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
解析入門〈3〉(松坂 和夫(著)、岩波書店)の第14章(多変数の関数)、14.1(微分可能性と勾配ベクトル)、問題5.を取り組んでみる。
コード(Emacs)
Python 3
#!/usr/bin/env python3 from sympy import pprint, symbols, sqrt, log, exp, Derivative, Function, Matrix n = 2 xs = symbols([f'x{i}' for i in range(1, n + 1)]) f = Function('f')(sum([x for x in xs])) g = Function('g')(sum([2 * x for x in xs])) c = symbols('c') grad1 = [Derivative(f + g, x, 1) for x in xs] grad2 = [Derivative(c * f, x, 1) for x in xs] grad3 = [Derivative(f * g, x, 1) for x in xs] grad4 = [Derivative(1 / f, x, 1) for x in xs] for grad in [grad1, grad2, grad3, grad3]: for t in [grad, [h.doit() for h in grad]]: pprint(t) print() print()
入出力結果(Terminal, Jupyter(IPython))
$ ./sample5.py ⎡ ∂ ∂ ⎤ ⎢───(f(x₁ + x₂) + g(2⋅x₁ + 2⋅x₂)), ───(f(x₁ + x₂) + g(2⋅x₁ + 2⋅x₂))⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡⎛ d ⎞│ ⎛ d ⎞│ ⎛ d ⎞│ ⎢⎜───(f(ξ₁))⎟│ + 2⋅⎜───(g(ξ₁))⎟│ , ⎜───(f(ξ₁))⎟│ ⎣⎝dξ₁ ⎠│ξ₁=x₁ + x₂ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂ ⎝dξ₁ ⎠│ξ₁=x₁ + ⎛ d ⎞│ ⎤ + 2⋅⎜───(g(ξ₁))⎟│ ⎥ x₂ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂⎦ ⎡ ∂ ∂ ⎤ ⎢───(c⋅f(x₁ + x₂)), ───(c⋅f(x₁ + x₂))⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ ⎛ d ⎞│ ⎛ d ⎞│ ⎤ ⎢c⋅⎜───(f(ξ₁))⎟│ , c⋅⎜───(f(ξ₁))⎟│ ⎥ ⎣ ⎝dξ₁ ⎠│ξ₁=x₁ + x₂ ⎝dξ₁ ⎠│ξ₁=x₁ + x₂⎦ ⎡ ∂ ∂ ⎤ ⎢───(f(x₁ + x₂)⋅g(2⋅x₁ + 2⋅x₂)), ───(f(x₁ + x₂)⋅g(2⋅x₁ + 2⋅x₂))⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ ⎛ d ⎞│ ⎛ d ⎞│ ⎢2⋅f(x₁ + x₂)⋅⎜───(g(ξ₁))⎟│ + g(2⋅x₁ + 2⋅x₂)⋅⎜───(f(ξ₁))⎟│ ⎣ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂ ⎝dξ₁ ⎠│ξ₁=x₁ ⎛ d ⎞│ ⎛ d ⎞│ , 2⋅f(x₁ + x₂)⋅⎜───(g(ξ₁))⎟│ + g(2⋅x₁ + 2⋅x₂)⋅⎜───(f(ξ₁))⎟│ + x₂ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂ ⎝dξ₁ ⎠│ξ ⎤ ⎥ ₁=x₁ + x₂⎦ ⎡ ∂ ∂ ⎤ ⎢───(f(x₁ + x₂)⋅g(2⋅x₁ + 2⋅x₂)), ───(f(x₁ + x₂)⋅g(2⋅x₁ + 2⋅x₂))⎥ ⎣∂x₁ ∂x₂ ⎦ ⎡ ⎛ d ⎞│ ⎛ d ⎞│ ⎢2⋅f(x₁ + x₂)⋅⎜───(g(ξ₁))⎟│ + g(2⋅x₁ + 2⋅x₂)⋅⎜───(f(ξ₁))⎟│ ⎣ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂ ⎝dξ₁ ⎠│ξ₁=x₁ ⎛ d ⎞│ ⎛ d ⎞│ , 2⋅f(x₁ + x₂)⋅⎜───(g(ξ₁))⎟│ + g(2⋅x₁ + 2⋅x₂)⋅⎜───(f(ξ₁))⎟│ + x₂ ⎝dξ₁ ⎠│ξ₁=2⋅x₁ + 2⋅x₂ ⎝dξ₁ ⎠│ξ ⎤ ⎥ ₁=x₁ + x₂⎦ $
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="k0">k0 = </label> <input id="k0" type="number" min="0" step="1" 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_k0 = document.querySelector('#k0'), inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2, input_k0], 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 c = 5, f = (x) => x, g = (x) => 2 * x, h1 = (x) => f(x) + g(x), h2 = (x) => c * f(x), h3 = (x) => f(x) * g(x), h4 = (x) => 1 / f(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 = [], fns = [[f, 'red'], [g, 'green'], [h1, 'blue'], [h2, 'brown'], [h3, 'orange'], [h4, 'purple']]; 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();
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