseFloat(input_c3.value);

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

    let points = [],
        lines = [],
        f1 = (x) => c1 * Math.cos(x) + c2 * Math.sin(x),
        f2 = (x) => c1 * Math.exp(x) + c2 * x ** 2,
        f3 = (x) =>
        c1 * Math.exp(x) + c2 * Math.exp(2 * x) + c3 * Math.exp(3 * x),
        fns = [[f1, 'red'],
               [f2, 'green'],
               [f3, 'blue']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });
    
    fns2
        .forEach((o) => {
            let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), 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('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.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.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
c1 = c2 = c3 =

Kamimura's blog

2017年9月2日土曜日

数学 - 線型代数 - ベクトル空間 - 1次従属と1次独立(三角関数(正弦、余弦)、指数関数(微分、導関数))

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