{"id":7634,"date":"2024-02-20T13:03:42","date_gmt":"2024-02-20T04:03:42","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=7634"},"modified":"2024-02-20T13:03:42","modified_gmt":"2024-02-20T04:03:42","slug":"%e5%8f%82%e8%80%83%ef%bc%9asympy-%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%9asympy-%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\uff1aSymPy \u3067\u30b9\u30ab\u30e9\u30fc\u5834\u30fb\u30d9\u30af\u30c8\u30eb\u5834\u306e\u5fae\u5206"},"content":{"rendered":"

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\u5fc5\u8981\u306a\u30e2\u30b8\u30e5\u30fc\u30eb\u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[1]:<\/div>\n
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from<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n\r\ninit_printing<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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<\/div>\n
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\u504f\u5fae\u5206<\/h3>\n
SymPy \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);<\/code> \u3084 diff(f, y);<\/code> \u306a\u3069\u3068\u66f8\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[2]:<\/div>\n
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# \u95a2\u6570 f(x, y) \u306e\u5b9a\u7fa9<\/span>\r\nf<\/span> =<\/span> Function<\/span>(<\/span>'f'<\/span>)(<\/span>x<\/span>,<\/span> y<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[3]:<\/div>\n
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diff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[3]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial x} f{\\left(x,y \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[4]:<\/div>\n
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diff<\/span>(<\/span>f<\/span>,<\/span> y<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[4]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial y} f{\\left(x,y \\right)}$<\/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
$$\\frac{\\partial^2}{\\partial y \\partial x} f(x, y) – \\frac{\\partial^2}{\\partial x \\partial y} f(x, y) = 0$$<\/p>\n
\u3092\u793a\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[5]:<\/div>\n
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Eq<\/span>(<\/span>Derivative<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>,<\/span> x<\/span>),<\/span> y<\/span>)<\/span> -<\/span> Derivative<\/span>(<\/span>Derivative<\/span>(<\/span>f<\/span>,<\/span> y<\/span>),<\/span> x<\/span>),<\/span>\r\n   diff<\/span>(<\/span>diff<\/span>(<\/span>f<\/span>,<\/span> x<\/span>),<\/span> y<\/span>)<\/span> -<\/span> diff<\/span>(<\/span>diff<\/span>(<\/span>f<\/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[5]:<\/div>\n
$\\displaystyle \\frac{\\partial^{2}}{\\partial y\\partial x} f{\\left(x,y \\right)} – \\frac{\\partial^{2}}{\\partial x\\partial y} f{\\left(x,y \\right)} = 0$<\/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[6]:<\/div>\n
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def<\/span> grad<\/span>(<\/span>phi<\/span>):<\/span>\r\n    return<\/span> Matrix<\/span>([<\/span>diff<\/span>(<\/span>phi<\/span>,<\/span>x<\/span>),<\/span> diff<\/span>(<\/span>phi<\/span>,<\/span>y<\/span>),<\/span> diff<\/span>(<\/span>phi<\/span>,<\/span>z<\/span>)])<\/span>\r\n<\/pre>\n<\/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[7]:<\/div>\n
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rvec<\/span> =<\/span> Matrix<\/span>([<\/span>x<\/span>,<\/span> y<\/span>,<\/span> z<\/span>])<\/span>\r\nr<\/span> =<\/span> sqrt<\/span>(<\/span>rvec<\/span>.<\/span>dot<\/span>(<\/span>rvec<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[8]:<\/div>\n
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grad<\/span>(<\/span>r<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[8]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}\\frac{x}{\\sqrt{x^{2} + y^{2} + z^{2}}}\\\\\\frac{y}{\\sqrt{x^{2} + y^{2} + z^{2}}}\\\\\\frac{z}{\\sqrt{x^{2} + y^{2} + z^{2}}}\\end{matrix}\\right]$<\/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[9]:<\/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[9]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$<\/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[10]:<\/div>\n
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grad<\/span>(<\/span>1<\/span>\/<\/span>r<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[10]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}- \\frac{x}{\\left(x^{2} + y^{2} + z^{2}\\right)^{\\frac{3}{2}}}\\\\- \\frac{y}{\\left(x^{2} + y^{2} + z^{2}\\right)^{\\frac{3}{2}}}\\\\- \\frac{z}{\\left(x^{2} + y^{2} + z^{2}\\right)^{\\frac{3}{2}}}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[11]:<\/div>\n
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_<\/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[11]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$<\/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[12]:<\/div>\n
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def<\/span> div<\/span>(<\/span>v<\/span>):<\/span>\r\n    return<\/span> diff<\/span>(<\/span>v<\/span>[<\/span>0<\/span>],<\/span> x<\/span>)<\/span> +<\/span> diff<\/span>(<\/span>v<\/span>[<\/span>1<\/span>],<\/span> y<\/span>)<\/span> +<\/span> diff<\/span>(<\/span>v<\/span>[<\/span>2<\/span>],<\/span> z<\/span>)<\/span>\r\n<\/pre>\n<\/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[13]:<\/div>\n
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div<\/span>(<\/span>rvec<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[13]:<\/div>\n
$\\displaystyle 3$<\/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[14]:<\/div>\n
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def<\/span> rot<\/span>(<\/span>v<\/span>):<\/span>\r\n    vx<\/span> =<\/span> v<\/span>[<\/span>0<\/span>]<\/span>\r\n    vy<\/span> =<\/span> v<\/span>[<\/span>1<\/span>]<\/span>\r\n    vz<\/span> =<\/span> v<\/span>[<\/span>2<\/span>]<\/span>\r\n    return<\/span> Matrix<\/span>(<\/span>\r\n            [<\/span>diff<\/span>(<\/span>vz<\/span>,<\/span> y<\/span>)<\/span> -<\/span> diff<\/span>(<\/span>vy<\/span>,<\/span> z<\/span>),<\/span> \r\n             diff<\/span>(<\/span>vx<\/span>,<\/span> z<\/span>)<\/span> -<\/span> diff<\/span>(<\/span>vz<\/span>,<\/span> x<\/span>),<\/span> \r\n             diff<\/span>(<\/span>vy<\/span>,<\/span> x<\/span>)<\/span> -<\/span> diff<\/span>(<\/span>vx<\/span>,<\/span> y<\/span>)]<\/span>\r\n            )<\/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[15]:<\/div>\n
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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[15]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$<\/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[16]:<\/div>\n
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phi<\/span> =<\/span> Function<\/span>(<\/span>'phi'<\/span>)(<\/span>x<\/span>,<\/span> y<\/span>,<\/span> z<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[17]:<\/div>\n
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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[17]:<\/div>\n
$\\displaystyle \\frac{\\partial^{2}}{\\partial x^{2}} \\phi{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial y^{2}} \\phi{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial z^{2}} \\phi{\\left(x,y,z \\right)}$<\/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[18]:<\/div>\n
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def<\/span> Laplacian<\/span>(<\/span>phi<\/span>):<\/span>\r\n    # \u30b9\u30ab\u30e9\u30fc\u95a2\u6570 phi \u306e\u307f\u306b\u9069\u7528<\/span>\r\n    return<\/span> div<\/span>(<\/span>grad<\/span>(<\/span>phi<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[19]:<\/div>\n
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Laplacian<\/span>(<\/span>1<\/span>\/<\/span>r<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[19]:<\/div>\n
$\\displaystyle \\frac{3 x^{2}}{\\left(x^{2} + y^{2} + z^{2}\\right)^{\\frac{5}{2}}} + \\frac{3 y^{2}}{\\left(x^{2} + y^{2} + z^{2}\\right)^{\\frac{5}{2}}} + \\frac{3 z^{2}}{\\left(x^{2} + y^{2} + z^{2}\\right)^{\\frac{5}{2}}} – \\frac{3}{\\left(x^{2} + y^{2} + z^{2}\\right)^{\\frac{3}{2}}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[20]:<\/div>\n
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simplify<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[20]:<\/div>\n
$\\displaystyle 0$<\/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[21]:<\/div>\n
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grad<\/span>(<\/span>phi<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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Out[21]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}\\frac{\\partial}{\\partial x} \\phi{\\left(x,y,z \\right)}\\\\\\frac{\\partial}{\\partial y} \\phi{\\left(x,y,z \\right)}\\\\\\frac{\\partial}{\\partial z} \\phi{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
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In\u00a0[22]:<\/div>\n
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rot<\/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[22]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$<\/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[23]:<\/div>\n
\n
\n
\n
a1<\/span> =<\/span> Function<\/span>(<\/span>'a1'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\na2<\/span> =<\/span> Function<\/span>(<\/span>'a2'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\na3<\/span> =<\/span> Function<\/span>(<\/span>'a3'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[24]:<\/div>\n
\n
\n
\n
a<\/span> =<\/span> [<\/span>a1<\/span>,<\/span> a2<\/span>,<\/span> a3<\/span>]<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[25]:<\/div>\n
\n
\n
\n
rot<\/span>(<\/span>a<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[25]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}- \\frac{\\partial}{\\partial z} a_{2}{\\left(x,y,z \\right)} + \\frac{\\partial}{\\partial y} a_{3}{\\left(x,y,z \\right)}\\\\\\frac{\\partial}{\\partial z} a_{1}{\\left(x,y,z \\right)} – \\frac{\\partial}{\\partial x} a_{3}{\\left(x,y,z \\right)}\\\\- \\frac{\\partial}{\\partial y} a_{1}{\\left(x,y,z \\right)} + \\frac{\\partial}{\\partial x} a_{2}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[26]:<\/div>\n
\n
\n
\n
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[26]:<\/div>\n
$\\displaystyle 0$<\/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[27]:<\/div>\n
\n
\n
\n
# \u30d9\u30af\u30c8\u30eb\u306b\u4f5c\u7528\u3059\u308b Laplacian \u3092\u5b9a\u7fa9<\/span>\r\ndef<\/span> Lapv<\/span>(<\/span>v<\/span>):<\/span>\r\n    return<\/span> Matrix<\/span>([<\/span>Laplacian<\/span>(<\/span>v<\/span>[<\/span>0<\/span>]),<\/span>\r\n                   Laplacian<\/span>(<\/span>v<\/span>[<\/span>1<\/span>]),<\/span>\r\n                   Laplacian<\/span>(<\/span>v<\/span>[<\/span>2<\/span>])])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[28]:<\/div>\n
\n
\n
\n
rot<\/span>(<\/span>rot<\/span>(<\/span>a<\/span>))<\/span> -<\/span> (<\/span>grad<\/span>(<\/span>div<\/span>(<\/span>a<\/span>))<\/span> -<\/span> Lapv<\/span>(<\/span>a<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[28]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$<\/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[29]:<\/div>\n
\n
\n
\n
E1<\/span> =<\/span> Function<\/span>(<\/span>'E1'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nE2<\/span> =<\/span> Function<\/span>(<\/span>'E2'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nE3<\/span> =<\/span> Function<\/span>(<\/span>'E3'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nB1<\/span> =<\/span> Function<\/span>(<\/span>'B1'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nB2<\/span> =<\/span> Function<\/span>(<\/span>'B2'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nB3<\/span> =<\/span> Function<\/span>(<\/span>'B3'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\n\r\nE<\/span> =<\/span> Matrix<\/span>([<\/span>E1<\/span>,<\/span> E2<\/span>,<\/span> E3<\/span>])<\/span>\r\nB<\/span> =<\/span> Matrix<\/span>([<\/span>B1<\/span>,<\/span> B2<\/span>,<\/span> B3<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[30]:<\/div>\n
\n
\n
\n
# \u5de6\u8fba\u304b\u3089\u53f3\u8fba\u3092\u5f15\u3044\u3066\u30bc\u30ed\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059<\/span>\r\n\r\ndiv<\/span>(<\/span>E<\/span>.<\/span>cross<\/span>(<\/span>B<\/span>))<\/span> -<\/span> (<\/span>rot<\/span>(<\/span>E<\/span>)<\/span>.<\/span>dot<\/span>(<\/span>B<\/span>)<\/span> -<\/span> E<\/span>.<\/span>dot<\/span>(<\/span>rot<\/span>(<\/span>B<\/span>)));<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[31]:<\/div>\n
\n
\n
\n
simplify<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[31]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$<\/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[32]:<\/div>\n
\n
\n
\n
D1<\/span> =<\/span> Function<\/span>(<\/span>'D1'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nD2<\/span> =<\/span> Function<\/span>(<\/span>'D2'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nD3<\/span> =<\/span> Function<\/span>(<\/span>'D3'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\n\r\nD<\/span> =<\/span> Matrix<\/span>([<\/span>D1<\/span>,<\/span> D2<\/span>,<\/span> D3<\/span>])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[33]:<\/div>\n
\n
\n
\n
# \u5de6\u8fba\u304b\u3089\u53f3\u8fba\u3092\u5f15\u3044\u3066\u30bc\u30ed\u3068\u306a\u308b\u3053\u3068\u3092\u793a\u3059<\/span>\r\n\r\ndiv<\/span>(<\/span>phi<\/span> *<\/span> D<\/span>)<\/span> -<\/span> (<\/span>grad<\/span>(<\/span>phi<\/span>)<\/span>.<\/span>dot<\/span>(<\/span>D<\/span>)<\/span> +<\/span> div<\/span>(<\/span>D<\/span>)<\/span> *<\/span> phi<\/span>);<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[34]:<\/div>\n
\n
\n
\n
simplify<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[34]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$<\/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":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-7634","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\/7634","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=7634"}],"version-history":[{"count":1,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7634\/revisions"}],"predecessor-version":[{"id":7635,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/7634\/revisions\/7635"}],"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=7634"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}