数学 - Python - JavaScript - 線形代数学 - R^n におけるベクトル – ベクトルのノルム(三角関数(正弦、余弦)、3倍角、連続関数、スカラー積、定積分)

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

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1章(R^n におけるベクトル)、3(ベクトルのノルム)、練習問題6.を取り組んでみる。

- $\begin{array}{l} ‖ \sin 3 x ‖ \\ = \sqrt{\int_{- π}^{π} \sin (3 x) \sin (3 x) d x} \\ = \sqrt{\int_{- π}^{π} \sin^{2} (3 x) d x} \\ \int_{- π}^{π} \sin^{2} (3 x) d x \\ = {[- \frac{1}{3} \cos (3 x) \sin (3 x)]}_{- π}^{π} - \int_{- π}^{π} (- \cos^{2} (3 x)) d x \\ = - \frac{1}{3} {[\cos (3 x) \sin (3 x)]}_{- π}^{π} + \int_{- π}^{π} (1 - \sin^{2} (3 x)) d x \\ = - \frac{1}{3} ((- \cos (3 π) \sin (3 π)) - (- \cos (- 3 π) \sin (- 3 π))) + \int_{- π}^{π} 1 d x - \int_{- π}^{π} \sin^{2} (3 x) d x \\ = \int_{- π}^{π} 1 d x - \int_{- π}^{π} \sin^{2} (3 x) d x \\ = 2 \int_{0}^{π} 1 d x - \int_{- π}^{π} \sin^{2} (3 x) d x \\ = 2 {[x]}_{0}^{π} - \int_{- π}^{π} \sin^{2} (3 x) d x \\ = 2 π - \int_{- π}^{π} \sin^{2} (3 x) d x \\ 2 \int_{- π}^{π} \sin^{2} (3 x) d x = 2 π \\ \int_{- π}^{π} \sin^{2} (3 x) d x = π \\ \sqrt{\int_{- π}^{π} \sin^{2} (3 x) d x} \\ = \sqrt{π} \\ ‖ \sin 3 x ‖ = \sqrt{π} \end{array}$
- $\begin{array}{l} ‖ \cos x ‖ \\ = \sqrt{\int_{- π}^{π} \cos x \cos x d x} \\ = \sqrt{\int_{- π}^{π} \cos^{2} x d x} \\ \int_{- π}^{π} \cos^{2} x d x \\ = {[\sin x \cos x]}_{- π}^{π} - \int_{- π}^{π} (- \sin^{2} x) d x \\ = (\sin π \cos π - \sin (- π) \cos (- π)) + \int_{- π}^{π} (1 - \cos^{2} x) d x \\ = \int_{- π}^{π} 1 d x - \int_{- π}^{π} \cos^{2} x d x \\ = 2 \int_{0}^{π} 1 d x - \int_{- π}^{π} \cos^{2} x d x \\ = 2 {[x]}_{0}^{π} - \int_{- π}^{π} \cos^{2} x d x \\ = 2 π - \int_{- π}^{π} \cos^{2} x d x \\ 2 \int_{- π}^{π} \cos^{2} x d x = 2 π \\ \sqrt{\int_{- π}^{π} \cos^{2} x d x} = \sqrt{π} \\ ‖ \cos x ‖ = \sqrt{π} \end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, Integral, sin, cos, pi, sqrt

print('6.')
x = symbols('x')
f = sin(3 * x)
g = cos(x)


def dot(f, g):
    return Integral(f * g, (x, -pi, pi))

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

入出力結果(Terminal, Jupyter(IPython))

$ ./sample6.py
6.
sin(3⋅x)
π              
⌠              
⎮     2        
⎮  sin (3⋅x) dx
⌡              
-π             

π

√π


sin(3⋅x)
π            
⌠            
⎮     2      
⎮  cos (x) dx
⌡            
-π           

π

√π


$

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">

<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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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 f = (x) => Math.sin(3 * x),
    g = (x) => Math.cos(x),
    h1 = (x) => f(x) * f(x),
    h2 = (x) => g(x) * g(x);

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);

    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']],
        fns = [[f, 'green'],
               [g, 'blue'],
               [h1, 'orange'],
               [h2, '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 =

Kamimura's blog

2017年9月25日月曜日

数学 - Python - JavaScript - 線形代数学 - R^n におけるベクトル – ベクトルのノルム(三角関数(正弦、余弦)、3倍角、連続関数、スカラー積、定積分)

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