2017年12月13日水曜日

学習環境

解析入門〈3〉(松坂 和夫(著)、岩波書店)の第12章(距離空間の位相)、12.4(n次元実数空間における曲線)、問題5.を取り組んでみる。


  1. 曲線のパラメーター表示を

    t , log t

    と考える。

    γ t = t , log t γ ' t = 1 , 1 t γ ' t = 1 + 1 t 2

    もとめる曲線の長さ。

    1 3 γ ' t dt = 1 3 1 + 1 t 2 dt = 1 3 t 2 + 1 t 2 dt = 1 3 t 2 + 1 t dt
    u = t 2 + 1 d u dt = 2 t 2 t 2 + 1 d u dt = t t 2 + 1 dt = t 2 + 1 t d u 1 t 3 2 u 10
    L t = 2 10 t 2 + 1 t · t 2 + 1 t d u = 2 10 t 2 + 1 t 2 d u = 2 10 1 + 1 t 2 d u = 2 10 1 d u + 2 10 1 t 2 d u
    u 2 = t 2 + 1 t 2 = u 2 - 1
    L t = 10 - 2 + 2 10 1 u 2 - 1 d u 1 u 2 - 1 = 1 u + 1 · 1 u - 1 A u + 1 + B u - 1 = A + B u - A + B u 2 - 1 A + B = 0 - A + B = 1 B = 1 2 , A = - 1 2 1 u 2 - 1 = - 1 2 u + 1 + 1 2 u - 1
    L t = 10 - 2 + 1 2 2 10 - 1 u + 1 + 1 u - 1 d u = 10 - 2 + 1 2 - log u + 1 + log u - 1 2 10 = 10 - 2 + 1 2 log 10 - 1 10 + 1 - log 2 - 1 2 + 1 = 10 - 2 + 1 2 log 10 - 1 2 + 1 10 + 1 2 - 1

コード(Emacs)

Python 3

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

t = symbols('t')
f = sqrt(1 + 1 / t ** 2)
I = Integral(f, (t, 1, 3))
for o in [I, I.doit()]:
    pprint(o)
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample5.py
3                 
⌠                 
⎮      ________   
⎮     ╱     1     
⎮    ╱  1 + ──  dt
⎮   ╱        2    
⎮ ╲╱        t     
⌡                 
1                 

-√2 - asinh(1/3) + log(1 + √2) + √10

$

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="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'),
    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.log(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 = [[1, y1, 1, y2, 'red'],
                 [3, y1, 3, y2, 'blue']],
        fns = [[f, '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();







0 コメント:

コメントを投稿

関連コンテンツ