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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