数学 - Python - JavaScript - 線形代数学 - ベクトル空間 – 基底(連続実数関数のなすベクトル空間、スカラー積(内積)、積分、三角関数(正弦)、整数倍、垂直)

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

ラング線形代数学(上)(S.ラング (著)、芹沢正三 (翻訳)、ちくま学芸文庫)の2章(ベクトル空間)、3(基底)、練習問題10.を取り組んでみる。

$\begin{array}{l} \int_{- π}^{π} \sin (n t) \sin (m t) dt \\ = \frac{1}{2} \int_{- π}^{π} (\cos (n t - m t) - \cos (n t + m t)) dt \\ = \frac{1}{2} \int_{- π}^{π} (\cos ((n - m) t) - \cos ((n + n) t)) dt \\ = \int_{0}^{π} (\cos ((n - m) t) - \cos ((n - m) t)) dt \\ = {[\sin ((n - m) t) - \sin ((n - m) t)]}_{0}^{π} = 0 \end{array}$

よって 2つの関数の内積は0、すなわちたがいに垂直である。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, sin, pi, Integral

print('10.')
n, m = symbols('n, m', integer=True, positive=True)
t = symbols('t')
f = sin(n * t)
g = sin(m * t)
I = Integral(f * g, (t, -pi, pi))

for s in [f, g, I, I.doit()]:
    pprint(s)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample10.py 
10.
sin(n⋅t)

sin(m⋅t)

π                      
⌠                      
⎮  sin(m⋅t)⋅sin(n⋅t) dt
⌡                      
-π                     

⎧π  for m = n
⎨            
⎩0  otherwise

$

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="n0">n = </label>
<input id="n0" type="number" min="1" step="1" value="2">
<label for="m0">m = </label>
<input id="m0" type="number" min="1" step="1" value="3">

<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,
    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_n0 = document.querySelector('#n0'),
    input_m0 = document.querySelector('#m0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0, input_m0],
    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),
        n0 = parseInt(input_n0.value, 10),
        m0 = parseInt(input_m0.value, 10);

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

    let points = [],
        lines = [[-Math.PI, y1, -Math.PI, y2, 'red'],
                 [Math.PI, y1, Math.PI, y2, 'red']],
        f = (x) => Math.sin(n0 * x),
        g = (x) => Math.sin(m0 * x),
        h = (x) => f(x) * g(x),
        fns = [[f, 'green'],
               [g, 'blue'],
               [h, 'brown']],
        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 =
n = m =

Kamimura's blog

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2017年11月9日木曜日

数学 - Python - JavaScript - 線形代数学 - ベクトル空間 – 基底(連続実数関数のなすベクトル空間、スカラー積(内積)、積分、三角関数(正弦)、整数倍、垂直)

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