{"id":2398,"date":"2022-02-26T19:52:33","date_gmt":"2022-02-26T10:52:33","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2398"},"modified":"2023-04-14T08:29:33","modified_gmt":"2023-04-13T23:29:33","slug":"maxima-jupyter-%e3%81%a6%e3%82%99%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\/maxima-%e3%81%a7%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/maxima-jupyter-%e3%81%a6%e3%82%99%e5%81%8f%e5%be%ae%e5%88%86\/","title":{"rendered":"Maxima \u3066\u3099\u504f\u5fae\u5206"},"content":{"rendered":"

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\u504f\u5fae\u5206\uff1a\u591a\u5909\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h3>\n

Maxima \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

Maxima-Jupyter \u3067\u306f\u504f\u5fae\u5206\u306e\u8868\u793a\u304c $\\displaystyle \\frac{\\partial}{\\partial x}$ \u3084 $\\displaystyle \\frac{\\partial}{\\partial y}$ \u3067\u306f\u306a\u304f\uff0c$\\displaystyle \\frac{d}{dx}$ \u3084 $\\displaystyle \\frac{d}{dy}$ \u3068\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u304c\uff0c\u3054\u4e86\u627f\u304f\u3060\u3055\u3044\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>, x<\/span>)<\/span>;\r\ndiff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>, y<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}\\frac{d}{d\\,x}\\,f\\left(x , y\\right)\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{2}$}\\frac{d}{d\\,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$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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z<\/span>:<\/span> x<\/span>**<\/span>3<\/span> -<\/span> 4<\/span> *<\/span> x<\/span>**<\/span>2<\/span> *<\/span> y<\/span> +<\/span> x<\/span> *<\/span> y<\/span> +<\/span> 3<\/span> *<\/span> y<\/span>**<\/span>2<\/span>;\r\n\r\ndiff<\/span>(<\/span>z<\/span>, x<\/span>)<\/span>;\r\ndiff<\/span>(<\/span>z<\/span>, y<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{3}$}3\\,y^2-4\\,x^2\\,y+x\\,y+x^3\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{4}$}-8\\,x\\,y+y+3\\,x^2\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{5}$}6\\,y-4\\,x^2+x\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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(2) $\\displaystyle z = \\tan^{-1} \\frac{y}{x} $<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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z<\/span>:<\/span> atan<\/span>(<\/span>y<\/span>\/<\/span>x<\/span>)<\/span>;\r\n\r\ndiff<\/span>(<\/span>z<\/span>, x<\/span>)<\/span>;\r\ndiff<\/span>(<\/span>z<\/span>, y<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{6}$}\\arctan \\left(\\frac{y}{x}\\right)\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{7}$}-\\frac{y}{x^2\\,\\left(\\frac{y^2}{x^2}+1\\right)}\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{8}$}\\frac{1}{x\\,\\left(\\frac{y^2}{x^2}+1\\right)}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u504f\u5fae\u5206\u306f\u3057\u3066\u304f\u308c\u307e\u3057\u305f\u304c\uff0c\u5206\u6bcd\u306f\u3082\u3046\u3061\u3087\u3063\u3068\u304c\u3093\u3070\u3063\u3066\u7c21\u5358\u5316\u3057\u3066\u3044\u305f\u3060\u304d\u305f\u3044\u3068\u3053\u308d\u3002
\nratsimp()<\/code> \u95a2\u6570\u3067\u300c\u7c21\u5358\u5316\u300d\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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dzdx<\/span>:<\/span> diff<\/span>(<\/span>z<\/span>, x<\/span>)<\/span>$\r\ndzdx<\/span> =<\/span> ratsimp<\/span>(<\/span>dzdx<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{10}$}-\\frac{y}{x^2\\,\\left(\\frac{y^2}{x^2}+1\\right)}=-\\frac{y}{y^2+x^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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dzdy<\/span>:<\/span> diff<\/span>(<\/span>z<\/span>, y<\/span>)<\/span>$\r\ndzdy<\/span> =<\/span> ratsimp<\/span>(<\/span>dzdy<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{12}$}\\frac{1}{x\\,\\left(\\frac{y^2}{x^2}+1\\right)}=\\frac{x}{y^2+x^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
Maxima \u3067\u5168\u5fae\u5206\u3092\u6c42\u3081\u308b\u3068\u304d\u306f $df(x,y) = $ diff(f(x, y));<\/code> \u3068\u66f8\u304d\u307e\u3059\u3002<\/p>\n
\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 $<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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z<\/span>:<\/span> sqrt<\/span>(<\/span>x<\/span>**<\/span>2<\/span> -<\/span> y<\/span>**<\/span>2<\/span>)<\/span>$\r\n\r\n'<\/span>diff<\/span>(<\/span>z<\/span>)<\/span> =<\/span> diff<\/span>(<\/span>z<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{14}$}d\\left(\\sqrt{x^2-y^2}\\right)=\\frac{x\\,dx}{\\sqrt{x^2-y^2}}-\\frac{y\\,dy}{\\sqrt{x^2-y^2}}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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z<\/span>:<\/span> x<\/span> *<\/span> y<\/span>**<\/span>2$\r\n\r\n'<\/span>diff<\/span>(<\/span>z<\/span>)<\/span> =<\/span> diff<\/span>(<\/span>z<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{16}$}d\\left(x\\,y^2\\right)=2\\,x\\,y\\,dy+y^2\\,dx\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u9ad8\u968e\u504f\u5c0e\u95a2\u6570<\/h3>\n
Maxima \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[8]:<\/div>\n
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diff<\/span>(<\/span>diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>, y<\/span>)<\/span>, x<\/span>)<\/span> -<\/span> diff<\/span>(<\/span>diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>, x<\/span>)<\/span>, y<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{17}$}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\uff0cMaxima \u3067\u306f
\ntaylor(f(x), x, a, 5);<\/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[9]:<\/div>\n
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f<\/span>(<\/span>x<\/span>)<\/span>:=<\/span> exp<\/span>(<\/span>x<\/span>)<\/span>;\r\n\r\ntaylor<\/span>(<\/span>f<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 5<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{18}$}f\\left(x\\right):=\\exp x\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{19}$}\\frac{x^5}{120}+\\frac{x^4}{24}+\\frac{x^3}{6}+\\frac{x^2}{2}+x+1\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Maxima-Jupyter \u3067\u306f\u306a\u305c\u304b\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306e\u7d50\u679c\u8868\u793a\u304c\u9805\u306e\u9806\u756a\u304c\u964d\u3079\u304d\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u6607\u3079\u304d\u306e\u9806\u306b\u8868\u793a\u3055\u305b\u305f\u3044\u3068\u304d\u306f\uff0ctrunc()<\/code> \u3092\u4f7f\u3063\u3066…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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trunc<\/span>(<\/span>taylor<\/span>(<\/span>f<\/span>(<\/span>x<\/span>)<\/span>, x<\/span>, 0<\/span>, 5<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{20}$}1+x+\\frac{x^2}{2}+\\frac{x^3}{6}+\\frac{x^4}{24}+\\frac{x^5}{120}+\\cdots \\]<\/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\u3082 taylor()<\/code> \u95a2\u6570\u3092\u4f7f\u3063\u3066\u66f8\u304f\u3002 $f(x, y)$ \u3092$x = a, y = b$ \u306e\u307e\u308f\u308a\u3067$3$\u6b21\u307e\u3067\u5c55\u958b\u3059\u308b\u3068\u304d\u306f\uff0c
\ntaylor(f(x, y), [x, y], [a, b], [3, 3]);<\/code> \u306e\u3088\u3046\u306b\u3002<\/p>\n
\u4f8b\u3068\u3057\u3066\uff0c$\\displaystyle f(x, y) = \\frac{1-y}{1 + \\sin x}$ \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[11]:<\/div>\n
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f<\/span>(<\/span>x<\/span>,y<\/span>)<\/span>:=<\/span> (<\/span>1-<\/span>y<\/span>)<\/span>\/<\/span>(<\/span>1<\/span> +<\/span> sin<\/span>(<\/span>x<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{21}$}f\\left(x , y\\right):=\\frac{1-y}{1+\\sin x}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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trunc<\/span>(<\/span>taylor<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>, [<\/span>x<\/span>, y<\/span>]<\/span>, [<\/span>0<\/span>, 0]<\/span>, [<\/span>2<\/span>, 2]))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{22}$}1-x+x^2-y+x\\,y+\\cdots \\]<\/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)
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\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[13]:<\/div>\n
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x<\/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
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\\[\\tag{${\\it \\%o}_{23}$}u\\,\\cos v\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{24}$}u\\,\\sin v\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{25}$}u^2\\,\\cos v\\,\\sin v\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{\\partial z}{\\partial u}$ \u3092\u8a08\u7b97\u3059\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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diff<\/span>(<\/span>z<\/span>, u<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{26}$}2\\,u\\,\\cos v\\,\\sin v\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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\/* \u4e0a\u306e\u7d50\u679c\u3092\u7c21\u5358\u5316\u3002*\/<\/span>\r\n\/* trigreduce() \u306f\u4e09\u89d2\u95a2\u6570\u306e\u7a4d\u3092\u306a\u308b\u3079\u304f\u6e1b\u3089\u3057\u3066\u7c21\u5358\u5316\u3057\u307e\u3059\u3002*\/<\/span>\r\ntrigreduce<\/span>(<\/span>%<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[15]:<\/div>\n
\\[\\tag{${\\it \\%o}_{27}$}u\\,\\sin \\left(2\\,v\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u540c\u69d8\u306b\u3057\u3066\uff0c$\\displaystyle \\frac{\\partial z}{\\partial v}$ \u3092\u8a08\u7b97\u3059\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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diff(z, u)<\/code> \u306f $\\displaystyle \\frac{\\partial z}{\\partial u}$ \u3067\u3059\u304c\uff0cMaxima \u306f $\\partial $ \u3068\u306f\u66f8\u3051\u306a\u3044\u3088\u3046\u3067\u3059\u3002\u8a08\u7b97\u305d\u306e\u3082\u306e\u306f\u3061\u3083\u3093\u3068\u504f\u5fae\u5206\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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'<\/span>diff<\/span>(<\/span>z<\/span>, v<\/span>)<\/span> =<\/span> diff<\/span>(<\/span>z<\/span>, v<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[16]:<\/div>\n
\\[\\tag{${\\it \\%o}_{28}$}\\frac{d}{d\\,v}\\,\\left(u^2\\,\\cos v\\,\\sin v\\right)=u^2\\,\\cos ^2v-u^2\\,\\sin ^2v\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[17]:<\/div>\n
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\/* \u4e0a\u306e\u7d50\u679c\u306e\u53f3\u8fba rhs \u3092\u7c21\u5358\u5316\u3002*\/<\/span>\r\n\/* trigreduce() \u306f\u4e09\u89d2\u95a2\u6570\u306e\u7a4d\u3092\u306a\u308b\u3079\u304f\u6e1b\u3089\u3057\u3066\u7c21\u5358\u5316\u3057\u307e\u3059\u3002*\/<\/span>\r\ntrigreduce<\/span>(<\/span>rhs<\/span>(<\/span>%<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[17]:<\/div>\n
\\[\\tag{${\\it \\%o}_{29}$}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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In\u00a0[18]:<\/div>\n
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\/* \u3053\u308c\u307e\u3067\u306e x, y \u306e\u5b9a\u7fa9\u3092\u5ff5\u306e\u305f\u3081\u306b\u6d88\u53bb\u3002*\/<\/span>\r\nkill<\/span>(<\/span>x<\/span>, y<\/span>, z<\/span>)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\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\u3063\u3066\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
\u3053\u308c\u3092 Maxima \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
\u3053\u308c\u3092 Maxima \u3067\u8868\u73fe\u3059\u308b\u306b\u306f\uff0cy<\/code> \u306f x<\/code> \u306b\u4f9d\u5b58\u3059\u308b (depends()<\/code>) \u3068\u3044\u3046\u610f\u5473\u3067\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304d\u307e\u3059\u3002
\ndepends(y, x);<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[19]:<\/div>\n
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depends<\/span>(<\/span>y<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[19]:<\/div>\n
\\[\\tag{${\\it \\%o}_{31}$}\\left[ y\\left(x\\right) \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$y$ \u306e $x$ \u4f9d\u5b58\u6027\u3092\u5ba3\u8a00\u3057\u305f\u4e0a\u3067\uff0c$f(x, y) = x^2 + y^2 -1 = 0$ \u3092 $x$ \u3067\u5fae\u5206\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[20]:<\/div>\n
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f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span>:=<\/span> x<\/span>**<\/span>2<\/span> +<\/span> y<\/span>**<\/span>2<\/span> -<\/span> 1<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[20]:<\/div>\n
\\[\\tag{${\\it \\%o}_{32}$}f\\left(x , y\\right):=x^2+y^2-1\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$f(x, y) = 0$ \u306e\u4e21\u8fba\u3092 $x$ \u3067\u5fae\u5206\u3057\uff0c\u5909\u6570 df<\/code> \u306b\u4ee3\u5165\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[21]:<\/div>\n
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df<\/span>:<\/span> diff<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span> =<\/span> 0<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[21]:<\/div>\n
\\[\\tag{${\\it \\%o}_{33}$}2\\,y\\,\\left(\\frac{d}{d\\,x}\\,y\\right)+2\\,x=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u5f0f\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[22]:<\/div>\n
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ans<\/span>:<\/span> solve<\/span>(<\/span>df<\/span>, diff<\/span>(<\/span>y<\/span>,x<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[22]:<\/div>\n
\\[\\tag{${\\it \\%o}_{34}$}\\left[ \\frac{d}{d\\,x}\\,y=-\\frac{x}{y} \\right] \\]<\/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
\nMaxima \u3067\u306f\uff0csolve(f(x, y) = 0, y);<\/code> \u3068\u66f8\u304f\u3002\u89e3\u3092 sol<\/code> \u306b\u4ee3\u5165\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[23]:<\/div>\n
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sol<\/span>:<\/span> solve<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span> =<\/span> 0<\/span>, y<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[23]:<\/div>\n
\\[\\tag{${\\it \\%o}_{35}$}\\left[ y=-\\sqrt{1-x^2} , y=\\sqrt{1-x^2} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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1\u3064\u76ee\u306e\u89e3\u306f sol[1]<\/code> \u306e\u3088\u3046\u306b\u53d6\u308a\u51fa\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[24]:<\/div>\n
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sol<\/span>[<\/span>1]<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[24]:<\/div>\n
\\[\\tag{${\\it \\%o}_{36}$}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[25]:<\/div>\n
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dy1<\/span>:<\/span> '<\/span>diff<\/span>(<\/span>y<\/span>, x<\/span>)<\/span> =<\/span> diff<\/span>(<\/span>rhs<\/span>(<\/span>sol<\/span>[<\/span>1])<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[25]:<\/div>\n
\\[\\tag{${\\it \\%o}_{37}$}\\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 (dy1<\/code>)\u306e\u53f3\u8fba (rhs()<\/code>) \u306e\u5206\u6bcd\u3092 $\\sqrt{1-x^2} = -y$ \u3092\u4ee3\u5165\u3057\u3066\u8a55\u4fa1 (ev()<\/code>) \u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[26]:<\/div>\n
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dai1<\/span>:<\/span> -<\/span>rhs<\/span>(<\/span>sol<\/span>[<\/span>1])<\/span> =<\/span> -<\/span>lhs<\/span>(<\/span>sol<\/span>[<\/span>1])<\/span>;\r\nlhs<\/span>(<\/span>dy1<\/span>)<\/span> =<\/span> ev<\/span>(<\/span>rhs<\/span>(<\/span>dy1<\/span>)<\/span>, dai1<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[26]:<\/div>\n
\\[\\tag{${\\it \\%o}_{38}$}\\sqrt{1-x^2}=-y\\]<\/div>\n<\/div>\n
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Out[26]:<\/div>\n
\\[\\tag{${\\it \\%o}_{39}$}\\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 sol[2]<\/code> \u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[27]:<\/div>\n
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sol<\/span>[<\/span>2]<\/span>;\r\ndy2<\/span>:<\/span> '<\/span>diff<\/span>(<\/span>y<\/span>, x<\/span>)<\/span> =<\/span> diff<\/span>(<\/span>rhs<\/span>(<\/span>sol<\/span>[<\/span>2])<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[27]:<\/div>\n
\\[\\tag{${\\it \\%o}_{40}$}y=\\sqrt{1-x^2}\\]<\/div>\n<\/div>\n
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Out[27]:<\/div>\n
\\[\\tag{${\\it \\%o}_{41}$}\\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[28]:<\/div>\n
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dai2<\/span>:<\/span> rhs<\/span>(<\/span>sol<\/span>[<\/span>2])<\/span>=<\/span>lhs<\/span>(<\/span>sol<\/span>[<\/span>2])<\/span>;\r\nlhs<\/span>(<\/span>dy2<\/span>)<\/span> =<\/span> ev<\/span>(<\/span>rhs<\/span>(<\/span>dy2<\/span>)<\/span>, dai2<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[28]:<\/div>\n
\\[\\tag{${\\it \\%o}_{42}$}\\sqrt{1-x^2}=y\\]<\/div>\n<\/div>\n
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Out[28]:<\/div>\n
\\[\\tag{${\\it \\%o}_{43}$}\\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 sol[1]<\/code> \u306b\u3064\u3044\u3066\u30822\u3064\u76ee\u306e\u89e3 sol[2]<\/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 ans[1]<\/code> \u306b\u683c\u7d0d\u3057\u3066\u3044\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[29]:<\/div>\n
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ans<\/span>[<\/span>1]<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[29]:<\/div>\n
\\[\\tag{${\\it \\%o}_{44}$}\\frac{d}{d\\,x}\\,y=-\\frac{x}{y}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u9670\u95a2\u6570\u306e2\u968e\u5fae\u5206<\/h5>\n
ans[1]<\/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[30]:<\/div>\n
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ddy<\/span>:<\/span> diff<\/span>(<\/span>ans<\/span>[<\/span>1]<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[30]:<\/div>\n
\\[\\tag{${\\it \\%o}_{45}$}\\frac{d^2}{d\\,x^2}\\,y=\\frac{x\\,\\left(\\frac{d}{d\\,x}\\,y\\right)}{y^2}-\\frac{1}{y}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u5f0f\u306e\u53f3\u8fba\u306b ans[1]<\/code> \u3064\u307e\u308a $\\displaystyle \\frac{dy}{dx} = – \\frac{x}{y}$ \u3092\u4ee3\u5165\u3057\u3066\u8a55\u4fa1\uff08ev()<\/code>\uff09\u3057\u3066\u3084\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[31]:<\/div>\n
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ddy<\/span>:<\/span> ev<\/span>(<\/span>ddy<\/span>, ans<\/span>[<\/span>1])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[31]:<\/div>\n
\\[\\tag{${\\it \\%o}_{46}$}\\frac{d^2}{d\\,x^2}\\,y=-\\frac{1}{y}-\\frac{x^2}{y^3}\\]<\/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[32]:<\/div>\n
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sols<\/span>:<\/span> solve<\/span>([<\/span>x<\/span> =<\/span> 0<\/span>, f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span> =<\/span> 0]<\/span>, [<\/span>x<\/span>, y<\/span>])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[32]:<\/div>\n
\\[\\tag{${\\it \\%o}_{47}$}\\left[ \\left[ x=0 , y=-1 \\right] , \\left[ x=0 , y=1 \\right] \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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sols[1]<\/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[33]:<\/div>\n
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rddy<\/span>:<\/span> rhs<\/span>(<\/span>ddy<\/span>)<\/span>$\r\nlhs<\/span>(<\/span>ddy<\/span>)<\/span> =<\/span> ev<\/span>(<\/span>rddy<\/span>, sols<\/span>[<\/span>1])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[33]:<\/div>\n
\\[\\tag{${\\it \\%o}_{49}$}\\frac{d^2}{d\\,x^2}\\,y=1\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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… \u3068\u306a\u308a\uff0c$\\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 sols[2]<\/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[34]:<\/div>\n
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lhs<\/span>(<\/span>ddy<\/span>)<\/span> =<\/span> ev<\/span>(<\/span>rddy<\/span>, sols<\/span>[<\/span>2])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[34]:<\/div>\n
\\[\\tag{${\\it \\%o}_{50}$}\\frac{d^2}{d\\,x^2}\\,y=-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
Maxima \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[35]:<\/div>\n
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\/* plot2d() \u3067\u63cf\u304f\u5834\u5408 *\/<\/span>\r\nplot2d<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span> =<\/span> 0<\/span>, \r\n       [<\/span>x<\/span>, -<\/span>1.2<\/span>, 1.<\/span>2]<\/span>, [<\/span>y<\/span>, -<\/span>1.2<\/span>, 1.<\/span>2]<\/span>, \r\n       [<\/span>same_xy<\/span>]<\/span>, \r\n       [<\/span>title<\/span>, \"f(x, y) = 0\"<\/span>]<\/span>\r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\"\"<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[36]:<\/div>\n
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\/* draw2d() \u3067\u63cf\u304f\u5834\u5408 *\/<\/span>\r\n\/* \u5197\u9577\u3060\u304c\u3044\u308d\u3044\u308d\u8a2d\u5b9a\u304c\u53ef\u80fd *\/<\/span>\r\n\r\ndraw2d<\/span>(<\/span>\r\n  \/* \u6ed1\u3089\u304b\u306b\u3059\u308b\u305f\u3081\u306b ip_grid \u3092\u591a\u3081\u306b <\/span>\r\n  ip_grid = [200, 200], *\/<\/span>\r\n  \/* \u7e26\u6a2a\u6bd4 *\/<\/span>\r\n  proportional_axes<\/span> =<\/span> xy<\/span>, yrange<\/span> =<\/span> [<\/span>-<\/span>1.2<\/span>, 1.<\/span>5]<\/span>,\r\n  \/* x \u8ef8 y \u8ef8\u306e\u8868\u793a *\/<\/span>\r\n  xaxis<\/span> =<\/span> true<\/span>, yaxis<\/span> =<\/span> true<\/span>, \r\n  \r\n  title<\/span> =<\/span> \"f(x, y) = 0\"<\/span>,\r\n  implicit<\/span>(<\/span>f<\/span>(<\/span>x<\/span>, y<\/span>)<\/span> =<\/span> 0<\/span>, x<\/span>, -<\/span>1.2<\/span>, 1.2<\/span>, y<\/span>, -<\/span>1.2<\/span>, 1.2<\/span>)<\/span>, \r\n\r\n  point_type<\/span> =<\/span> 7<\/span>, point_size<\/span> =<\/span> 0.7<\/span>,\r\n  key<\/span> =<\/span> \"y(x) \u306e\u6700\u5927\u5024\"<\/span>,\r\n  color<\/span> =<\/span> \"red\"<\/span>,\r\n  points<\/span>([[<\/span>0<\/span>,1]])<\/span>, \r\n\r\n  key<\/span> =<\/span> \"y(x) \u306e\u6700\u5c0f\u5024\"<\/span>,\r\n  color<\/span> =<\/span> \"blue\"<\/span>,\r\n  points<\/span>([[<\/span>0<\/span>,-<\/span>1]])<\/span>\r\n)<\/span>$\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":6118,"menu_order":20,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2398","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\/2398","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=2398"}],"version-history":[{"count":4,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2398\/revisions"}],"predecessor-version":[{"id":6162,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2398\/revisions\/6162"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6118"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2398"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}