{"id":2789,"date":"2022-03-30T11:37:40","date_gmt":"2022-03-30T02:37:40","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=2789"},"modified":"2024-02-20T13:01:17","modified_gmt":"2024-02-20T04:01:17","slug":"%e5%8f%82%e8%80%83%ef%bc%9amaxima-jupyter-%e3%81%a7%e3%82%b9%e3%82%ab%e3%83%a9%e3%83%bc%e5%a0%b4%e3%83%bb%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e5%a0%b4%e3%81%ae%e5%be%ae%e5%88%86","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%82%b9%e3%82%ab%e3%83%a9%e3%83%bc%e5%a0%b4%e3%83%bb%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e5%a0%b4%e3%81%ae%e5%be%ae%e5%88%86\/%e5%8f%82%e8%80%83%ef%bc%9amaxima-jupyter-%e3%81%a7%e3%82%b9%e3%82%ab%e3%83%a9%e3%83%bc%e5%a0%b4%e3%83%bb%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e5%a0%b4%e3%81%ae%e5%be%ae%e5%88%86\/","title":{"rendered":"\u53c2\u8003\uff1aMaxima-Jupyter \u3067\u30b9\u30ab\u30e9\u30fc\u5834\u30fb\u30d9\u30af\u30c8\u30eb\u5834\u306e\u5fae\u5206"},"content":{"rendered":"

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\u504f\u5fae\u5206<\/h3>\n

Maxima \u3067\u306f\uff0c\u504f\u5fae\u5206\u306f\uff11\u5909\u6570\u306e\u5834\u5408\u306e\u5fae\u5206\u3068\u540c\u3058 diff()<\/code> \u95a2\u6570\u3092\u4f7f\u3063\u3066
\ndiff(f(x, y), x);<\/code> \u3084 diff(f(x, y), y);<\/code> \u306a\u3069\u3068\u66f8\u304d\u307e\u3059\u3002<\/p>\n

\uff08Maxima-Jupyter \u306e\u8868\u793a\u306f $\\displaystyle \\frac{\\partial}{\\partial x} f(x,y)$ \u3067\u306f\u306a\u304f\uff0c$\\displaystyle \\frac{d}{dx} f(x,y)$ \u3068\u306a\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u304c\uff0c\u305d\u3053\u306f\u3054\u4e86\u627f\u9858\u3044\u307e\u3059\u3002\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n

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In\u00a0[1]:<\/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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\u504f\u5fae\u5206\u306f\u4ea4\u63db\u53ef\u80fd<\/h4>\n
$$\\frac{\\partial^2}{\\partial x \\partial y} f(x, y) = \\frac{\\partial^2}{\\partial y \\partial x} f(x, y)$$<\/p>\n
\u3067\u3059\u3002<\/p>\n
$$\\frac{\\partial^2}{\\partial x \\partial y} f(x, y) – \\frac{\\partial^2}{\\partial y \\partial x} f(x, y) = 0$$<\/p>\n
\u3092\u793a\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/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>, y<\/span>)<\/span>, x<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[2]:<\/div>\n
\\[\\tag{${\\it \\%o}_{3}$}0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d\uff08grad\uff09<\/h3>\n
\u30b9\u30ab\u30e9\u30fc\u5834 $\\varphi(\\boldsymbol{r})$ \u306e\u52fe\u914d\u3092\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n
$$\\nabla \\varphi \\equiv \\mbox{grad}\\, \\varphi = \\left(\\frac{\\partial \\varphi}{\\partial x}, \\frac{\\partial \\varphi}{\\partial y},\\frac{\\partial \\varphi}{\\partial z}\\right)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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grad<\/span>(<\/span>phi<\/span>)<\/span>:=<\/span> [<\/span>diff<\/span>(<\/span>phi<\/span>, x<\/span>)<\/span>, diff<\/span>(<\/span>phi<\/span>, y<\/span>)<\/span>, diff<\/span>(<\/span>phi<\/span>, z<\/span>)]<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
\\[\\tag{${\\it \\%o}_{4}$}{\\it grad}\\left(\\varphi\\right):=\\left[ {\\it diff}\\left(\\varphi , x\\right) , {\\it diff}\\left(\\varphi , y\\right) , {\\it diff}\\left(\\varphi , z\\right) \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u52fe\u914d\uff08grad\uff09\u306e\u8a08\u7b97\u4f8b<\/h4>\n
$r = \\sqrt{x^2 + y^2 + z^2}$ \u306e\u3068\u304d\uff0c\u4ee5\u4e0b\u306e\u8a08\u7b97\u3092\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n
$$\\nabla r = \\frac{\\partial r}{\\partial x} \\boldsymbol{i}
\n+\\frac{\\partial r}{\\partial y} \\boldsymbol{j}
\n+\\frac{\\partial r}{\\partial z} \\boldsymbol{k} $$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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'<\/span>rvec<\/span> =<\/span> rvec<\/span>:<\/span> [<\/span>x<\/span>, y<\/span>, z<\/span>]<\/span>;\r\n'<\/span>r<\/span> =<\/span> r<\/span>:<\/span> sqrt<\/span>(<\/span>rvec<\/span>.<\/span>rvec<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
\\[\\tag{${\\it \\%o}_{5}$}{\\it rvec}=\\left[ x , y , z \\right] \\]<\/div>\n<\/div>\n
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\\[\\tag{${\\it \\%o}_{6}$}r=\\sqrt{z^2+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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'<\/span>grad<\/span>(<\/span>'<\/span>r<\/span>)<\/span> =<\/span> grad<\/span>(<\/span>r<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[5]:<\/div>\n
\\[\\tag{${\\it \\%o}_{7}$}{\\it grad}\\left(r\\right)=\\left[ \\frac{x}{\\sqrt{z^2+y^2+x^2}} , \\frac{y}{\\sqrt{z^2+y^2+x^2}} , \\frac{z}{\\sqrt{z^2+y^2+x^2}} \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u4e0a\u306e\u7d50\u679c\u304c<\/p>\n
$$\\nabla r =\\frac{x}{r} \\,\\boldsymbol{i} + \\frac{y}{r} \\,\\boldsymbol{j} + \\frac{z}{r} \\,\\boldsymbol{k} = \\frac{\\boldsymbol{r}}{r}$$<\/p>\n
\u3068\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n
\u5de6\u8fba\u304b\u3089\u53f3\u8fba\u3092\u5f15\u3044\u3066\u30bc\u30ed\u30d9\u30af\u30c8\u30eb\u306b\u306a\u3063\u3066\u3044\u308c\u3070\u3044\u3044\u3067\u3059\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[6]:<\/div>\n
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grad<\/span>(<\/span>r<\/span>)<\/span> -<\/span> rvec<\/span>\/<\/span>r<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[6]:<\/div>\n
\\[\\tag{${\\it \\%o}_{8}$}\\left[ 0 , 0 , 0 \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u307e\u305f\uff0c\u96fb\u78c1\u6c17\u5b66\u3067\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u8a08\u7b97\u3092\u3059\u308b\u5fc5\u8981\u3082\u51fa\u3066\u304f\u308b\u306e\u3067\uff0c\u3084\u3063\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n
$$\\nabla \\left(\\frac{1}{r}\\right) = – \\frac{\\boldsymbol{r}}{r^3}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[7]:<\/div>\n
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'<\/span>grad<\/span>(<\/span>1\/'<\/span>r<\/span>)<\/span> =<\/span> grad<\/span>(<\/span>1\/<\/span>r<\/span>)<\/span>;\r\ngrad<\/span>(<\/span>1\/<\/span>r<\/span>)<\/span> +<\/span> rvec<\/span>\/<\/span>r<\/span>**<\/span>3<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{9}$}{\\it grad}\\left(\\frac{1}{r}\\right)=\\left[ -\\frac{x}{\\left(z^2+y^2+x^2\\right)^{\\frac{3}{2}}} , -\\frac{y}{\\left(z^2+y^2+x^2\\right)^{\\frac{3}{2}}} , -\\frac{z}{\\left(z^2+y^2+x^2\\right)^{\\frac{3}{2}}} \\right] \\]<\/div>\n<\/div>\n
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Out[7]:<\/div>\n
\\[\\tag{${\\it \\%o}_{10}$}\\left[ 0 , 0 , 0 \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u5834\u306e\u767a\u6563 (div)<\/h3>\n
\u30d9\u30af\u30c8\u30eb\u5834\u306e\u767a\u6563\u3092\u5b9a\u7fa9\u3059\u308b\u3002<\/p>\n
$$\\nabla\\cdot\\boldsymbol{v} \\equiv \\mbox{div}\\, \\boldsymbol{v} = \\frac{\\partial v_x}{\\partial x} + \\frac{\\partial v_y}{\\partial y} + \\frac{\\partial v_z}{\\partial z}$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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div<\/span>(<\/span>v<\/span>)<\/span>:=<\/span> diff<\/span>(<\/span>v<\/span>[<\/span>1]<\/span>, x<\/span>)<\/span> +<\/span> diff<\/span>(<\/span>v<\/span>[<\/span>2]<\/span>, y<\/span>)<\/span> +<\/span> diff<\/span>(<\/span>v<\/span>[<\/span>3]<\/span>, z<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
\\[\\tag{${\\it \\%o}_{11}$}{\\it div}\\left(v\\right):={\\it diff}\\left(v_{1} , x\\right)+{\\it diff}\\left(v_{2} , y\\right)+{\\it diff}\\left(v_{3} , z\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u767a\u6563 (div) \u306e\u8a08\u7b97\u4f8b<\/h4>\n
$$\\nabla\\cdot \\boldsymbol{r} = \\frac{\\partial x}{\\partial x} + \\frac{\\partial y}{\\partial y} +\\frac{\\partial z}{\\partial z} = 1+1+1 = 3$$<\/p>\n
\uff08\u7b54\u3048\u306f\u7a7a\u9593\u306e\u6b21\u5143\u306e\u6570\uff09<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[9]:<\/div>\n
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'<\/span>div<\/span>(<\/span>'<\/span>rvec<\/span>)<\/span> =<\/span> div<\/span>(<\/span>rvec<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[9]:<\/div>\n
\\[\\tag{${\\it \\%o}_{12}$}{\\it div}\\left({\\it rvec}\\right)=3\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2 (rot)<\/h3>\n
\u30d9\u30af\u30c8\u30eb\u306e\u56de\u8ee2\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3057\u3066\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n
\\begin{eqnarray}
\n\\nabla\\times\\boldsymbol{v} \\equiv \\mbox{rot}\\, \\boldsymbol{v}
\n&=&\\left( \\frac{\\partial}{\\partial y} v_z – \\frac{\\partial}{\\partial z} v_y, \\frac{\\partial}{\\partial z} v_x – \\frac{\\partial}{\\partial x} v_z, \\frac{\\partial}{\\partial x} v_y – \\frac{\\partial}{\\partial y} v_x\\right)
\n\\end{eqnarray}<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[10]:<\/div>\n
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rot<\/span>(<\/span>v<\/span>)<\/span>:=<\/span> [<\/span>diff<\/span>(<\/span>v<\/span>[<\/span>3]<\/span>, y<\/span>)<\/span> -<\/span> diff<\/span>(<\/span>v<\/span>[<\/span>2]<\/span>, z<\/span>)<\/span>, \r\n          diff<\/span>(<\/span>v<\/span>[<\/span>1]<\/span>, z<\/span>)<\/span> -<\/span> diff<\/span>(<\/span>v<\/span>[<\/span>3]<\/span>, x<\/span>)<\/span>, \r\n          diff<\/span>(<\/span>v<\/span>[<\/span>2]<\/span>, x<\/span>)<\/span> -<\/span> diff<\/span>(<\/span>v<\/span>[<\/span>1]<\/span>, y<\/span>)]<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u56de\u8ee2 (rot) \u306e\u8a08\u7b97\u4f8b<\/h4>\n
$$\\nabla\\times \\left(\\frac{\\boldsymbol{r}}{r}\\right)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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'<\/span>rot<\/span>(<\/span>'<\/span>rvec<\/span>\/'<\/span>r<\/span>)<\/span> =<\/span> rot<\/span>(<\/span>rvec<\/span>\/<\/span>r<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[11]:<\/div>\n
\\[\\tag{${\\it \\%o}_{14}$}{\\it rot}\\left(\\frac{{\\it rvec}}{r}\\right)=\\left[ 0 , 0 , 0 \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30b9\u30ab\u30e9\u30fc\u5834\u3084\u30d9\u30af\u30c8\u30eb\u5834\u306e2\u968e\u5fae\u5206\u3068\u6052\u7b49\u5f0f<\/h3>\n
\u52fe\u914d (grad) \u306e\u767a\u6563 (div)<\/h4>\n
\u30b9\u30ab\u30e9\u30fc\u5834 $\\varphi$ \u306e\u52fe\u914d $\\nabla \\varphi$ \u306e\u767a\u6563 $\\nabla\\cdot(\\nabla \\varphi) $<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[12]:<\/div>\n
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div<\/span>(<\/span>grad<\/span>(<\/span>phi<\/span>(<\/span>x<\/span>,y<\/span>,z<\/span>)))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[12]:<\/div>\n
\\[\\tag{${\\it \\%o}_{15}$}\\frac{d^2}{d\\,z^2}\\,\\varphi\\left(x , y , z\\right)+\\frac{d^2}{d\\,y^2}\\,\\varphi\\left(x , y , z\\right)+\\frac{d^2}{d\\,x^2}\\,\\varphi\\left(x , y , z\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\u306e\u8a08\u7b97\u4f8b<\/h5>\n
$$\\nabla^2 \\left(\\frac{1}{r}\\right) = 0 \\quad (r \\neq 0)$$<\/p>\n
\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[13]:<\/div>\n
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Laplacian<\/span>(<\/span>phi<\/span>)<\/span>:=<\/span> div<\/span>(<\/span>grad<\/span>(<\/span>phi<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
\\[\\tag{${\\it \\%o}_{16}$}{\\it Laplacian}\\left(\\varphi\\right):={\\it div}\\left({\\it grad}\\left(\\varphi\\right)\\right)\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[14]:<\/div>\n
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'<\/span>Laplacian<\/span>(<\/span>1\/<\/span>r<\/span>)<\/span> =<\/span> Laplacian<\/span>(<\/span>1\/<\/span>r<\/span>)<\/span>, ratsimp<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[14]:<\/div>\n
\\[\\tag{${\\it \\%o}_{17}$}{\\it Laplacian}\\left(\\frac{1}{\\sqrt{z^2+y^2+x^2}}\\right)=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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\u52fe\u914d (grad) \u306e\u56de\u8ee2 (rot)<\/h4>\n
\u30b9\u30ab\u30e9\u30fc\u5834 $\\varphi$ \u306e\u52fe\u914d $\\nabla \\varphi$ \u306e\u56de\u8ee2 $\\nabla\\times(\\nabla \\varphi) $\u3002<\/p>\n
\u6052\u7b49\u7684\u306b
\n$$\\nabla\\times(\\nabla \\varphi) = \\boldsymbol{0}$$\u3067\u3042\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[15]:<\/div>\n
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\/* phi \u304c x, y, z \u306e\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u3092\u5ba3\u8a00\u3059\u308b\u3002*\/<\/span>\r\ndepends<\/span>(<\/span>phi<\/span>, [<\/span>x<\/span>, y<\/span>, z<\/span>])<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[15]:<\/div>\n
\\[\\tag{${\\it \\%o}_{18}$}\\left[ \\varphi\\left(x , y , z\\right) \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[16]:<\/div>\n
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'<\/span>grad<\/span>(<\/span>phi<\/span>)<\/span> =<\/span> grad<\/span>(<\/span>phi<\/span>)<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[16]:<\/div>\n
\\[\\tag{${\\it \\%o}_{19}$}{\\it grad}\\left(\\varphi\\right)=\\left[ \\frac{d}{d\\,x}\\,\\varphi , \\frac{d}{d\\,y}\\,\\varphi , \\frac{d}{d\\,z}\\,\\varphi \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[17]:<\/div>\n
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'<\/span>rot<\/span>(<\/span>'<\/span>grad<\/span>(<\/span>phi<\/span>))<\/span> =<\/span> rot<\/span>(<\/span>grad<\/span>(<\/span>phi<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[17]:<\/div>\n
\\[\\tag{${\\it \\%o}_{20}$}{\\it rot}\\left({\\it grad}\\left(\\varphi\\right)\\right)=\\left[ 0 , 0 , 0 \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u56de\u8ee2 (rot) \u306e\u767a\u6563 (div)<\/h4>\n
\u30d9\u30af\u30c8\u30eb\u5834 $\\boldsymbol{a}$ \u306e\u56de\u8ee2 $\\nabla \\times\\boldsymbol{a} $ \u306e\u767a\u6563 $\\nabla\\cdot(\\nabla \\times\\boldsymbol{a}) $<\/p>\n
\u6052\u7b49\u7684\u306b
\n\\begin{eqnarray}
\n\\nabla\\cdot(\\nabla \\times\\boldsymbol{a}) &=&
\n0 \\end{eqnarray}
\n\u3067\u3042\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[18]:<\/div>\n
\n
\n
\n
\/* \u30d9\u30af\u30c8\u30eb a \u306e\u5b9a\u7fa9\u3002*\/<\/span>\r\na<\/span>:<\/span> [<\/span>a1<\/span>, a2<\/span>, a3<\/span>]<\/span>$\r\n\/* a \u306e\u5404\u6210\u5206\u306f x, y, z \u306b\u4f9d\u5b58\u3059\u308b\u3002*\/<\/span>\r\ndepends<\/span>([<\/span>a1<\/span>, a2<\/span>, a3<\/span>]<\/span>, [<\/span>x<\/span>, y<\/span>, z<\/span>])<\/span>$\r\n\r\n'<\/span>rot<\/span>(<\/span>a<\/span>)<\/span> =<\/span> rot<\/span>(<\/span>a<\/span>)<\/span>;\r\n'<\/span>div<\/span>(<\/span>'<\/span>rot<\/span>(<\/span>a<\/span>))<\/span> =<\/span> div<\/span>(<\/span>rot<\/span>(<\/span>a<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[18]:<\/div>\n
\\[\\tag{${\\it \\%o}_{23}$}{\\it rot}\\left(\\left[ a_{1} , a_{2} , a_{3} \\right] \\right)=\\left[ \\frac{d}{d\\,y}\\,a_{3}-\\frac{d}{d\\,z}\\,a_{2} , \\frac{d}{d\\,z}\\,a_{1}-\\frac{d}{d\\,x}\\,a_{3} , \\frac{d}{d\\,x}\\,a_{2}-\\frac{d}{d\\,y}\\,a_{1} \\right] \\]<\/div>\n<\/div>\n
\n
Out[18]:<\/div>\n
\\[\\tag{${\\it \\%o}_{24}$}{\\it div}\\left({\\it rot}\\left(\\left[ a_{1} , a_{2} , a_{3} \\right] \\right)\\right)=0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u56de\u8ee2 (rot) \u306e\u56de\u8ee2 (rot)<\/h4>\n
\u30d9\u30af\u30c8\u30eb\u5834 $\\boldsymbol{a}$ \u306e\u56de\u8ee2 $\\nabla \\times\\boldsymbol{a} $ \u306e\u56de\u8ee2 $\\nabla\\times(\\nabla \\times\\boldsymbol{a}) $<\/p>\n
$$ \\nabla\\times(\\nabla \\times\\boldsymbol{a}) = \\nabla\\left( \\nabla\\cdot\\boldsymbol{a} \\right) – \\nabla^2 \\boldsymbol{a} $$<\/p>\n
\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[19]:<\/div>\n
\n
\n
\n
rot<\/span>(<\/span>rot<\/span>(<\/span>a<\/span>))<\/span> -<\/span> \r\n  (<\/span>grad<\/span>(<\/span>div<\/span>(<\/span>a<\/span>))<\/span> -<\/span> Laplacian<\/span>(<\/span>a<\/span>))<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[19]:<\/div>\n
\\[\\tag{${\\it \\%o}_{25}$}\\left[ 0 , 0 , 0 \\right] \\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u8ffd\u8a18\uff1a\u5916\u7a4d\u306e\u767a\u6563<\/h3>\n
\u4e00\u822c\u306b\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{E}, \\ \\boldsymbol{B}$ \u306e\u5916\u7a4d $\\boldsymbol{E}\\times\\boldsymbol{B}$ \u306e\u767a\u6563\u306f\uff0c\u305d\u308c\u305e\u308c\u306e\u30d9\u30af\u30c8\u30eb\u306e\u56de\u8ee2\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u3002<\/p>\n
$$\\nabla\\cdot\\left( \\boldsymbol{E}\\times\\boldsymbol{B}\\right)
\n= \\left(\\nabla\\times\\boldsymbol{E} \\right)\\cdot\\boldsymbol{B}
\n– \\boldsymbol{E}\\cdot\\left(\\nabla\\times\\boldsymbol{B} \\right)$$<\/p>\n
\u3053\u306e\u5f0f\u306f\uff0c\u96fb\u78c1\u5834\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6\u30fb\u30a8\u30cd\u30eb\u30ae\u30fc\u6d41\u675f\u306e\u3068\u3053\u308d\u3067\u51fa\u3066\u304f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[20]:<\/div>\n
\n
\n
\n
\/* \u5916\u7a4d\u306e\u5b9a\u7fa9 *\/<\/span>\r\ncross<\/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>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[21]:<\/div>\n
\n
\n
\n
E<\/span> :<\/span> [<\/span>Ex<\/span>, Ey<\/span>, Ez<\/span>]<\/span>$\r\nB<\/span> :<\/span> [<\/span>Bx<\/span>, By<\/span>, Bz<\/span>]<\/span>$\r\ndepends<\/span>([<\/span>Ex<\/span>, Ey<\/span>, Ez<\/span>, Bx<\/span>, By<\/span>, Bz<\/span>]<\/span>, [<\/span>x<\/span>, y<\/span>, z<\/span>])<\/span>$\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[22]:<\/div>\n
\n
\n
\n
\/* \u53f3\u8fba\u304b\u3089\u5de6\u8fba\u3092\u5f15\u3044\u3066\u30bc\u30ed\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059 *\/<\/span>\r\n\r\n(<\/span>rot<\/span>(<\/span>E<\/span>)<\/span>.<\/span>B<\/span> -<\/span> E<\/span>.<\/span>rot<\/span>(<\/span>B<\/span>))<\/span> -<\/span> div<\/span>(<\/span>cross<\/span>(<\/span>E<\/span>, B<\/span>))<\/span>, expand<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[22]:<\/div>\n
\\[\\tag{${\\it \\%o}_{30}$}0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u8ffd\u8a18\uff1a\u30b9\u30ab\u30e9\u30fc\u3068\u30d9\u30af\u30c8\u30eb\u306e\u7a4d\u306e\u767a\u6563<\/h3>\n
$$\\nabla\\cdot\\left( \\phi\\,\\boldsymbol{D}\\right) = \\left(\\nabla\\phi\\right)\\cdot\\boldsymbol{D} +\\left(\\nabla\\cdot\\boldsymbol{D}\\right)\\,\\phi$$<\/p>\n
\u3053\u306e\u5f0f\u3082\uff0c\u96fb\u78c1\u5834\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5bc6\u5ea6\u30fb\u30a8\u30cd\u30eb\u30ae\u30fc\u6d41\u675f\u306e\u3068\u3053\u308d\u3067\u51fa\u3066\u304f\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[23]:<\/div>\n
\n
\n
\n
\/* \u53f3\u8fba\u304b\u3089\u5de6\u8fba\u3092\u5f15\u3044\u3066\u30bc\u30ed\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3059 *\/<\/span>\r\n\r\ndepends<\/span>(<\/span>phi<\/span>, [<\/span>x<\/span>,y<\/span>,z<\/span>])<\/span>$\r\nD<\/span>:<\/span> [<\/span>Dx<\/span>, Dy<\/span>, Dy<\/span>]<\/span>$\r\ndepends<\/span>([<\/span>Dx<\/span>, Dy<\/span>, Dy<\/span>]<\/span>, [<\/span>x<\/span>, y<\/span>, z<\/span>])<\/span>$\r\n\r\ngrad<\/span>(<\/span>phi<\/span>)<\/span>.<\/span>D<\/span> +<\/span> div<\/span>(<\/span>D<\/span>)<\/span> *<\/span> phi<\/span> -<\/span> div<\/span>(<\/span>phi<\/span>*<\/span>D<\/span>)<\/span>, expand<\/span>;\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[23]:<\/div>\n
\\[\\tag{${\\it \\%o}_{34}$}0\\]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2605,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-2789","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\/2789","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=2789"}],"version-history":[{"count":6,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2789\/revisions"}],"predecessor-version":[{"id":3772,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2789\/revisions\/3772"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2605"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=2789"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}