{"id":2444,"date":"2022-03-07T11:08:23","date_gmt":"2022-03-07T02:08:23","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?p=2444"},"modified":"2023-03-14T16:36:22","modified_gmt":"2023-03-14T07:36:22","slug":"maxima-jupyter-%e3%81%a7%e6%a5%95%e5%86%86%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b","status":"publish","type":"post","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/2444\/","title":{"rendered":"Maxima-Jupyter \u3067\u6955\u5186\u306e\u9762\u7a4d\u3092\u6c42\u3081\u308b"},"content":{"rendered":"

\u9762\u7a4d\u901f\u5ea6\u4e00\u5b9a\u5247\u306e\u969b\u306b\u6955\u5186\u306e\u9762\u7a4d\u3092\u4f7f\u3063\u305f\u306e\u3067\u5ff5\u306e\u305f\u3081\u3002\u307e\u305f\uff0c\u7a4d\u5206\u306e\u6388\u696d\u306e\u4f8b\u984c\u7528\u3068\u3057\u3066\u3002<\/p>\n

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\u6955\u5186\u306e\u4e2d\u5fc3\u3092\u539f\u70b9\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3067\u9762\u7a4d\u3092\u6c42\u3081\u308b<\/h3>\n

\u6955\u5186\u306e\u4e2d\u5fc3\u3092\u539f\u70b9\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3092 $X, Y$ \u3068\u3059\u308b\u3068\uff0c\u9577\u534a\u5f84 $a$\uff0c\u77ed\u534a\u5f84 $b$ \uff08\u96e2\u5fc3\u7387 $e$ \u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068 $b = a \\sqrt{1-e^2}$\uff09\u306e\u6955\u5186\u306e\u5f0f\u306f<\/p>\n

$$ \\frac{X^2}{a^2} + \\frac{Y^2}{b^2} = 1$$<\/p>\n

\u3053\u306e\u5f0f\u3092 $Y$ \u306b\u3064\u3044\u3066\u89e3\u304d\uff0c2\u6b21\u65b9\u7a0b\u5f0f\u3060\u304b\u3089\u89e3\u304c2\u3064\u3042\u308b\u306e\u3067 $Y_1(X), Y_2(X)\\\u00a0 (> Y_1(X))$ \u3068\u304a\u304d\uff0c
\n$$ S = \\int_{-a}^{a} (Y_2 – Y_1)\\, dX$$
\n\u304b\u3089\u6955\u5186\u306e\u9762\u7a4d\u3092\u6c42\u3081\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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eq<\/span>:<\/span> X<\/span>**<\/span>2\/<\/span>a<\/span>**<\/span>2<\/span> +<\/span> Y<\/span>**<\/span>2\/<\/span>b<\/span>**<\/span>2<\/span> =<\/span> 1<\/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{Y^2}{b^2}+\\frac{X^2}{a^2}=1\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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sol<\/span>:<\/span> solve<\/span>(<\/span>eq<\/span>, Y<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}\\left[ Y=-\\frac{\\sqrt{a^2-X^2}\\,b}{a} , Y=\\frac{\\sqrt{a^2-X^2}\\,b}{a} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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Y1<\/span>:<\/span> rhs<\/span>(<\/span>sol<\/span>[<\/span>1])<\/span>;\r\nY2<\/span>:<\/span> rhs<\/span>(<\/span>sol<\/span>[<\/span>2])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{3}$}-\\frac{\\sqrt{a^2-X^2}\\,b}{a}\\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{4}$}\\frac{\\sqrt{a^2-X^2}\\,b}{a}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6955\u5186\u306e\u9762\u7a4d\u306f\uff0c$Y = Y_2, Y = Y_1, X = -a, X = a$ \u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u9762\u7a4d\u3067\u3042\u308b\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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assume<\/span>(<\/span>a<\/span> ><\/span> 0<\/span>)<\/span>$ \/* a \u306f\u6b63\u3068\u4eee\u5b9a\u3002 *\/<\/span>\r\nS<\/span>:<\/span> integrate<\/span>(<\/span>Y2<\/span> -<\/span> Y1<\/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[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}\\pi\\,a\\,b\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u77ed\u534a\u5f84 $b$ \u3092 $b = a \\sqrt{1-e^2}$ \u3067\u8868\u3059\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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ev<\/span>(<\/span>S<\/span>, b<\/span> =<\/span> a<\/span>*<\/span>sqrt<\/span>(<\/span>1-<\/span>e<\/span>**<\/span>2<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}\\pi\\,a^2\\,\\sqrt{1-e^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2 $u$ \u3092\u4f7f\u3063\u305f\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3067\u9762\u7a4d\u3092\u6c42\u3081\u308b<\/h3>\n
\u6955\u5186\u306e\u4e2d\u5fc3\u3092\u539f\u70b9\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $X, Y$ \u3092\u5a92\u4ecb\u5909\u6570 $u$ \u3067\u8868\u3059\u3068<\/p>\n
\\begin{eqnarray}
\nX &=& a \\cos u\\\\
\nY &=& b \\sin u
\n\\end{eqnarray}$$ S = 2 \\int_{-a}^a Y\\, dX = 2 \\int_{\\pi}^0 Y(u) \\frac{dX}{du} du$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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X<\/span>:<\/span> a<\/span> *<\/span> cos<\/span>(<\/span>u<\/span>)<\/span>$\r\nY<\/span>:<\/span> b<\/span> *<\/span> sin<\/span>(<\/span>u<\/span>)<\/span>$\r\n\r\nS<\/span>:<\/span> 2<\/span> *<\/span> integrate<\/span>(<\/span>Y<\/span>*<\/span>diff<\/span>(<\/span>X<\/span>, u<\/span>)<\/span>, u<\/span>, %pi<\/span>, 0<\/span>)<\/span>;\r\nev<\/span>(<\/span>S<\/span>, b<\/span> =<\/span> a<\/span>*<\/span>sqrt<\/span>(<\/span>1-<\/span>e<\/span>**<\/span>2<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{10}$}\\pi\\,a\\,b\\]<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}\\pi\\,a^2\\,\\sqrt{1-e^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u6955\u5186\u306e\u7126\u70b9\u3092\u539f\u70b9\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y$ \u3092\u5a92\u4ecb\u5909\u6570 $u$ \u3067\u8868\u3059\u3068<\/p>\n
\\begin{eqnarray}
\nx &=& X – a e = a(\\cos u – e)\\\\
\ny &=& Y = b \\sin u
\n\\end{eqnarray}$$ S = 2 \\int_{-a-ae}^{a-ae} y \\, dx = 2 \\int_{\\pi}^0 y(u) \\frac{dx}{du} du = 2 \\int_{\\pi}^0 Y(u) \\frac{dX}{du} du$$<\/p>\n
\u3068\u306a\u308a\uff0c\u4e0a\u8a18\u306e\u6955\u5186\u306e\u4e2d\u5fc3\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $X, Y$ \u306e\u5a92\u4ecb\u5909\u6570\u8868\u793a\u7248\u3068\u7b54\u3048\u306f\u540c\u3058\u306b\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
\u9762\u7a4d\u901f\u5ea6\u4e00\u5b9a\u5247\u306e\u969b\u306b\u6955\u5186\u306e\u9762\u7a4d\u3092\u4f7f\u3063\u305f\u306e\u3067\u5ff5\u306e\u305f\u3081\u3002\u307e\u305f\uff0c\u7a4d\u5206\u306e\u6388\u696d\u306e\u4f8b\u984c\u7528\u3068\u3057\u3066\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[14],"tags":[],"class_list":["post-2444","post","type-post","status-publish","format-standard","hentry","category-maxima","nodate","item-wrap"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t