{"id":6196,"date":"2023-04-19T18:08:19","date_gmt":"2023-04-19T09:08:19","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6196"},"modified":"2024-03-15T14:51:28","modified_gmt":"2024-03-15T05:51:28","slug":"sympy-%e3%81%a7%e5%81%8f%e5%be%ae%e5%88%86","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/sympy-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/sympy-%e3%81%a7%e5%81%8f%e5%be%ae%e5%88%86\/","title":{"rendered":"SymPy \u3067\u504f\u5fae\u5206"},"content":{"rendered":"

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\u5fc5\u8981\u306a\u30d1\u30c3\u30b1\u30fc\u30b8\u3092 import \u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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from sympy.abc<\/span> import *<\/span> \r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n\r\n# SymPy Plotting Backends (SPB): \u30b0\u30e9\u30d5\u3092\u63cf\u304f\u969b\u306b\u5229\u7528<\/span>\r\nfrom<\/span> spb<\/span> import<\/span> *<\/span>\r\n\r\n# \u30b0\u30e9\u30d5\u3092 SVG \u3067 Notebook \u306b\u30a4\u30f3\u30e9\u30a4\u30f3\u8868\u793a<\/span>\r\n%<\/span>config<\/span> InlineBackend.figure_formats = ['svg']\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u504f\u5fae\u5206\uff1a\u591a\u5909\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n
SymPy \u3067\u306e\u504f\u5fae\u5206\u306f\uff08\u5e38\u5fae\u5206\u3068\u540c\u69d8\u306e\u66f8\u304d\u65b9\u3067\u3059\u304c\uff09\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002<\/p>\n
$\\displaystyle \\frac{\\partial}{\\partial x} f(x, y) = $ diff(f(x, y), x);<\/code><\/p>\n
$\\displaystyle \\frac{\\partial}{\\partial y} f(x, y) = $ diff(f(x, y), y);<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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f<\/span> =<\/span> Function<\/span>(<\/span>'f'<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial x} f{\\left(x,y \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> y<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial y} f{\\left(x,y \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b\u984c<\/h4>\n
\u6b21\u306e\u95a2\u6570 $z$ \u306e\u504f\u5c0e\u95a2\u6570\u3092\u6c42\u3081\u3088\u3002<\/p>\n
(1) $ z = x^3 – 4 x^2 y + x y + 3 y^2$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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z<\/span> =<\/span> x<\/span>**<\/span>3<\/span> -<\/span> 4<\/span>*<\/span>x<\/span>**<\/span>2<\/span>*<\/span>y<\/span> +<\/span> x<\/span>*<\/span>y<\/span> +<\/span> 3<\/span>*<\/span>y<\/span>**<\/span>2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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diff<\/span>(<\/span>z<\/span>,<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle 3 x^{2} – 8 x y + y$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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diff<\/span>(<\/span>z<\/span>,<\/span> y<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle – 4 x^{2} + x + 6 y$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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(2) $\\displaystyle z = \\tan^{-1} \\frac{y}{x} $<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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z<\/span> =<\/span> atan<\/span>(<\/span>y<\/span>\/<\/span>x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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diff<\/span>(<\/span>z<\/span>,<\/span> x<\/span>)<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
$\\displaystyle – \\frac{y}{x^{2} + y^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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diff<\/span>(<\/span>z<\/span>,<\/span> y<\/span>)<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
$\\displaystyle \\frac{x}{x^{2} + y^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5168\u5fae\u5206<\/h3>\n
2\u5909\u6570\u95a2\u6570 $z = f(x, y)$ \u306e\u5168\u5fae\u5206 $dz$ \u3068\u306f<\/p>\n
$$ dz = df(x,y) = \\frac{\\partial f}{\\partial x} dx + \\frac{\\partial f}{\\partial y} dy$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b\u984c<\/h4>\n
\u6b21\u306e\u95a2\u6570 $z$ \u306e\u5168\u5fae\u5206\u3092\u6c42\u3081\u3088\u3002<\/p>\n
(1) $ z = \\sqrt{x^2 – y^2} $<\/p>\n
(2) $ z = x y^2 $<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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z<\/span> =<\/span> sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span> -<\/span> y<\/span>**<\/span>2<\/span>)<\/span>\r\nvar<\/span>(<\/span>'dx dy'<\/span>)<\/span>\r\n\r\ndiff<\/span>(<\/span>z<\/span>,<\/span> x<\/span>)<\/span> *<\/span> dx<\/span> +<\/span> diff<\/span>(<\/span>z<\/span>,<\/span> y<\/span>)<\/span> *<\/span> dy<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
$\\displaystyle \\frac{dx x}{\\sqrt{x^{2} – y^{2}}} – \\frac{dy y}{\\sqrt{x^{2} – y^{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u8a18\u306e\u7b54\u3048\uff0c
\n$\\displaystyle dz = \\frac{x}{\\sqrt{x^{2} – y^{2}}} dx – \\frac{y}{\\sqrt{x^{2} – y^{2}}}dy $ \u3068\u8868\u793a\u3057\u3066\u307b\u3057\u3044\u3068\u3053\u308d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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z<\/span> =<\/span> x<\/span>*<\/span>y<\/span>**<\/span>2<\/span>\r\ndiff<\/span>(<\/span>z<\/span>,<\/span> x<\/span>)<\/span> *<\/span> dx<\/span> +<\/span> diff<\/span>(<\/span>z<\/span>,<\/span> y<\/span>)<\/span> *<\/span> dy<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
$\\displaystyle dx y^{2} + 2 dy x y$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u8a18\u306e\u7b54\u3048\uff0c
\n$\\displaystyle dz = y^{2}\\,dx + 2 x y\\, dy $ \u3068\u8868\u793a\u3057\u3066\u307b\u3057\u3044\u3068\u3053\u308d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u9ad8\u968e\u504f\u5c0e\u95a2\u6570<\/h3>\n
SymPy \u304c2\u968e\u504f\u5c0e\u95a2\u6570\u306b\u95a2\u3057\u3066<\/p>\n
$$ \\frac{\\partial^2 f}{\\partial x \\partial y} = \\frac{\\partial^2 f}{\\partial y \\partial x} $$<\/p>\n
\u306e\u3088\u3046\u306b\uff0c\u504f\u5fae\u5206\u306e\u9806\u5e8f\u3092\u5909\u3048\u3066\u3082\u3088\u3044\u3053\u3068\u3092\u77e5\u3063\u3066\u3044\u308b\u304b\u78ba\u8a8d\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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f<\/span> =<\/span> Function<\/span>(<\/span>'f'<\/span>)<\/span>\r\n\r\nEq<\/span>((<\/span>Derivative<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> x<\/span>),<\/span> y<\/span>)<\/span> -<\/span> \r\n    Derivative<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> y<\/span>),<\/span> x<\/span>)),<\/span> \r\n   (<\/span>Derivative<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> x<\/span>),<\/span> y<\/span>)<\/span> -<\/span> \r\n    Derivative<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>(<\/span>x<\/span>,<\/span> y<\/span>),<\/span> y<\/span>),<\/span> x<\/span>))<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
$\\displaystyle \\frac{\\partial^{2}}{\\partial y\\partial x} f{\\left(x,y \\right)} – \\frac{\\partial^{2}}{\\partial x\\partial y} f{\\left(x,y \\right)} = 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\uff082\u5909\u6570\uff09<\/h3>\n
1\u5909\u6570\u95a2\u6570 $f(x)$ \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e\u5fa9\u7fd2<\/h4>\n
$f(x)$ \u3092 $x = a$ \u306e\u307e\u308f\u308a\u3067 $n = 5$ \u6b21\uff08$x^5$\uff09\u307e\u3067\u5c55\u958b\u3059\u308b\u3068\u304d\u306f\uff0cSymPy \u3067\u306f
\nseries(f(x), x, a, 6)<\/code> \u306e\u3088\u3046\u306b\u66f8\u304f\u3002<\/p>\n
\u4f8b\u3068\u3057\u3066\uff0c$f(x) = e^{x}$ \u3092 $x = 0$ \u307e\u308f\u308a\u3067 $5$ \u6b21\u307e\u3067\u5c55\u958b\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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f<\/span> =<\/span> exp<\/span>(<\/span>x<\/span>)<\/span>\r\n\r\nseries<\/span>(<\/span>f<\/span>,<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 6<\/span>)<\/span> # 5 \u6b21\u307e\u3067\u306a\u3089 6 \u3068\u66f8\u304f\u3002\u305d\u308c\u304c Python \u6d41\u3002<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[14]:<\/div>\n
$\\displaystyle 1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\sin x$ \u3092 $x = 0$ \u306e\u307e\u308f\u308a\u3067 $x^3$ \u307e\u3067\u5c55\u958b\u3059\u308b\u3068\u304d\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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series<\/span>(<\/span>sin<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 0<\/span>,<\/span> 4<\/span>)<\/span> # 3 \u6b21\u307e\u3067\u306a\u3089 4 \u3068\u66f8\u304f\u3002<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle x – \\frac{x^{3}}{6} + O\\left(x^{4}\\right)$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2\u5909\u6570\u95a2\u6570 $f(x, y)$ \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b<\/h4>\n
\u672c\u984c\u3067\u3042\u308b2\u5909\u6570\u95a2\u6570 $f(x, y)$ \u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3002series()<\/code> \u306f1\u5909\u6570\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e\u307f\u306e\u3088\u3046\u3060\u3002<\/p>\n
\u4f8b\u3068\u3057\u3066\uff0c$\\displaystyle f(x, y) = \\frac{e^x}{1-x+2 y}$ \u3092 $x = 0, y = 0$ \u306e\u307e\u308f\u308a\u3067 $2$ \u6b21\u307e\u3067\u5c55\u958b\u3057\u3066\u307f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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f<\/span> =<\/span> exp<\/span>(<\/span>x<\/span>)<\/span>\/<\/span>(<\/span>1<\/span>-<\/span>x<\/span>+<\/span>2<\/span>*<\/span>y<\/span>)<\/span>\r\nf<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{e^{x}}{- x + 2 y + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$x \\rightarrow \\epsilon x, \\ y \\rightarrow \\epsilon y$ \u3068\u304a\u304d…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[17]:<\/div>\n
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var<\/span>(<\/span>'epsilon'<\/span>)<\/span>\r\nF<\/span> =<\/span> f<\/span>.<\/span>subs<\/span>(<\/span>x<\/span>,<\/span> epsilon<\/span>*<\/span>x<\/span>)<\/span>.<\/span>subs<\/span>(<\/span>y<\/span>,<\/span> epsilon<\/span>*<\/span>y<\/span>)<\/span>\r\nF<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{e^{\\epsilon x}}{- \\epsilon x + 2 \\epsilon y + 1}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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1.$\\epsilon$ \u306b\u3064\u3044\u30662\u6b21\u307e\u3067\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3057\uff0c$\\dots$ series()<\/code><\/p>\n
2.\u30ab\u30c3\u30b3\u3092\u306f\u305a\u3057\u3066\u5c55\u958b\u3002$\\dots$ expand()<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[18]:<\/div>\n
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series<\/span>(<\/span>F<\/span>,<\/span> epsilon<\/span>,<\/span> 0<\/span>,<\/span> 3<\/span>)<\/span>.<\/span>expand<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[18]:<\/div>\n
$\\displaystyle 1 – 2 \\epsilon y + 2 \\epsilon x + 4 \\epsilon^{2} y^{2} – 6 \\epsilon^{2} x y + \\frac{5 \\epsilon^{2} x^{2}}{2} + O\\left(\\epsilon^{3}\\right)$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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3.\u9ad8\u6b21\u306e\u9805\u3092\u793a\u3059 $O()$ \u3092\u53d6\u308a\u53bb\u308a\uff0c$\\dots$ removeO()<\/code><\/p>\n
\uff08\u3053\u3053\u3067\u964d\u3079\u304d\u306e\u9806\u306b\u4e26\u3076\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[19]:<\/div>\n
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_<\/span>.<\/span>removeO<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[19]:<\/div>\n
$\\displaystyle \\frac{5 \\epsilon^{2} x^{2}}{2} – 6 \\epsilon^{2} x y + 4 \\epsilon^{2} y^{2} + 2 \\epsilon x – 2 \\epsilon y + 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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4.\u6700\u5f8c\u306b $\\epsilon \\rightarrow 1$ \u3068\u304a\u304d\u306a\u304a\u3059\u3002$\\dots$ subs()<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[20]:<\/div>\n
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_<\/span>.<\/span>subs<\/span>(<\/span>epsilon<\/span>,<\/span> 1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{5 x^{2}}{2} – 6 x y + 2 x + 4 y^{2} – 2 y + 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5408\u6210\u95a2\u6570\u306e\u504f\u5fae\u5206<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b\u984c<\/h4>\n
\\begin{eqnarray}
\nz &=& f(x, y) = x y \\\\
\nx &=& x(u, v) = u\\cos v \\\\
\ny &=& y(u, v) = u \\sin v \\\\
\n\\therefore\\ \\ z &=& f\\left(x(u, v), y(u, v)\\right) = z(u, v)
\n\\end{eqnarray}<\/p>\n
\u306b\u3064\u3044\u3066\uff0c\u4ee5\u4e0b\u3092\u6c42\u3081\u308b\u3002<\/p>\n
$$ 1. \\ \\frac{\\partial z}{\\partial u} \\quad 2. \\ \\frac{\\partial z}{\\partial v} $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[21]:<\/div>\n
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var<\/span>(<\/span>'x y z u v'<\/span>)<\/span>\r\n\r\nx<\/span> =<\/span> u<\/span>*<\/span>cos<\/span>(<\/span>v<\/span>)<\/span>\r\ny<\/span> =<\/span> u<\/span>*<\/span>sin<\/span>(<\/span>v<\/span>)<\/span>\r\n\r\nz<\/span> =<\/span> x<\/span> *<\/span> y<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{\\partial z}{\\partial u} = \\cdots$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[22]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>z<\/span>,<\/span> u<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>z<\/span>,<\/span> u<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[22]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial u} u^{2} \\sin{\\left(v \\right)} \\cos{\\left(v \\right)} = 2 u \\sin{\\left(v \\right)} \\cos{\\left(v \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u53f3\u8fba\u3092\u7c21\u5358\u5316\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[23]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>z<\/span>,<\/span> u<\/span>)<\/span>.<\/span>doit<\/span>(),<\/span>\r\n   Derivative<\/span>(<\/span>z<\/span>,<\/span> u<\/span>)<\/span>.<\/span>doit<\/span>()<\/span>.<\/span>simplify<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[23]:<\/div>\n
$\\displaystyle 2 u \\sin{\\left(v \\right)} \\cos{\\left(v \\right)} = u \\sin{\\left(2 v \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{\\partial z}{\\partial v} = \\cdots$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[24]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>z<\/span>,<\/span> v<\/span>),<\/span> \r\n   Derivative<\/span>(<\/span>z<\/span>,<\/span> v<\/span>)<\/span>.<\/span>doit<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[24]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial v} u^{2} \\sin{\\left(v \\right)} \\cos{\\left(v \\right)} = – u^{2} \\sin^{2}{\\left(v \\right)} + u^{2} \\cos^{2}{\\left(v \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[25]:<\/div>\n
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# \u53f3\u8fba\u3092\u7c21\u5358\u5316<\/span>\r\n\r\nEq<\/span>(<\/span>diff<\/span>(<\/span>z<\/span>,<\/span> v<\/span>),<\/span> \r\n   diff<\/span>(<\/span>z<\/span>,<\/span> v<\/span>)<\/span>.<\/span>simplify<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[25]:<\/div>\n
$\\displaystyle – u^{2} \\sin^{2}{\\left(v \\right)} + u^{2} \\cos^{2}{\\left(v \\right)} = u^{2} \\cos{\\left(2 v \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u9670\u95a2\u6570\u5b9a\u7406<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b\u984c<\/h4>\n
$f(x, y) = x^2 + y^2 -1 = 0$ \u3092\u6e80\u305f\u3059\u9670\u95a2\u6570 $y=y(x)$ \u306e\u5fae\u5206\u3092\u6c42\u3081\u308b\u3002<\/p>\n
\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u308f\u305a\u306b\u89e3\u304f<\/h5>\n
\u9670\u95a2\u6570\u5b9a\u7406\u3088\u308a\uff0c<\/p>\n
$$ \\frac{dy}{dx} = – \\frac{\\frac{\\partial f}{\\partial x}}{\\frac{\\partial f}{\\partial y}}$$<\/p>\n
\u307e\u305a\u306f\u3053\u308c\u3092\u4f7f\u308f\u305a\u306b SymPy \u3067\u5c0e\u3044\u3066\u307f\u307e\u3059\u3002<\/p>\n
\u307e\u305a\uff0c$f(x, y) = 0$ \u3088\u308a $y$ \u306f $x$ \u306e\uff08\u9670\uff09\u95a2\u6570\u3068\u306a\u308b\u306e\u3067\uff0c$y(x)$\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[26]:<\/div>\n
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var<\/span>(<\/span>'x y f'<\/span>)<\/span>\r\n\r\ny<\/span> =<\/span> Function<\/span>(<\/span>'y'<\/span>)<\/span>\r\nf<\/span> =<\/span> x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span> -<\/span> 1<\/span>\r\nf<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[26]:<\/div>\n
$\\displaystyle x^{2} + y^{2}{\\left(x \\right)} – 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[27]:<\/div>\n
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df<\/span> =<\/span> diff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>)<\/span>\r\ndf<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[27]:<\/div>\n
$\\displaystyle 2 x + 2 y{\\left(x \\right)} \\frac{d}{d x} y{\\left(x \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{df}{dx} = 0$ \u3092 $\\displaystyle \\frac{dy}{dx}$ \u306b\u3064\u3044\u3066\u89e3\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[28]:<\/div>\n
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sol<\/span> =<\/span> solve<\/span>(<\/span>df<\/span>,<\/span> diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>))<\/span>\r\n\r\nEq<\/span>(<\/span>diff<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> sol<\/span>[<\/span>0<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[28]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} = – \\frac{x}{y{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u9670\u95a2\u6570\u5b9a\u7406\u3092\u4f7f\u3063\u3066…<\/h5>\n
\u9670\u95a2\u6570\u5b9a\u7406\u3088\u308a\uff0c<\/p>\n
$$ \\frac{dy}{dx} = – \\frac{\\frac{\\partial f}{\\partial x}}{\\frac{\\partial f}{\\partial y}}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[29]:<\/div>\n
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var<\/span>(<\/span>'x y f'<\/span>)<\/span>\r\nf<\/span> =<\/span> x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span> -<\/span> 1<\/span>\r\n\r\ndydx<\/span> =<\/span> -<\/span> diff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>)<\/span>\/<\/span>diff<\/span>(<\/span>f<\/span>,<\/span> y<\/span>)<\/span>\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>,<\/span> x<\/span>),<\/span> dydx<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[29]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y = – \\frac{x}{y}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u967d\u95a2\u6570\u3068\u3057\u3066\u8868\u3057\u3066\u5fae\u5206\u3059\u308b<\/h5>\n
\u5225\u89e3\u3068\u3057\u3066\uff0c\u967d\u95a2\u6570\u3068\u3057\u3066\u8868\u3057\u3066\u89e3\u304f\u5834\u5408\u3002<\/p>\n
$f(x, y) = x^2 + y^2 – 1 = 0$ \u3092 $y$ \u306b\u3064\u3044\u3066\u89e3\u304f\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[30]:<\/div>\n
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var<\/span>(<\/span>'x y f'<\/span>)<\/span>\r\nf<\/span> =<\/span> x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span> -<\/span> 1<\/span>\r\nf<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[30]:<\/div>\n
$\\displaystyle x^{2} + y^{2} – 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[31]:<\/div>\n
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sols<\/span> =<\/span> solve<\/span>(<\/span>f<\/span>,<\/span> y<\/span>)<\/span>\r\nsols<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[31]:<\/div>\n
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[-sqrt(1 - x**2), sqrt(1 - x**2)]<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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1\u3064\u76ee\u306e\u89e3\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u53d6\u308a\u51fa\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[32]:<\/div>\n
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Eq<\/span>(<\/span>y<\/span>,<\/span> sols<\/span>[<\/span>0<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[32]:<\/div>\n
$\\displaystyle y = – \\sqrt{1 – x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u305d\u3057\u3066\u305d\u308c\u3092\u5fae\u5206\u3059\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[33]:<\/div>\n
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dy0<\/span> =<\/span> diff<\/span>(<\/span>sols<\/span>[<\/span>0<\/span>],<\/span> x<\/span>)<\/span>\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>,<\/span> x<\/span>),<\/span> dy0<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[33]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y = \\frac{x}{\\sqrt{1 – x^{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5fae\u5206\u3057\u305f\u7d50\u679c\u306e\u53f3\u8fba\u306e\u5206\u6bcd\u306b $\\sqrt{1-x^2} = -y$ \u3092\u4ee3\u5165\u3057\u3066 $y$ \u3067\u66f8\u304d\u76f4\u3059\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[34]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>,<\/span> x<\/span>),<\/span> \r\n   dy0<\/span>.<\/span>subs<\/span>(<\/span>-<\/span>sols<\/span>[<\/span>0<\/span>],<\/span> -<\/span>y<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[34]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y = – \\frac{x}{y}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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2\u3064\u76ee\u306e\u89e3\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[35]:<\/div>\n
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Eq<\/span>(<\/span>y<\/span>,<\/span> sols<\/span>[<\/span>1<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[35]:<\/div>\n
$\\displaystyle y = \\sqrt{1 – x^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[36]:<\/div>\n
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dy1<\/span> =<\/span> diff<\/span>(<\/span>sols<\/span>[<\/span>1<\/span>],<\/span> x<\/span>)<\/span>\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>,<\/span> x<\/span>),<\/span> dy1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[36]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y = – \\frac{x}{\\sqrt{1 – x^{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[37]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>,<\/span> x<\/span>),<\/span> \r\n   dy1<\/span>.<\/span>subs<\/span>(<\/span>sols<\/span>[<\/span>1<\/span>],<\/span> y<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[37]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y = – \\frac{x}{y}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4ee5\u4e0a\u306e\u3053\u3068\u304b\u3089\uff0c1\u3064\u76ee\u306e\u89e3 sols[0]<\/code> \u306b\u3064\u3044\u3066\u30822\u3064\u76ee\u306e\u89e3 sols[1]<\/code> \u306b\u3064\u3044\u3066\u3082\uff0c<\/p>\n
$$\\frac{dy}{dx} = – \\frac{x}{y}$$<\/p>\n
\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002\u3053\u308c\u306f\u9670\u95a2\u6570\u5b9a\u7406\u3067\u6c42\u3081\u305f\u89e3\u3068\u4e00\u81f4\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u4f8b\u984c<\/h4>\n
$f(x,y) = x^2 + y^2 -1 = 0$ \u3092\u6e80\u305f\u3059\u9670\u95a2\u6570 $y = y(x)$ \u306e\u6975\u5927\u5024\u3068\u6975\u5c0f\u5024\u3092\u6c42\u3081\u3088\u3002<\/p>\n
\u6975\u5024\u3092\u6c42\u3081\u308b\u306b\u306f $\\displaystyle \\frac{dy}{dx}$ \u3068\u6975\u5927\u30fb\u6975\u5c0f\uff08\u4e0a\u306b\u51f8\u30fb\u4e0b\u306b\u51f8\uff09\u3092\u5224\u65ad\u3059\u308b\u305f\u3081\u306b $\\displaystyle \\frac{d^2y}{dx^2}$ \u304c\u5fc5\u8981\u3002<\/p>\n
\u9670\u95a2\u6570\u306e1\u968e\u5fae\u5206<\/h5>\n
\u3059\u3067\u306b\uff0c$\\displaystyle \\frac{dy}{dx} = – \\frac{x}{y}$ \u306f\u6c42\u3081\u3066\u3044\u3066\uff0c\u5909\u6570 sol[0]<\/code> \u306b\u683c\u7d0d\u3057\u3066\u3044\u308b\u3002\u3053\u308c\u3092 dy<\/code> \u3068\u3057\u3066\u4f7f\u3063\u3066\u3044\u3053\u3046\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[38]:<\/div>\n
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y<\/span> =<\/span> Function<\/span>(<\/span>'y'<\/span>)<\/span>\r\ndy<\/span> =<\/span> sol<\/span>[<\/span>0<\/span>]<\/span>\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> dy<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[38]:<\/div>\n
$\\displaystyle \\frac{d}{d x} y{\\left(x \\right)} = – \\frac{x}{y{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u9670\u95a2\u6570\u306e2\u968e\u5fae\u5206<\/h5>\n
dy<\/code> \u3059\u306a\u308f\u3061 sol[0]<\/code> \u3092\u3082\u30461\u968e $x$ \u3067\u5fae\u5206\u3057\u3066 $\\displaystyle \\frac{d^2y}{dx^2}$ \u3092\u6c42\u3081\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[39]:<\/div>\n
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ddy<\/span> =<\/span> diff<\/span>(<\/span>dy<\/span>,<\/span> x<\/span>)<\/span>\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>),<\/span> ddy<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[39]:<\/div>\n
$\\displaystyle \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = \\frac{x \\frac{d}{d x} y{\\left(x \\right)}}{y^{2}{\\left(x \\right)}} – \\frac{1}{y{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u5f0f\u306e\u53f3\u8fba\u306b dy<\/code> \u3064\u307e\u308a $\\displaystyle \\frac{dy(x)}{dx} = – \\frac{x}{y(x)}$ \u3092\u4ee3\u5165\u3057\u3066\u3084\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[40]:<\/div>\n
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ddy<\/span> =<\/span> ddy<\/span>.<\/span>subs<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>),<\/span> dy<\/span>)<\/span>\r\nEq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>),<\/span> ddy<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[40]:<\/div>\n
$\\displaystyle \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = – \\frac{x^{2}}{y^{3}{\\left(x \\right)}} – \\frac{1}{y{\\left(x \\right)}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6975\u5024\u3068\u306a\u308a\u305d\u3046\u306a $x$ \u306e\u5024\u306f<\/p>\n
$\\displaystyle \\frac{dy}{dx} = – \\frac{x}{y} = 0$ \u3088\u308a $x = 0$.<\/p>\n
\u305d\u306e\u3068\u304d\u306e $y$ \u306e\u5024\u306f\u9023\u7acb\u65b9\u7a0b\u5f0f $x = 0, \\ f(x, y)=0$ \u3092\u89e3\u3044\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[41]:<\/div>\n
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ans2<\/span> =<\/span> solve<\/span>([<\/span>x<\/span>,<\/span> x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>(<\/span>x<\/span>)<\/span>**<\/span>2<\/span> -<\/span> 1<\/span>],<\/span> [<\/span>x<\/span>,<\/span> y<\/span>(<\/span>x<\/span>)])<\/span>\r\nans2<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[41]:<\/div>\n
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[(0, -1), (0, 1)]<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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ans2[0]<\/code> \u3064\u307e\u308a $(x, y) = (0, -1)$ \u306e\u3068\u304d\uff0c$\\displaystyle \\frac{d^2y}{dx^2}$ \u3064\u307e\u308a ddy<\/code> \u306e\u53f3\u8fba\u306e\u5024\u3092\u8a55\u4fa1\u3059\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[42]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>),<\/span>\r\n   ddy<\/span>.<\/span>subs<\/span>(<\/span>x<\/span>,<\/span> ans2<\/span>[<\/span>0<\/span>][<\/span>0<\/span>])<\/span>.<\/span>subs<\/span>(<\/span>y<\/span>(<\/span>0<\/span>),<\/span> ans2<\/span>[<\/span>0<\/span>][<\/span>1<\/span>]))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[42]:<\/div>\n
$\\displaystyle \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = 1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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… \u3068\u306a\u308a\uff0c$\\displaystyle \\frac{d^2y}{dx^2} = 1 > 0$ \u3060\u304b\u3089\u3053\u3053\u3067\u306f\u4e0b\u306b\u51f8\uff0c\u3064\u307e\u308a\u6975\u5c0f\u5024\uff08\u6700\u5c0f\u5024\uff09\u3002<\/p>\n
\u540c\u69d8\u306b ans2[1]<\/code> \u3064\u307e\u308a $(x, y) = (0, 1)$ \u306e\u5834\u5408\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[43]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>y<\/span>(<\/span>x<\/span>),<\/span> x<\/span>,<\/span> 2<\/span>),<\/span>\r\n   ddy<\/span>.<\/span>subs<\/span>(<\/span>x<\/span>,<\/span> ans2<\/span>[<\/span>1<\/span>][<\/span>0<\/span>])<\/span>.<\/span>subs<\/span>(<\/span>y<\/span>(<\/span>0<\/span>),<\/span> ans2<\/span>[<\/span>1<\/span>][<\/span>1<\/span>]))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[43]:<\/div>\n
$\\displaystyle \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)} = -1$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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… \u3068\u306a\u308a\uff0c$\\displaystyle \\frac{d^2y}{dx^2} = -1 < 0$ \u3060\u304b\u3089\u3053\u3053\u3067\u306f\u4e0a\u306b\u51f8\uff0c\u3064\u307e\u308a\u6975\u5927\u5024\uff08\u6700\u5927\u5024\uff09\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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\u53c2\u8003\uff1a\u9670\u95a2\u6570\u306e\u30b0\u30e9\u30d5<\/h5>\n
SymPy Plotting Backends \u3067\u306f $f(x, y) = 0$ \u3068\u3044\u3046\u9670\u95a2\u6570\u8868\u793a\u306e\u307e\u307e\u3067\u30b0\u30e9\u30d5\u3092\u63cf\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[44]:<\/div>\n
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var<\/span>(<\/span>'x y f'<\/span>)<\/span>\r\nf<\/span> =<\/span> x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span> -<\/span> 1<\/span>\r\n\r\np1<\/span> =<\/span> plot_implicit<\/span>(<\/span>f<\/span>,<\/span> \"$f(x,y)=0$\"<\/span>,<\/span> \r\n                   (<\/span>x<\/span>,<\/span> -<\/span>1.2<\/span>,<\/span> 1.2<\/span>),<\/span> (<\/span>y<\/span>,<\/span> -<\/span>1.2<\/span>,<\/span> 1.5<\/span>),<\/span> \r\n                   legend<\/span> =<\/span> True<\/span>,<\/span>\r\n                   size<\/span>=<\/span>(<\/span>6<\/span>,<\/span>6<\/span>),<\/span>\r\n                   aspect<\/span> =<\/span> \"equal\"<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$y(x)$ \u306e\u6700\u5927\u5024\uff0c\u6700\u5c0f\u5024\u306e\u70b9\u3082\u3042\u308f\u305b\u3066\u30b0\u30e9\u30d5\u306b\u3002\uff08\u4ee5\u524d\u306e\u30d0\u30fc\u30b8\u30e7\u30f3\u3067\u306f plot_implicit()<\/code> \u4ee5\u5916\u306e\u51e1\u4f8b\u304c\u8868\u793a\u3055\u308c\u306a\u3044\u3068\u3044\u3046\u4e0d\u5177\u5408\u304c\u3042\u3063\u305f\u304c\uff0cv2.1.0 \u3067\u4fee\u6b63\u3055\u308c\u305f\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[45]:<\/div>\n
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p2<\/span> =<\/span> plot_list<\/span>([<\/span>0<\/span>],[<\/span>1<\/span>],<\/span> \"y(x) \u306e\u6700\u5927\u5024\"<\/span>,<\/span>\r\n               is_point<\/span> =<\/span> True<\/span>,<\/span> legend<\/span> =<\/span> True<\/span>,<\/span>\r\n               xlim<\/span>=<\/span>(<\/span>-<\/span>1.2<\/span>,<\/span> 1.2<\/span>),<\/span> ylim<\/span>=<\/span>(<\/span>-<\/span>1.2<\/span>,<\/span> 1.5<\/span>),<\/span> \r\n               size<\/span>=<\/span>(<\/span>6<\/span>,<\/span>6<\/span>),<\/span>\r\n               aspect<\/span> =<\/span> \"equal\"<\/span>,<\/span> show<\/span> =<\/span> False<\/span>)<\/span>\r\n\r\np3<\/span> =<\/span> plot_list<\/span>([<\/span>0<\/span>],[<\/span>-<\/span>1<\/span>],<\/span> \"y(x) \u306e\u6700\u5c0f\u5024\"<\/span>,<\/span>\r\n               is_point<\/span> =<\/span> True<\/span>,<\/span> legend<\/span> =<\/span> True<\/span>,<\/span>\r\n               xlim<\/span>=<\/span>(<\/span>-<\/span>1.2<\/span>,<\/span> 1.2<\/span>),<\/span> ylim<\/span>=<\/span>(<\/span>-<\/span>1.2<\/span>,<\/span> 1.5<\/span>),<\/span> \r\n               size<\/span>=<\/span>(<\/span>6<\/span>,<\/span>6<\/span>),<\/span>\r\n               aspect<\/span> =<\/span> \"equal\"<\/span>,<\/span> show<\/span> =<\/span> False<\/span>)<\/span>\r\n\r\np<\/span> =<\/span> p1<\/span>+<\/span>p2<\/span>+<\/span>p3<\/span>\r\np<\/span>.<\/span>show<\/span>();<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":6176,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6196","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6196","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=6196"}],"version-history":[{"count":5,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6196\/revisions"}],"predecessor-version":[{"id":8092,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6196\/revisions\/8092"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6176"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6196"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}