2017年8月29日火曜日

学習環境

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第19章(細分による加法 - 積分法)、19.3(定積分の性質と計算)、簡単な例、問24.を取り組んでみる。


    1. sin( mx+nx )=sinmxcosnx+cosmxsinnx sin( mxnx )=sinmxcosnxcosmxsinnx sin( mx+nx )+sin( mxnx )=2sinmxcosnx sinmxcosnx= sin( mx+nx )+sin( mxnx ) 2 = sin( ( m+n )x )+sin( ( mn )x ) 2 π π sinmxcosnxdx = π π sin( ( m+n )x )+sin( ( mn )x ) 2 dx = 1 2 ( π π sin( ( m+n )x )dx + π π sin( ( mn )x )dx ) = 1 2 ( 0+0 ) =0

    2. cos( mx+nx )=cosmxcosnxsinmxsinnx cos( mxnx )=cosmxcosnx+sinmxsinnx cos( mxnx )cos( mx+nx )=2sinmxsinnx sinmxsinnx= cos( mxnx )cos( mx+nx ) 2 = cos( ( mn )x )cos( ( m+n )x ) 2 π π sinmxsinnxdx = π π cos( ( mn )x )cos( ( m+n )x ) 2 dx = 1 2 ( π π cos( ( mn )x )dx π π cos( ( m+n )x )dx ) mn mn0 m,n>0 m+n0 1 2 ( π π cos( ( mn )x )dx π π cos( ( m+n )x )dx ) = 1 2 ( 0+0 ) =0 m=n mn=0 m,n>0 m+n0 1 2 ( π π cos( ( mn )x )dx π π cos( ( m+n )x )dx ) = 1 2 ( 2π0 ) =π

コード(Emacs)

Python 3

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

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

print('24.')
print('(1)')
x = symbols('x')
m, n = symbols('m n', integer=True, positive=True)
f = sin(m * x) * cos(n * x)
I = Integral(f, (x, -pi, pi))
for g in [I, I.doit()]:
    pprint(g)

try:
    p = plot(f.subs({m: 2, n: 3}), show=False, legend=True)
    p.save(f'sample24_1.svg')
except Exception as err:
    print(type(err), err)

print()

print('(2)')
f = sin(m * x) * sin(n * x)
I = Integral(f, (x, -pi, pi))
for g in [I, I.doit()]:
    pprint(g)

try:
    p = plot(f.subs({m: 2, n: 3}), show=False, legend=True)
    p.save(f'sample24_2_1.svg')
except Exception as err:
    print(type(err), err)

print()

f = sin(m * x) * sin(m * x)
I = Integral(f, (x, -pi, pi))
for g in [I, I.doit()]:
    pprint(g)

try:
    p = plot(f.subs({m: 2}), show=False, legend=True)
    p.save(f'sample24_2_2.svg')
except Exception as err:
    print(type(err), err)

入出力結果(Terminal, IPython)

$ ./sample24.py
24.
(1)
π                      
⌠                      
⎮  sin(m⋅x)⋅cos(n⋅x) dx
⌡                      
-π                     
0

(2)
π                      
⌠                      
⎮  sin(m⋅x)⋅sin(n⋅x) dx
⌡                      
-π                     
⎧π  for m = n
⎨            
⎩0  otherwise

π              
⌠              
⎮     2        
⎮  sin (m⋅x) dx
⌡              
-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">
<br>
<label for="m0">m = </label>
<input id="m0" type="number" min="1" step="1" value="2">
<label for="n0">n = </label>
<input id="n0" 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="sample24.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_m0 = document.querySelector('#m0'),
    input_n0 = document.querySelector('#n0'),    
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_m0, input_n0],
    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),
        m0 = parseInt(input_m0.value, 10),
        n0 = parseInt(input_n0.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(m0 * x) * Math.cos(n0 * x),
        g = (x) => Math.sin(m0 * x) * Math.sin(n0 * x),
        fns = [[f, 'green'],
               [g, 'blue']],
        fns1 = [],
        fns2 = [];

    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]);
            }
        }
    });
    fns1.forEach((o) => {
        let [fn, color] = o;
        
        lines.push([x1, fn(x1), x2, fn(x2), color]);
    });
    fns2.forEach((o) => {
        let [fn, color] = o;

        for (let x = x1; x <= x2; x += dx0) {
            let g = fn(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();








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