{"id":6441,"date":"2024-02-28T12:30:21","date_gmt":"2024-02-28T03:30:21","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6441"},"modified":"2024-02-28T14:48:19","modified_gmt":"2024-02-28T05:48:19","slug":"%e5%8f%82%e8%80%83%ef%bc%9amaxima-%e3%81%a7%e6%a5%95%e5%86%86%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%83%bb%e5%91%a8%ef%bc%8c%e5%9b%9e%e8%bb%a2%e6%a5%95%e5%86%86%e4%bd%93%e3%81%ae%e8%a1%a8%e9%9d%a2%e7%a9%8d","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%a6b\/%e7%a9%8d%e5%88%86%ef%bc%9a%e3%81%84%e3%81%8f%e3%81%a4%e3%81%8b%e3%81%ae%e5%bf%9c%e7%94%a8\/%e5%8f%82%e8%80%83%ef%bc%9amaxima-%e3%81%a7%e6%a5%95%e5%86%86%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%83%bb%e5%91%a8%ef%bc%8c%e5%9b%9e%e8%bb%a2%e6%a5%95%e5%86%86%e4%bd%93%e3%81%ae%e8%a1%a8%e9%9d%a2%e7%a9%8d\/","title":{"rendered":"\u53c2\u8003\uff1aMaxima \u3067\u6955\u5186\u306e\u9762\u7a4d\u30fb\u5468\uff0c\u56de\u8ee2\u6955\u5186\u4f53\u306e\u8868\u9762\u7a4d\u30fb\u4f53\u7a4d"},"content":{"rendered":"

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\u6955\u5186\u306e\u9762\u7a4d<\/h3>\n

Maxima \u3067\u9577\u534a\u5f84 $a$\uff0c\u5358\u534a\u5f84 $b$ \u306e\u6955\u5186\u3092\u63cf\u304f\u4f8b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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a<\/span>:<\/span> 2$\r\nb<\/span>:<\/span> 1$\r\n\r\ndraw2d<\/span>(<\/span>\r\n  font_size<\/span> =<\/span> 14<\/span>, dimensions<\/span>=<\/span>[<\/span>640<\/span>, 400]<\/span>,\r\n  \/* \u7e26\u6a2a\u6bd4 *\/<\/span>\r\n  proportional_axes<\/span> =<\/span> xy<\/span>, \r\n  \/* \u8868\u793a\u7bc4\u56f2 *\/<\/span>\r\n  xrange<\/span> =<\/span> [<\/span>-<\/span>2.5<\/span>, 2.<\/span>5]<\/span>, yrange<\/span> =<\/span> [<\/span>-<\/span>1.5<\/span>, 1.<\/span>5]<\/span>,\r\n  transparent<\/span> =<\/span> false<\/span>,\r\n  fill_color<\/span>  =<\/span> yellow<\/span>,\r\n  line_width<\/span>  =<\/span> 3<\/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  xtics<\/span> =<\/span> {[<\/span>\"-a\"<\/span>, -<\/span>a<\/span>]<\/span>, [<\/span>\"a\"<\/span>, a<\/span>]}<\/span>,\r\n  ytics<\/span> =<\/span> {[<\/span>\"-b\"<\/span>, -<\/span>b<\/span>]<\/span>, [<\/span>\"b\"<\/span>, b<\/span>]}<\/span>, user_preamble<\/span> =<\/span> \"set grid front;\"<\/span>,\r\n\r\n  ellipse<\/span>(<\/span>0<\/span>, 0<\/span>, a<\/span>, b<\/span>, 0<\/span>, 360<\/span>)<\/span>\r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\displaystyle \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ \u306e\u6955\u5186\u306e\u9762\u7a4d\u3002<\/p>\n
\u4fbf\u5b9c\u4e0a\uff0c$a > b > 0$ \u3068\u3057\u3066 $a$ \u306f\u9577\u534a\u5f84\uff0c$b$ \u306f\u77ed\u534a\u5f84\u3068\u547c\u3076\u3002\u96e2\u5fc3\u7387 $e$ \u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068\uff0c$b = a \\sqrt{1-e^2}$\u3002<\/p>\n
$\\displaystyle \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ \u3088\u308a $\\displaystyle y = \\pm b \\sqrt{1-\\frac{x^2}{a^2}}$\u3002\u4e0a\u534a\u5206\u306e\u9762\u7a4d\uff0c\u3064\u307e\u308a $\\displaystyle y = b \\sqrt{1-\\frac{x^2}{a^2}}$ \u3068 $x$ \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092\u6c42\u3081\u30662\u500d\u3059\u308c\u3070\u3088\u3044\u304b\u3089\uff0c<\/p>\n
\\begin{eqnarray}
\nS &=& 2 \\int_{-a}^a b \\sqrt{1 \u2013 \\frac{x^2}{a^2}} dx \\\\
\n&&\\qquad (x \\equiv a \\sin\\theta, \\ \\sqrt{1 \u2013 \\frac{x^2}{a^2}} = \\cos\\theta, \\ dx = a \\cos\\theta d\\theta)\\\\
\n&=& 2 a b \\int_{-\\pi\/2}^{\\pi\/2} \\cos^2\\theta d\\theta\\\\
\n&=& a b \\int_{-\\pi\/2}^{\\pi\/2} (1 + \\cos 2\\theta) d\\theta \\\\
\n&=&a b \\left[ \\theta + \\frac{1}{2}\\sin 2\\theta\\right]_{-\\pi\/2}^{\\pi\/2} \\\\
\n&=& \\pi a b
\n\\end{eqnarray}<\/p>\n
Maxima \u3067\u76f4\u63a5\u7a4d\u5206\u3057\u3066\u307f\u308b\u3068\uff0c\uff08$a>0$ \u3092\u4eee\u5b9a\u3057\u3066\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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kill<\/span>(<\/span>a<\/span>, b<\/span>, e<\/span>, x<\/span>, y<\/span>)<\/span>$\r\nassume<\/span>(<\/span>a<\/span> ><\/span> 0<\/span>, b<\/span> ><\/span> 0<\/span>, e<\/span> ><\/span> 0<\/span>, e<\/span> <<\/span> 1<\/span>)<\/span>$\r\n\r\n2<\/span> *'<\/span>integrate<\/span>(<\/span>b<\/span>*<\/span>sqrt<\/span>(<\/span>1-<\/span>(<\/span>x<\/span>\/<\/span>a<\/span>)<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>a<\/span>, a<\/span>)<\/span> =<\/span>\r\n2<\/span> *<\/span> integrate<\/span>(<\/span>b<\/span>*<\/span>sqrt<\/span>(<\/span>1-<\/span>(<\/span>x<\/span>\/<\/span>a<\/span>)<\/span>**<\/span>2<\/span>)<\/span>, x<\/span>, -<\/span>a<\/span>, a<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}2\\,b\\,\\int_{-a}^{a}{\\sqrt{1-\\frac{x^2}{a^2}}\\;dx}=\\pi\\,a\\,b\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6955\u5186\u306e\u5468<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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a<\/span>:<\/span> 2$\r\nb<\/span>:<\/span> 1$\r\n\r\ndraw2d<\/span>(<\/span>\r\n  font_size<\/span> =<\/span> 14<\/span>, dimensions<\/span>=<\/span>[<\/span>640<\/span>, 400]<\/span>,\r\n  \/* \u7e26\u6a2a\u6bd4 *\/<\/span>\r\n  proportional_axes<\/span> =<\/span> xy<\/span>, \r\n  \/* \u8868\u793a\u7bc4\u56f2 *\/<\/span>\r\n  xrange<\/span> =<\/span> [<\/span>-<\/span>2.5<\/span>, 2.<\/span>5]<\/span>, yrange<\/span> =<\/span> [<\/span>-<\/span>1.5<\/span>, 1.<\/span>5]<\/span>,\r\n  transparent<\/span> =<\/span> true<\/span>,\r\n  line_width<\/span>  =<\/span> 3<\/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  xtics<\/span> =<\/span> {[<\/span>\"-a\"<\/span>, -<\/span>a<\/span>]<\/span>, [<\/span>\"a\"<\/span>, a<\/span>]}<\/span>,\r\n  ytics<\/span> =<\/span> {[<\/span>\"-b\"<\/span>, -<\/span>b<\/span>]<\/span>, [<\/span>\"b\"<\/span>, b<\/span>]}<\/span>, user_preamble<\/span> =<\/span> \"set grid front;\"<\/span>,\r\n\r\n  ellipse<\/span>(<\/span>0<\/span>, 0<\/span>, a<\/span>, b<\/span>, 0<\/span>, 360<\/span>)<\/span>\r\n)<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6955\u5186\u306e\u5f0f\u3092\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3059\u308b\u3068\uff0c\uff08$x = a \\cos\\theta, \\ y = b \\sin\\theta$ \u3068\u3059\u308b\u3068\u3053\u308d\u3092\u8da3\u5411\u3092\u5909\u3048\u3066\uff09 $x = a \\sin\\theta, \\ y = b \\cos\\theta$\u3002$0 \\le \\theta \\le \\pi\/2$ \u306e\u9577\u3055\u30924\u500d\u3059\u308c\u3070\u3088\u3044\u304b\u3089\uff0c<\/p>\n
\\begin{eqnarray}
\nL &=& 4 \\int_0^{\\frac{\\pi}{2}} \\sqrt{\\left(\\frac{dx}{d\\theta}\\right)^2 + \\left(\\frac{dy}{d\\theta}\\right)^2} \\,d\\theta\\\\
\n&=& 4 \\int_0^{\\frac{\\pi}{2}} \\sqrt{a^2 \\cos^2\\theta + b^2 \\sin^2\\theta} \\,d\\theta \\\\
\n&=& 4 a \\int_0^{\\frac{\\pi}{2}} \\sqrt{\\cos^2\\theta + (1-e^2) \\sin^2\\theta} \\,d\\theta \\\\
\n&=& 4 a \\int_0^{\\frac{\\pi}{2}} \\sqrt{1 -e^2 \\sin^2\\theta} \\,d\\theta \\\\
\n&=& 4 a E(e)
\n\\end{eqnarray}<\/p>\n
\u3053\u3053\u3067
\n$\\displaystyle E(k) \\equiv \\int_0^{\\frac{\\pi}{2}} \\sqrt{1 -k^2 \\sin^2\\theta} \\,d\\theta$ \u306f\u7b2c\u4e8c\u7a2e\u5b8c\u5168\u6955\u5186\u7a4d\u5206\u3067\u3042\u308b\u3002<\/p>\n