多変数の関数グラフと等位線だ円、直線、2次方程式の解

続解析入門 (原書第2版)
楽天ブックス
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au PAY マーケット
原著: Calculus of Several Variables

学習環境

続解析入門 (原書第2版) (S.ラング(著)、松坂和夫(翻訳)、片山孝次(翻訳)、岩波書店)の第3章(多変数の関数)、1(グラフと等位線)の練習問題9、10、11.の解答を求めてみる。

\frac{x^{2}}{4} + \frac{y^{2}}{16} = c

c = 0

のとき、

(x, y) = (0, 0)

c > 0

のときはだ円

\frac{x^{2}}{4 c} + \frac{y^{2}}{16 c} = 1

2 x - 3 y = c

y = \frac{2}{3} x - \frac{1}{3} c

直線。

\frac{x y}{x^{2} + y^{2}} = c

x y = c (x^{2} + y^{2})

c x^{2} + c y^{2} - x y = 0

c = 0

のとき、

x = 0

または

y = 0

c \neq 0

のとき、

c y^{2} - x y + c x^{2} = 0

y = \frac{x \pm \sqrt{x^{2} - 4 c^{2} x^{2}}}{2 c}

y = \frac{x \pm x \sqrt{1 - 4 c^{2}}}{2 c} = \frac{1 \pm \sqrt{1 - 4 c^{2}}}{2 c} x

よって等位線は直線で、

1 - 4 c^{2} \geq 0

0 < c^{2} \leq \frac{1}{4}

0 < | c | \leq \frac{1}{2}

コード

#!/usr/bin/env python3
from sympy import plot, solve, Rational
from sympy.plotting import plot3d
from sympy.abc import x, y

print('9, 10, 11.')

fs = [
    x ** 2 / 4 + y ** 2 / 16,
    2 * x - 3 * y,
    x * y / (x ** 2 + y ** 2),
]
css = [
    range(-2, 3),
    range(-5, 5),
    [-Rational(1, 2), -Rational(1, 3), Rational(1, 3), Rational(1, 2)]
]
colors = ['red', 'green', 'blue', 'brown', 'orange',
          'purple', 'pink', 'gray', 'skyblue', 'yellow']

for i, (f, cs) in enumerate(zip(fs, css), 9):
    print(f'{i}.')
    p = plot3d(f, show=False)
    p.save(f'sample{i}_1.png')
    ys = []
    for c in cs:
        ys += solve(f - c, y)
    p = plot(*ys,
             (x, -10, 10),
             ylim=(-10, 10),
             legend=True,
             show=False)
    for o, color in zip(p, colors):
        o.line_color = color
        print(o, color)
    p.save(f'sample{i}_2.png')

p.show()

入出力結果

% ./sample9.py
9, 10, 11.
9.
cartesian line: -2*sqrt(-x**2 - 8) for x over (-10.0, 10.0) red
cartesian line: 2*sqrt(-x**2 - 8) for x over (-10.0, 10.0) green
cartesian line: -2*sqrt(-x**2 - 4) for x over (-10.0, 10.0) blue
cartesian line: 2*sqrt(-x**2 - 4) for x over (-10.0, 10.0) brown
cartesian line: -2*I*x for x over (-10.0, 10.0) orange
cartesian line: 2*I*x for x over (-10.0, 10.0) purple
cartesian line: -2*sqrt(4 - x**2) for x over (-10.0, 10.0) pink
cartesian line: 2*sqrt(4 - x**2) for x over (-10.0, 10.0) gray
cartesian line: -2*sqrt(8 - x**2) for x over (-10.0, 10.0) skyblue
cartesian line: 2*sqrt(8 - x**2) for x over (-10.0, 10.0) yellow
10.
cartesian line: 2*x/3 + 5/3 for x over (-10.0, 10.0) red
cartesian line: 2*x/3 + 4/3 for x over (-10.0, 10.0) green
cartesian line: 2*x/3 + 1 for x over (-10.0, 10.0) blue
cartesian line: 2*x/3 + 2/3 for x over (-10.0, 10.0) brown
cartesian line: 2*x/3 + 1/3 for x over (-10.0, 10.0) orange
cartesian line: 2*x/3 for x over (-10.0, 10.0) purple
cartesian line: 2*x/3 - 1/3 for x over (-10.0, 10.0) pink
cartesian line: 2*x/3 - 2/3 for x over (-10.0, 10.0) gray
cartesian line: 2*x/3 - 1 for x over (-10.0, 10.0) skyblue
cartesian line: 2*x/3 - 4/3 for x over (-10.0, 10.0) yellow
11.
cartesian line: -x for x over (-10.0, 10.0) red
cartesian line: x*(-3 + sqrt(5))/2 for x over (-10.0, 10.0) green
cartesian line: -x*(sqrt(5) + 3)/2 for x over (-10.0, 10.0) blue
cartesian line: x*(3 - sqrt(5))/2 for x over (-10.0, 10.0) brown
cartesian line: x*(sqrt(5) + 3)/2 for x over (-10.0, 10.0) orange
cartesian line: x for x over (-10.0, 10.0) purple
%