{"id":2785,"date":"2022-03-30T11:31:45","date_gmt":"2022-03-30T02:31:45","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2785"},"modified":"2024-02-20T12:04:21","modified_gmt":"2024-02-20T03:04:21","slug":"%e5%8f%82%e8%80%83%ef%bc%9amaxima-jupyter-%e3%81%a7%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%ae%e6%bc%94%e7%ae%97","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e9%9b%bb%e7%a3%81%e6%b0%97%e5%ad%a6-i\/%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%ae%e6%bc%94%e7%ae%97\/%e5%8f%82%e8%80%83%ef%bc%9amaxima-jupyter-%e3%81%a7%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%81%ae%e6%bc%94%e7%ae%97\/","title":{"rendered":"\u53c2\u8003\uff1aMaxima-Jupyter \u3067\u30d9\u30af\u30c8\u30eb\u306e\u6f14\u7b97"},"content":{"rendered":"

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Maxima \u306b\u304a\u3051\u308b\u30d9\u30af\u30c8\u30eb\u306e\u8868\u8a18<\/h3>\n

Maxima \u3067\u306f\uff0c\u30ea\u30b9\u30c8\uff08[<\/code> \u3068 ]<\/code> \u306e\u9593\u306b\u6210\u5206\u3092\u30ab\u30f3\u30de\u3067\u533a\u5207\u3063\u3066\u4e26\u3079\u305f\u3082\u306e\uff09\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\uff08matrix()<\/code> \u3092\u4f7f\u3063\u305f\u8868\u3057\u65b9\u3082\u3042\u308a\u307e\u3059\u304c\uff0c\u3053\u3053\u3067\u306f\u7701\u7565\u3002\uff09<\/p>\n

Maxima \u306e\u30ea\u30b9\u30c8\u3067\u306f\uff0c$x$ \u6210\u5206\uff0c$y$ \u6210\u5206… \u306e\u3088\u3046\u306b\u6307\u5b9a\u3059\u308b\u3088\u308a\u3082\uff0c1\u756a\u76ee\u306e\u6210\u5206\uff0c2\u756a\u76ee\u306e\u6210\u5206… \u306e\u3088\u3046\u306b\u6307\u5b9a\u3057\u305f\u307b\u3046\u304c\u4fbf\u5229\u3067\u3059\u3002<\/p>\n

[a1, a2, a3]<\/code> \u306e\u3088\u3046\u306b\u66f8\u304f\u3068\u6dfb\u5b57\u3068\u3057\u3066\u8868\u793a\u3057\u3066\u304f\u308c\u307e\u3059\u3057\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/div>\n
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a<\/span>:<\/span> [<\/span>a1<\/span>, a2<\/span>, a3<\/span>]<\/span>;\r\nb<\/span>:<\/span> [<\/span>b1<\/span>, b2<\/span>, b3<\/span>]<\/span>;\r\nc<\/span>:<\/span> [<\/span>c1<\/span>, c2<\/span>, c3<\/span>]<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{1}$}\\left[ a_{1} , a_{2} , a_{3} \\right] \\]<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{2}$}\\left[ b_{1} , b_{2} , b_{3} \\right] \\]<\/div>\n<\/div>\n
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Out[1]:<\/div>\n
\\[\\tag{${\\it \\%o}_{3}$}\\left[ c_{1} , c_{2} , c_{3} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u6f14\u7b97<\/h3>\n
\u8db3\u7b97\uff0c\u5f15\u304d\u7b97<\/h4>\n
\u30d9\u30af\u30c8\u30eb\u306e\u8db3\u7b97\u5f15\u304d\u7b97\u3092\u3057\u3066\u307f\u308b\u3068\uff0c\u6210\u5206\u540c\u58eb\u306e\u8db3\u7b97\u5f15\u304d\u7b97\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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a<\/span> +<\/span> b<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{4}$}\\left[ b_{1}+a_{1} , b_{2}+a_{2} , b_{3}+a_{3} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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a<\/span> -<\/span> b<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}\\left[ a_{1}-b_{1} , a_{2}-b_{2} , a_{3}-b_{3} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u6570\u500d<\/h4>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u6570\u500d\u306f\uff0c\u5404\u6210\u5206\u3092\u305d\u308c\u305e\u308c\u5b9a\u6570\u500d\u3057\u305f\u3082\u306e\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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C<\/span> *<\/span> a<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{6}$}\\left[ C\\,a_{1} , C\\,a_{2} , C\\,a_{3} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d<\/h3>\n
Maxima \u3067\u306f\uff0c\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306f .<\/code>\uff08\u30c9\u30c3\u30c8\uff0c\u30d4\u30ea\u30aa\u30c9\uff09\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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a<\/span> .<\/span> b<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}a_{3}\\,b_{3}+a_{2}\\,b_{2}+a_{1}\\,b_{1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055<\/h4>\n
\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}$ \u306e\u5927\u304d\u3055 $|\\boldsymbol{a}|$ \u306f\u81ea\u5206\u81ea\u8eab\u3068\u306e\u5185\u7a4d\u306e\u5e73\u65b9\u6839\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002
\n$$|\\boldsymbol{a}| \\equiv \\sqrt{\\boldsymbol{a}\\cdot\\boldsymbol{a}}$$<\/p>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u3092\u8fd4\u3059\u95a2\u6570 norm()<\/code> \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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norm<\/span>(<\/span>v<\/span>)<\/span>:=<\/span> sqrt<\/span>(<\/span>v<\/span> .<\/span> v<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{8}$}{\\it norm}\\left(v\\right):=\\sqrt{v\\cdot v}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4eca\u5b9a\u7fa9\u3057\u305f norm()<\/code> \u3092\u4f7f\u3063\u3066\uff0c\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}$ \u306e\u5927\u304d\u3055 $|\\boldsymbol{a}|$ \u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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norm<\/span>(<\/span>a<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{9}$}\\sqrt{a_{3}^2+a_{2}^2+a_{1}^2}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d<\/h3>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u3092\u8a08\u7b97\u3059\u308b\u95a2\u6570 cross()<\/code> \u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\uff0c\u81ea\u5206\u3067\u5b9a\u7fa9\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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cross<\/span>(<\/span>u<\/span>, v<\/span>)<\/span>:=<\/span>\r\n           [<\/span>u<\/span>[<\/span>2]<\/span>*<\/span>v<\/span>[<\/span>3]<\/span> -<\/span> u<\/span>[<\/span>3]<\/span>*<\/span>v<\/span>[<\/span>2]<\/span>,\r\n            u<\/span>[<\/span>3]<\/span>*<\/span>v<\/span>[<\/span>1]<\/span> -<\/span> u<\/span>[<\/span>1]<\/span>*<\/span>v<\/span>[<\/span>3]<\/span>,\r\n            u<\/span>[<\/span>1]<\/span>*<\/span>v<\/span>[<\/span>2]<\/span> -<\/span> u<\/span>[<\/span>2]<\/span>*<\/span>v<\/span>[<\/span>1]]<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{10}$}{\\it cross}\\left(u , v\\right):=\\left[ u_{2}\\,v_{3}-u_{3}\\,v_{2} , u_{3}\\,v_{1}-u_{1}\\,v_{3} , u_{1}\\,v_{2}-u_{2}\\,v_{1} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\boldsymbol{c} = \\boldsymbol{a}\\times \\boldsymbol{b}$ \u306e\u8a08\u7b97\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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c<\/span>:<\/span> cross<\/span>(<\/span>a<\/span>, b<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}\\left[ a_{2}\\,b_{3}-a_{3}\\,b_{2} , a_{3}\\,b_{1}-a_{1}\\,b_{3} , a_{1}\\,b_{2}-a_{2}\\,b_{1} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\boldsymbol{c}$ \u306e\u6210\u5206\u3054\u3068\u306b\u8868\u793a\u3055\u305b\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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'<\/span>c<\/span>[<\/span>1]<\/span> =<\/span> c<\/span>[<\/span>1]<\/span>;\r\n'<\/span>c<\/span>[<\/span>2]<\/span> =<\/span> c<\/span>[<\/span>2]<\/span>;\r\n'<\/span>c<\/span>[<\/span>3]<\/span> =<\/span> c<\/span>[<\/span>3]<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
\\[\\tag{${\\it \\%o}_{12}$}c_{1}=a_{2}\\,b_{3}-a_{3}\\,b_{2}\\]<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
\\[\\tag{${\\it \\%o}_{13}$}c_{2}=a_{3}\\,b_{1}-a_{1}\\,b_{3}\\]<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
\\[\\tag{${\\it \\%o}_{14}$}c_{3}=a_{1}\\,b_{2}-a_{2}\\,b_{1}\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u5916\u7a4d\u306f\u300c\u53cd\u4ea4\u63db\u300d<\/h4>\n
$\\boldsymbol{a}\\times\\boldsymbol{b} {\\color{red}{\\neq}} \\boldsymbol{b}\\times\\boldsymbol{a}$ \u3067\u3042\u308b\u3053\u3068\uff0c\u5177\u4f53\u7684\u306b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u300c\u53cd\u4ea4\u63db\u300d\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002
\n$$\\boldsymbol{a}\\times\\boldsymbol{b} = – \\boldsymbol{b}\\times\\boldsymbol{a}$$<\/p>\n
$\\boldsymbol{a}\\times\\boldsymbol{b} + \\boldsymbol{b}\\times\\boldsymbol{a} = \\boldsymbol{0}$ \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3044\u3044\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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cross<\/span>(<\/span>a<\/span>, b<\/span>)<\/span> +<\/span> cross<\/span>(<\/span>b<\/span>, a<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{15}$}\\left[ 0 , 0 , 0 \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u306e\u4e09\u91cd\u7a4d<\/h3>\n
\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d<\/h4>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}$ \u306b\u3088\u308b\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306f $\\boldsymbol{a} \\cdot (\\boldsymbol{b}\\times\\boldsymbol{c})$ \u3067\u3042\u308a\uff0c\u3053\u308c\u306f\u7d50\u5c40\u306f\u30b9\u30ab\u30e9\u30fc\u3068\u306a\u308a\u307e\u3059\u3002\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u516c\u5f0f\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n
$$\\boldsymbol{a} \\cdot (\\boldsymbol{b}\\times\\boldsymbol{c})
\n= \\boldsymbol{b} \\cdot (\\boldsymbol{c}\\times\\boldsymbol{a})
\n= \\boldsymbol{c} \\cdot (\\boldsymbol{a}\\times\\boldsymbol{b}) $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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$\\boldsymbol{a} \\cdot (\\boldsymbol{b}\\times\\boldsymbol{c})
\n= \\boldsymbol{b} \\cdot (\\boldsymbol{c}\\times\\boldsymbol{a})$ \u3064\u307e\u308a\uff0c
\n$\\boldsymbol{a} \\cdot (\\boldsymbol{b}\\times\\boldsymbol{c}) \\,-\\, \\boldsymbol{b} \\cdot (\\boldsymbol{c}\\times\\boldsymbol{a}) = 0$ \u3092\u793a\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
Maxima \u306e\u7c21\u7565\u8868\u8a18\u3067\uff0c1\u884c\u3067\uff08\u76f4\u524d\u306e\u7d50\u679c\u3092\uff09ratsimp<\/code> \u306b\u3088\u308b\u7c21\u5358\u5316\u307e\u3067\u884c\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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a<\/span> .<\/span> cross<\/span>(<\/span>b<\/span>, c<\/span>)<\/span> -<\/span> b<\/span> .<\/span> cross<\/span>(<\/span>c<\/span>, a<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{16}$}0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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$\\boldsymbol{b} \\cdot (\\boldsymbol{c}\\times\\boldsymbol{a}) \\,-\\, \\boldsymbol{c} \\cdot (\\boldsymbol{a}\\times\\boldsymbol{b}) = 0$ \u3092\u793a\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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b<\/span> .<\/span> cross<\/span>(<\/span>c<\/span>, a<\/span>)<\/span> -<\/span> c<\/span> .<\/span> cross<\/span>(<\/span>a<\/span>, b<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
\\[\\tag{${\\it \\%o}_{17}$}0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d<\/h4>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}$ \u306b\u3088\u308b\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u306f $\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c})$ \u3067\u3042\u308a\uff0c\u3053\u308c\u306f\u7d50\u5c40\u306f\u30d9\u30af\u30c8\u30eb\u3092\u3064\u304f\u308a\u307e\u3059\u3002\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u306b\u306f\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u516c\u5f0f\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n
$$\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c})
\n= (\\boldsymbol{a}\\cdot\\boldsymbol{c})\\,\\boldsymbol{b} –
\n(\\boldsymbol{a}\\cdot\\boldsymbol{b})\\,\\boldsymbol{c} $$<\/p>\n
\u4e00\u6319\u306b\u8a3c\u660e\u3059\u308b\u306b\u306f\uff0c<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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cross<\/span>(<\/span>a<\/span>, cross<\/span>(<\/span>b<\/span>, c<\/span>))<\/span> -<\/span> ((<\/span>a<\/span>.<\/span>c<\/span>)<\/span> *<\/span> b<\/span> -<\/span> (<\/span>a<\/span>.<\/span>b<\/span>)<\/span> *<\/span> c<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[14]:<\/div>\n
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