{"id":9689,"date":"2024-11-21T10:34:31","date_gmt":"2024-11-21T01:34:31","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=9689"},"modified":"2024-11-21T10:38:46","modified_gmt":"2024-11-21T01:38:46","slug":"5-sympy-%e3%81%a7%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%83%bb%e8%a1%8c%e5%88%97%e3%83%bb%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e8%a7%a3%e6%9e%90","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e3%82%b3%e3%83%b3%e3%83%94%e3%83%a5%e3%83%bc%e3%82%bf%e6%bc%94%e7%bf%92\/sympy-%e6%bc%94%e7%bf%92\/5-sympy-%e3%81%a7%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e3%83%bb%e8%a1%8c%e5%88%97%e3%83%bb%e3%83%99%e3%82%af%e3%83%88%e3%83%ab%e8%a7%a3%e6%9e%90\/","title":{"rendered":"5. SymPy \u3067\u30d9\u30af\u30c8\u30eb\u30fb\u884c\u5217\u30fb\u30d9\u30af\u30c8\u30eb\u89e3\u6790"},"content":{"rendered":"

SymPy \u3067 Matrix()<\/code> \u3092\u4f7f\u3063\u305f\u30d9\u30af\u30c8\u30eb<\/strong><\/span>\u30fb\u884c\u5217<\/strong><\/span>\u306e\u5b9a\u7fa9\u3068\u6f14\u7b97\uff0c\u304a\u3088\u3073\u52fe\u914d (grad)<\/strong><\/span>\uff0c\u767a\u6563 (div)<\/strong><\/span>\uff0c\u56de\u8ee2 (rot)<\/strong><\/span> \u3092\u5b9a\u7fa9\u3057\u3066\u30d9\u30af\u30c8\u30eb\u89e3\u6790\u3002
\n<\/p>\n

\n
<\/div>\n
\n
\n

SymPy \u306e import<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[1]:<\/div>\n
\n
\n
\n
from<\/span> sympy.abc<\/span> import<\/span> *<\/span>\r\nfrom<\/span> sympy<\/span> import<\/span> *<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
Matrix()<\/code> \u3092\u4f7f\u3063\u305f\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u7fa9<\/h3>\n
Matrix()<\/code> \u3092\u4f7f\u3063\u305f3\u6b21\u5143\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a} = (a_1, a_2, a_3)$ \u306e\u5b9a\u7fa9\u4f8b\u3002\u6210\u5206\u306f\u5b9f\u6570\u3068\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[2]:<\/div>\n
\n
\n
\n
# \u6210\u5206\u306b\u4f7f\u3046\u5909\u6570\u306e\u5ba3\u8a00<\/span>\r\nvar<\/span>(<\/span>'a1:4'<\/span>,<\/span> real<\/span>=<\/span>True<\/span>)<\/span> # var('a1 a2 a3') \u3068\u540c\u3058<\/span>\r\n\r\na<\/span> =<\/span> Matrix<\/span>([<\/span>a1<\/span>,<\/span> a2<\/span>,<\/span> a3<\/span>])<\/span>\r\na<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[2]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{1}\\\\a_{2}\\\\a_{3}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\boldsymbol{b} = (b_1, b_2, b_3), \\ \\boldsymbol{c} = (c_1, c_2, c_3)$ \u3082\u540c\u69d8\u306b\u5b9a\u7fa9\u3002\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u66f8\u304d\u65b9\u3082\u3067\u304d\u308b\u3088\u3046\u3060\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[3]:<\/div>\n
\n
\n
\n
# Matrix() \u306e\u4e2d\u306b\u5909\u6570\u306e\u5ba3\u8a00\u3092\u5165\u308c\u308b\u65b9\u6cd5<\/span>\r\nb<\/span> =<\/span> Matrix<\/span>(<\/span>var<\/span>(<\/span>'b1:4'<\/span>,<\/span> real<\/span>=<\/span>True<\/span>))<\/span>\r\nc<\/span> =<\/span> Matrix<\/span>(<\/span>var<\/span>(<\/span>'c1:4'<\/span>,<\/span> real<\/span>=<\/span>True<\/span>))<\/span>\r\n\r\ndisplay<\/span>(<\/span>b<\/span>)<\/span>\r\ndisplay<\/span>(<\/span>c<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle \\left[\\begin{matrix}b_{1}\\\\b_{2}\\\\b_{3}\\end{matrix}\\right]$<\/div>\n<\/div>\n
\n
<\/div>\n
$\\displaystyle \\left[\\begin{matrix}c_{1}\\\\c_{2}\\\\c_{3}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u306e\u6f14\u7b97<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u305f\u3057\u7b97\uff0c\u5f15\u304d\u7b97 +, -<\/code><\/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
\n
\n
In\u00a0[4]:<\/div>\n
\n
\n
\n
a<\/span> +<\/span> b<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[4]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{1} + b_{1}\\\\a_{2} + b_{2}\\\\a_{3} + b_{3}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[5]:<\/div>\n
\n
\n
\n
a<\/span> -<\/span> b<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[5]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{1} -b_{1}\\\\a_{2} -b_{2}\\\\a_{3} -b_{3}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u6570\u500d\uff08\u30b9\u30ab\u30e9\u30fc\u500d\uff09 *<\/code><\/h4>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u6570\u500d\uff08\u30b9\u30ab\u30e9\u30fc\u500d\uff09\u306f\uff0c\u5404\u6210\u5206\u306b\u305d\u308c\u305e\u308c\u5b9a\u6570\uff08\u30b9\u30ab\u30e9\u30fc\uff09\u3092\u304b\u3051\u305f\u3082\u306e\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[6]:<\/div>\n
\n
\n
\n
var<\/span>(<\/span>'C'<\/span>)<\/span>\r\n\r\nC<\/span>*<\/span>a<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[6]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}C a_{1}\\\\C a_{2}\\\\C a_{3}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d .dot()<\/code><\/h4>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u306f $\\boldsymbol{a}\\cdot\\boldsymbol{b} = $a.dot(b)<\/code> \u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[7]:<\/div>\n
\n
\n
\n
a<\/span>.<\/span>dot<\/span>(<\/span>b<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[7]:<\/div>\n
$\\displaystyle a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055 .norm()<\/code><\/h5>\n
\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}$ \u306e\u5927\u304d\u3055 $|\\boldsymbol{a}|$ \u306f
\n$|\\boldsymbol{a}| \\equiv \\sqrt{\\boldsymbol{a}\\cdot\\boldsymbol{a}} =$ a.norm()<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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
\n
\n
In\u00a0[8]:<\/div>\n
\n
\n
\n
a<\/span>.<\/span>norm<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[8]:<\/div>\n
$\\displaystyle \\sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d .cross()<\/code><\/h4>\n
\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u306f $\\boldsymbol{a}\\times\\boldsymbol{b} = $a.cross(b)<\/code> \u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[9]:<\/div>\n
\n
\n
\n
a<\/span>.<\/span>cross<\/span>(<\/span>b<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[9]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{2} b_{3} -a_{3} b_{2}\\\\-a_{1} b_{3} + a_{3} b_{1}\\\\a_{1} b_{2} -a_{2} b_{1}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u5916\u7a4d\u306f\u300c\u53cd\u4ea4\u63db\u300d<\/h5>\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
\n
\n
In\u00a0[10]:<\/div>\n
\n
\n
\n
a<\/span>.<\/span>cross<\/span>(<\/span>b<\/span>)<\/span> +<\/span> b<\/span>.<\/span>cross<\/span>(<\/span>a<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[10]:<\/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
\u3042\u308b\u3044\u306f\u6bd4\u8f03\u6f14\u7b97\u5b50 ==<\/code> \u3092\u4f7f\u3063\u3066 a.cross(b) == -b.cross(a)<\/code> \u304c True<\/code> \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3082\u3088\u3044\u3067\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[11]:<\/div>\n
\n
\n
\n
a<\/span>.<\/span>cross<\/span>(<\/span>b<\/span>)<\/span> ==<\/span> -<\/span>b<\/span>.<\/span>cross<\/span>(<\/span>a<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[11]:<\/div>\n
\n
True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u306e\u4e09\u91cd\u7a4d<\/h4>\n
\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d<\/h5>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}$ \u306b\u3088\u308b\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d<\/strong>\u306f<\/p>\n
$$\\boldsymbol{a} \\cdot (\\boldsymbol{b}\\times\\boldsymbol{c})$$<\/p>\n
\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
\n
<\/div>\n
\n
\n
\u25cb\u7df4\u7fd2\uff1a\u30b9\u30ab\u30e9\u30fc\u4e09\u91cd\u7a4d\u306e\u516c\u5f0f<\/h5>\n
$\\boldsymbol{a} \\cdot (\\boldsymbol{b}\\times\\boldsymbol{c})
\n= \\boldsymbol{b} \\cdot (\\boldsymbol{c}\\times\\boldsymbol{a})$ \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b\u3002<\/p>\n
\u30d2\u30f3\u30c8\uff1a<\/p>\n
a.dot(b.cross(c)) - b.dot(c.cross(a))<\/code> \u3092 expand()<\/code> \u3067\u304b\u3063\u3053\u3092\u306f\u305a\u3057\u3066\u5c55\u958b\u3057\u3066\u30bc\u30ed\u306b\u306a\u308b\u3053\u3068\u3092\u793a\u305b\u3070\u3088\u3044\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u540c\u69d8\u306b\u3057\u3066\uff0c$\\boldsymbol{b} \\cdot (\\boldsymbol{c}\\times\\boldsymbol{a})
\n= \\boldsymbol{c} \\cdot (\\boldsymbol{a}\\times\\boldsymbol{b}) $ \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3082\u793a\u305b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d<\/h5>\n
3\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\boldsymbol{b}, \\boldsymbol{c}$ \u306b\u3088\u308b\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d<\/strong>\u306f<\/p>\n
$$\\boldsymbol{a} \\times (\\boldsymbol{b}\\times\\boldsymbol{c})$$<\/p>\n
\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<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u25cb\u7df4\u7fd2\uff1a\u30d9\u30af\u30c8\u30eb\u4e09\u91cd\u7a4d\u306e\u516c\u5f0f<\/h5>\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} $ \u3092\u793a\u305b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u25cb\u7df4\u7fd2\uff1a\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u30fb\u5916\u7a4d<\/h5>\n
$\\displaystyle \\boldsymbol{a} = \\left(-\\frac{1}{\\sqrt{2}}, \\frac{1}{\\sqrt{2}}, \\sqrt{3} \\right), \\ \\boldsymbol{b} = \\left(-1, 1, -\\sqrt{2} \\right)$ \u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e\u91cf\u3092\u8a08\u7b97\u3057\u306a\u3055\u3044\u3002<\/p>\n
$$1. \\ |\\boldsymbol{a}|, \\quad 2. \\ |\\boldsymbol{b}|, \\quad 3. \\ \\boldsymbol{a}\\cdot\\boldsymbol{b}, \\quad 4. \\ \\boldsymbol{a}\\times\\boldsymbol{b}$$<\/p>\n
\u307e\u305f\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\boldsymbol{b}$ \u306e\u306a\u3059\u89d2\u3092 $\\theta$ \u3068\u3057\u305f\u3068\u304d\u306e $\\cos\\theta$ \u304a\u3088\u3073 $\\sin\\theta$ \u306e\u5024\u3092\u6c42\u3081\u306a\u3055\u3044\u3002<\/p>\n
\u307e\u305a\uff0c\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\boldsymbol{b}$ \u3092 Matrix()<\/code> \u3067\u5b9a\u7fa9\u3057…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[12]:<\/div>\n
\n
\n
\n
a<\/span> =<\/span> Matrix<\/span>([<\/span>-<\/span>1<\/span>\/<\/span>sqrt<\/span>(<\/span>2<\/span>),<\/span> 1<\/span>\/<\/span>sqrt<\/span>(<\/span>2<\/span>),<\/span> sqrt<\/span>(<\/span>3<\/span>)])<\/span>\r\nb<\/span> =<\/span> Matrix<\/span>([<\/span>-<\/span>1<\/span>,<\/span> 1<\/span>,<\/span> -<\/span>sqrt<\/span>(<\/span>2<\/span>)])<\/span>\r\n\r\ndisplay<\/span>(<\/span>a<\/span>)<\/span>\r\ndisplay<\/span>(<\/span>b<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle \\left[\\begin{matrix}-\\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\\\\\sqrt{3}\\end{matrix}\\right]$<\/div>\n<\/div>\n
\n
<\/div>\n
$\\displaystyle \\left[\\begin{matrix}-1\\\\1\\\\-\\sqrt{2}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$1. \\ |\\boldsymbol{a}| = $ a.norm()<\/code> \u3067\u3042\u308b\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$2. \\ |\\boldsymbol{b}| = $ b.norm()<\/code> \u3067\u3042\u308b\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$3. \\ \\boldsymbol{a}\\cdot\\boldsymbol{b} = $ a.dot(b)<\/code> \u3067\u3042\u308b\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$4. \\ \\boldsymbol{a}\\times\\boldsymbol{b} = $ a.cross(b)<\/code> \u3067\u3042\u308b\u304b\u3089…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\boldsymbol{a}\\cdot\\boldsymbol{b} = |\\boldsymbol{a}| |\\boldsymbol{b}| \\cos\\theta$ \u3088\u308a\uff0c$\\displaystyle \\cos\\theta = \\frac{\\boldsymbol{a}\\cdot\\boldsymbol{b}}{|\\boldsymbol{a}| |\\boldsymbol{b}|} = \\cdots$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[13]:<\/div>\n
\n
\n
\n
uhen1<\/span> =<\/span> a<\/span>.<\/span>dot<\/span>(<\/span>b<\/span>)<\/span>\/<\/span>(<\/span>a<\/span>.<\/span>norm<\/span>()<\/span>*<\/span>b<\/span>.<\/span>norm<\/span>())<\/span>\r\nEq<\/span>(<\/span>cos<\/span>(<\/span>theta<\/span>),<\/span> uhen1<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[13]:<\/div>\n
$\\displaystyle \\cos{\\left(\\theta \\right)} = -\\frac{\\sqrt{6}}{4} + \\frac{\\sqrt{2}}{4}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3061\u306a\u307f\u306b $\\theta = $ acos(uhen1)<\/code> \u3067\u3042\u308b\u304b\u3089\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\boldsymbol{b}$ \u306e\u306a\u3059\u89d2 $\\theta$ \u3092\u6c42\u3081\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[14]:<\/div>\n
\n
\n
\n
th1<\/span> =<\/span> acos<\/span>(<\/span>uhen1<\/span>)<\/span>\r\nth1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[14]:<\/div>\n
$\\displaystyle \\frac{7 \\pi}{12}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$|\\boldsymbol{a}\\times\\boldsymbol{b}| = |\\boldsymbol{a}| |\\boldsymbol{b}| \\sin\\theta$ \u3088\u308a $\\displaystyle \\sin\\theta = \\frac{|\\boldsymbol{a}\\times\\boldsymbol{b}|}{|\\boldsymbol{a}| |\\boldsymbol{b}|} = \\cdots$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[15]:<\/div>\n
\n
\n
\n
uhen2<\/span> =<\/span> (<\/span>a<\/span>.<\/span>cross<\/span>(<\/span>b<\/span>))<\/span>.<\/span>norm<\/span>()<\/span>\/<\/span>(<\/span>a<\/span>.<\/span>norm<\/span>()<\/span>*<\/span>b<\/span>.<\/span>norm<\/span>())<\/span>\r\nEq<\/span>(<\/span>sin<\/span>(<\/span>theta<\/span>),<\/span> uhen2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[15]:<\/div>\n
$\\displaystyle \\sin{\\left(\\theta \\right)} = \\frac{\\sqrt{2} \\cdot \\left(1 + \\sqrt{3}\\right)}{4}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\theta = $ asin(uhen2)<\/code> \u3067\u3042\u308b\u304b\u3089\uff0c2\u3064\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{a}, \\boldsymbol{b}$ \u306e\u306a\u3059\u89d2 $\\theta$ \u3092\u6c42\u3081\u308b\u3068…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[16]:<\/div>\n
\n
\n
\n
asin<\/span>(<\/span>uhen2<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[16]:<\/div>\n
$\\displaystyle \\frac{5 \\pi}{12}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
acos(uhen1)<\/code> \u304b\u3089\u6c42\u3081\u305f\u89d2\u5ea6 $\\theta$ \u3068 asin(uhen2)<\/code> \u304b\u3089\u6c42\u3081\u305f\u89d2\u5ea6 $\\theta$ \u306f\u7570\u306a\u3063\u3066\u3044\u308b\u304c\uff0c\u3053\u308c\u306f\u3069\u306e\u3088\u3046\u306b\u8003\u3048\u305f\u3089\u3088\u3044\u304b\u3002<\/p>\n
\u30d2\u30f3\u30c8\uff1a $\\arccos x$ \u3068 $\\arcsin x$ \u306e\u5b9a\u7fa9\u57df\u306f\u3069\u3061\u3089\u3082 $-1 \\leq x \\leq 1$ \u3060\u304c\uff0c\u5024\u57df\u306f\uff1f<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
Matrix()<\/code> \u3092\u4f7f\u3063\u305f\u884c\u5217\u306e\u5b9a\u7fa9<\/h3>\n
Matrix()<\/code> \u3092\u4f7f\u3063\u305f\u884c\u5217<\/p>\n
$$ A = \\begin{pmatrix}
\na_{11} & a_{12} \\\\
\na_{21} & a_{22}
\n\\end{pmatrix}$$<\/p>\n
\u306e\u5b9a\u7fa9\u4f8b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[17]:<\/div>\n
\n
\n
\n
var<\/span>(<\/span>'a11:13, a21:23'<\/span>)<\/span>\r\nA<\/span> =<\/span> Matrix<\/span>([[<\/span>a11<\/span>,<\/span> a12<\/span>],<\/span> \r\n            [<\/span>a21<\/span>,<\/span> a22<\/span>]])<\/span>\r\nA<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[17]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{11} & a_{12}\\\\a_{21} & a_{22}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u66f8\u304d\u65b9\u3082\u3067\u304d\u308b\u3088\u3046\u3060\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[18]:<\/div>\n
\n
\n
\n
# Matrix() \u306e\u4e2d\u306b\u5909\u6570\u306e\u5ba3\u8a00\u3092\u5165\u308c\u308b\u65b9\u6cd5<\/span>\r\nB<\/span> =<\/span> Matrix<\/span>([<\/span>var<\/span>(<\/span>'b11:13'<\/span>),<\/span> \r\n            var<\/span>(<\/span>'b21:23'<\/span>)])<\/span>\r\nB<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[18]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}b_{11} & b_{12}\\\\b_{21} & b_{22}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u884c\u5217\u306e\u6f14\u7b97<\/h3>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u305f\u3057\u7b97\uff0c\u5f15\u304d\u7b97 +, -<\/code><\/h4>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[19]:<\/div>\n
\n
\n
\n
A<\/span> +<\/span> B<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[19]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{11} + b_{11} & a_{12} + b_{12}\\\\a_{21} + b_{21} & a_{22} + b_{22}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[20]:<\/div>\n
\n
\n
\n
A<\/span> -<\/span> B<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[20]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{11} -b_{11} & a_{12} -b_{12}\\\\a_{21} -b_{21} & a_{22} -b_{22}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u884c\u5217\u306e\u7a4d *<\/code><\/h4>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[21]:<\/div>\n
\n
\n
\n
A<\/span> *<\/span> B<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[21]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{11} b_{11} + a_{12} b_{21} & a_{11} b_{12} + a_{12} b_{22}\\\\a_{21} b_{11} + a_{22} b_{21} & a_{21} b_{12} + a_{22} b_{22}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[22]:<\/div>\n
\n
\n
\n
B<\/span> *<\/span> A<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[22]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{11} b_{11} + a_{21} b_{12} & a_{12} b_{11} + a_{22} b_{12}\\\\a_{11} b_{21} + a_{21} b_{22} & a_{12} b_{21} + a_{22} b_{22}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u884c\u5217\u306e\u7a4d\u306b\u95a2\u3057\u3066\u306f\u4ea4\u63db\u5247\u306f\u6210\u308a\u7acb\u3061\u307e\u305b\u3093\u3002$A B \\neq B A$
\n\u6bd4\u8f03\u6f14\u7b97\u5b50 !=<\/code> \u3092\u4f7f\u3063\u3066\u300c\u5de6\u8fba\u300d\u3068\u300c\u53f3\u8fba\u300d\u7b49\u3057\u304f\u306a\u3044\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[23]:<\/div>\n
\n
\n
\n
A<\/span> *<\/span> B<\/span> !=<\/span> B<\/span> *<\/span> A<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[23]:<\/div>\n
\n
True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u8ee2\u7f6e\u884c\u5217 .transpose()<\/code><\/h4>\n
\u884c\u5217 $A$ \u306e\u8ee2\u7f6e\u884c\u5217\u306f $A^{T} = $ transpose(A)<\/code> \u307e\u305f\u306f A.transpose()<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[24]:<\/div>\n
\n
\n
\n
A<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[24]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{11} & a_{12}\\\\a_{21} & a_{22}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[25]:<\/div>\n
\n
\n
\n
A<\/span>.<\/span>transpose<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[25]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{11} & a_{21}\\\\a_{12} & a_{22}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u9006\u884c\u5217 .inv()<\/code><\/h4>\n
\u884c\u5217 $A$ \u306e\u9006\u884c\u5217 $\\displaystyle A^{-1}$ \u306f A**(-1)<\/code> \u307e\u305f\u306f A.inv()<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[26]:<\/div>\n
\n
\n
\n
A<\/span>.<\/span>inv<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[26]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}\\frac{a_{22}}{a_{11} a_{22} -a_{12} a_{21}} & -\\frac{a_{12}}{a_{11} a_{22} -a_{12} a_{21}}\\\\-\\frac{a_{21}}{a_{11} a_{22} -a_{12} a_{21}} & \\frac{a_{11}}{a_{11} a_{22} -a_{12} a_{21}}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$A^{-1} A$ \u3084 $A A^{-1}$ \u304c\u5358\u4f4d\u884c\u5217\u306b\u306a\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[27]:<\/div>\n
\n
\n
\n
A<\/span>**<\/span>(<\/span>-<\/span>1<\/span>)<\/span> *<\/span> A<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[27]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}\\frac{a_{11} a_{22}}{a_{11} a_{22} -a_{12} a_{21}} -\\frac{a_{12} a_{21}}{a_{11} a_{22} -a_{12} a_{21}} & 0\\\\0 & \\frac{a_{11} a_{22}}{a_{11} a_{22} -a_{12} a_{21}} -\\frac{a_{12} a_{21}}{a_{11} a_{22} -a_{12} a_{21}}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[28]:<\/div>\n
\n
\n
\n
# \u4e0a\u8a18\u306e\u7d50\u679c\u3092\u7c21\u5358\u5316<\/span>\r\nsimplify<\/span>(<\/span>_<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[28]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[29]:<\/div>\n
\n
\n
\n
# together \u3067\u4e00\u7dd2\u306b\uff08\u901a\u5206\uff09<\/span>\r\ntogether<\/span>(<\/span>A<\/span> *<\/span> (<\/span>A<\/span>.<\/span>inv<\/span>()))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[29]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & 1\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u884c\u5217\u5f0f .det()<\/code><\/h4>\n
$A$ \u306e\u884c\u5217\u5f0f $\\det A$ \u306f A.det()<\/code> \u307e\u305f\u306f det(A)<\/code><\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[30]:<\/div>\n
\n
\n
\n
A<\/span>.<\/span>det<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[30]:<\/div>\n
$\\displaystyle a_{11} a_{22} – a_{12} a_{21}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[31]:<\/div>\n
\n
\n
\n
det<\/span>(<\/span>A<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[31]:<\/div>\n
$\\displaystyle a_{11} a_{22} – a_{12} a_{21}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\det A = \\det A^T$ \u304a\u3088\u3073 $\\displaystyle \\det A^{-1} = \\frac{1}{\\det A}$ \u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[32]:<\/div>\n
\n
\n
\n
det<\/span>(<\/span>A<\/span>)<\/span> ==<\/span> det<\/span>(<\/span>A<\/span>.<\/span>transpose<\/span>())<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[32]:<\/div>\n
\n
True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[33]:<\/div>\n
\n
\n
\n
det<\/span>(<\/span>A<\/span>.<\/span>inv<\/span>())<\/span> ==<\/span> 1<\/span>\/<\/span>det<\/span>(<\/span>A<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[33]:<\/div>\n
\n
True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u25cb\u7df4\u7fd2\uff1a\u884c\u5217\u306e\u7a4d\u306e\u884c\u5217\u5f0f<\/h4>\n
$\\displaystyle \\det\\, (A B) = (\\det A) \\cdot (\\det B)$ \u3092\u78ba\u8a8d\u3059\u308b\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[34]:<\/div>\n
\n
\n
\n
display<\/span>(<\/span>A<\/span>)<\/span>\r\ndisplay<\/span>(<\/span>B<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{11} & a_{12}\\\\a_{21} & a_{22}\\end{matrix}\\right]$<\/div>\n<\/div>\n
\n
<\/div>\n
$\\displaystyle \\left[\\begin{matrix}b_{11} & b_{12}\\\\b_{21} & b_{22}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u25cb\u7df4\u7fd2\uff1a3\u6b21\u5143\u306e\u56de\u8ee2\u884c\u5217<\/h4>\n
3\u6b21\u5143\u7a7a\u9593\u3067\u306e $z$ \u8ef8\u5468\u308a\u306e\u89d2\u5ea6 $\\phi$ \u306e\u56de\u8ee2\u3092\u8868\u3059\u56de\u8ee2\u884c\u5217\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u307e\u3059\u3002<\/p>\n
$$R(\\phi) =
\n\\begin{pmatrix}
\n\\cos \\phi & -\\sin \\phi & 0\\\\
\n\\sin \\phi & \\cos \\phi & 0\\\\
\n0 & 0 & 1
\n\\end{pmatrix}
\n$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[35]:<\/div>\n
\n
\n
\n
def<\/span> R<\/span>(<\/span>phi<\/span>):<\/span>\r\n    return<\/span> Matrix<\/span>([[<\/span>cos<\/span>(<\/span>phi<\/span>),<\/span> -<\/span>sin<\/span>(<\/span>phi<\/span>),<\/span> 0<\/span>],<\/span>\r\n                   [<\/span>sin<\/span>(<\/span>phi<\/span>),<\/span> cos<\/span>(<\/span>phi<\/span>),<\/span>  0<\/span>],<\/span> \r\n                   [<\/span>0<\/span>,<\/span>        0<\/span>,<\/span>         1<\/span>]])<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\displaystyle 1.\\ R(\\phi)$ \u304c\u76f4\u4ea4\u884c\u5217\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b\u3002<\/p>\n
\u76f4\u4ea4\u884c\u5217<\/strong>\u3068\u306f\uff0c\u8ee2\u7f6e\u884c\u5217\u304c\u9006\u884c\u5217\u306b\u306a\u3063\u3066\u3044\u308b\u884c\u5217\u306e\u3053\u3068\u3067\u3059\u304b\u3089\uff0c\u4ee5\u4e0b\u3092\u793a\u305b\u3070\u3088\u3044\u3067\u3059\u3002<\/p>\n
$$R(\\phi) \\,R^T(\\phi) = I$$<\/p>\n
\u3053\u3053\u3067\uff0c$I$ \u306f\u5358\u4f4d\u884c\u5217\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3061\u306a\u307f\u306b\uff0c$R(\\phi)$ \u306e\u9006\u884c\u5217\u306f $R(-\\phi)$ \u3067\u3042\u308b\u3053\u3068\u3082\u78ba\u8a8d\u3067\u304d\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r} = (x, y, z)$ \u3092\u3053\u306e\u56de\u8ee2\u884c\u5217\u3067\u76f4\u4ea4\u5909\u63db\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[36]:<\/div>\n
\n
\n
\n
# \u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u7fa9<\/span>\r\n\r\nr<\/span> =<\/span> Matrix<\/span>(<\/span>var<\/span>(<\/span>'x y z'<\/span>,<\/span> real<\/span> =<\/span> True<\/span>))<\/span>\r\nr<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[36]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}x\\\\y\\\\z\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u76f4\u4ea4\u5909\u63db\u3055\u308c\u305f\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u306f\uff0c\u5143\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}$ \u306e\u5927\u304d\u3055\u3068\u540c\u3058\u3067\u3042\u308b\u3053\u3068\u3092\u78ba\u304b\u3081\u307e\u3059\u3002<\/p>\n
\u307e\u305a\uff0c\u5143\u306e\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}$ \u306e\u5927\u304d\u3055\uff08\u306e2\u4e57\uff09\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[37]:<\/div>\n
\n
\n
\n
r<\/span>.<\/span>dot<\/span>(<\/span>r<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[37]:<\/div>\n
$\\displaystyle x^{2} + y^{2} + z^{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u76f4\u4ea4\u5909\u63db\u3055\u308c\u305f\u30d9\u30af\u30c8\u30eb\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[38]:<\/div>\n
\n
\n
\n
r1<\/span> =<\/span> R<\/span>(<\/span>phi<\/span>)<\/span> *<\/span> r<\/span>\r\nr1<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[38]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}x \\cos{\\left(\\phi \\right)} – y \\sin{\\left(\\phi \\right)}\\\\x \\sin{\\left(\\phi \\right)} + y \\cos{\\left(\\phi \\right)}\\\\z\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u3053\u306e\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\uff08\u306e2\u4e57\uff09\u306f…<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[39]:<\/div>\n
\n
\n
\n
(<\/span>r1<\/span>.<\/span>dot<\/span>(<\/span>r1<\/span>))<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[39]:<\/div>\n
$\\displaystyle x^{2} + y^{2} + z^{2}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb $\\boldsymbol{r}$ \u306e\u5927\u304d\u3055 $|\\boldsymbol{r}|$ \u306f r.norm()<\/code> \u3067\u3059\u304c\uff0c\u306a\u305c\u3053\u308c\u3092\u4f7f\u308f\u305a\u306b\u5927\u304d\u3055\u306e2\u4e57 $\\boldsymbol{r}\\cdot\\boldsymbol{r}$ \u3092\u4f7f\u3063\u305f\u304b\uff0c\u308f\u304b\u308a\u307e\u3059\u304b\uff1f<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[40]:<\/div>\n
\n
\n
\n
r<\/span>.<\/span>norm<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[40]:<\/div>\n
$\\displaystyle \\sqrt{x^{2} + y^{2} + z^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[41]:<\/div>\n
\n
\n
\n
r1<\/span>.<\/span>norm<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[41]:<\/div>\n
$\\displaystyle \\sqrt{z^{2} + \\left|{x \\sin{\\left(\\phi \\right)} + y \\cos{\\left(\\phi \\right)}}\\right|^{2} + \\left|{x \\cos{\\left(\\phi \\right)} – y \\sin{\\left(\\phi \\right)}}\\right|^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\displaystyle 2. \\ R(\\beta) \\, R(\\alpha) = R(\\alpha+\\beta)$ \u3092\u793a\u305b\u3002<\/p>\n
\u3053\u308c\u306f\uff0c\u307e\u305a $z$ \u8ef8\u306e\u307e\u308f\u308a\u306b\u89d2\u5ea6 $\\alpha$ \u3060\u3051\u56de\u8ee2\u3055\u305b\uff0c\u5f15\u304d\u7d9a\u304d\u89d2\u5ea6 $\\beta$ \u3060\u3051\u56de\u8ee2\u3055\u305b\u308b\u5909\u63db\u306f\uff0c$z$ \u8ef8\u306e\u307e\u308f\u308a\u306b\u4e00\u5ea6\u306b\u89d2\u5ea6 $\\alpha+\\beta$ \u56de\u8ee2\u3055\u305b\u308b\u5909\u63db\u3068\u540c\u3058\u3060\u3068\u3044\u3046\u3053\u3068\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[42]:<\/div>\n
\n
\n
\n
# \u5909\u6570\u306e\u5ba3\u8a00<\/span>\r\nvar<\/span>(<\/span>'alpha beta'<\/span>)<\/span>\r\n\r\nR<\/span>(<\/span>beta<\/span>)<\/span> *<\/span> R<\/span>(<\/span>alpha<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[42]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}- \\sin{\\left(\\alpha \\right)} \\sin{\\left(\\beta \\right)} + \\cos{\\left(\\alpha \\right)} \\cos{\\left(\\beta \\right)} & – \\sin{\\left(\\alpha \\right)} \\cos{\\left(\\beta \\right)} – \\sin{\\left(\\beta \\right)} \\cos{\\left(\\alpha \\right)} & 0\\\\\\sin{\\left(\\alpha \\right)} \\cos{\\left(\\beta \\right)} + \\sin{\\left(\\beta \\right)} \\cos{\\left(\\alpha \\right)} & – \\sin{\\left(\\alpha \\right)} \\sin{\\left(\\beta \\right)} + \\cos{\\left(\\alpha \\right)} \\cos{\\left(\\beta \\right)} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u89e3\u6790<\/h3>\n
SymPy \u306b\u306f\u30d9\u30af\u30c8\u30eb\u89e3\u6790\u306e\u305f\u3081\u306e\u30e2\u30b8\u30e5\u30fc\u30eb sympy.vector<\/code> \u304c\u3042\u308b\u3088\u3046\u3067\u3059\u304c\uff0c\u7c21\u5358\u306a\u306e\u3067\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u305d\u306e\u90fd\u5ea6\u5b9a\u7fa9\u3057\u3066\u3084\u3063\u3066\u307f\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30b9\u30ab\u30e9\u30fc\u5834\u306e\u52fe\u914d\uff08grad()<\/code> \u306e\u5b9a\u7fa9\uff09<\/h4>\n
\u30b9\u30ab\u30e9\u30fc\u5834 $\\varphi(\\boldsymbol{r})$ \u306e\u52fe\u914d\u3092\u4e0e\u3048\u308b\u95a2\u6570 grad()<\/code> \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
\n
\n
In\u00a0[43]:<\/div>\n
\n
\n
\n
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
\n
<\/div>\n
\n
\n
\u30b9\u30ab\u30e9\u30fc\u5834 $\\varphi(x,y,z)$ \u306e\u52fe\u914d $\\nabla\\varphi$ \u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[44]:<\/div>\n
\n
\n
\n
# varphi \u3092 x, y, z \u306e\u95a2\u6570\u3068\u3057\u3066\u5ba3\u8a00<\/span>\r\nvarphi<\/span> =<\/span> Function<\/span>(<\/span>'varphi'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\ngrad<\/span>(<\/span>varphi<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[44]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}\\frac{\\partial}{\\partial x} \\varphi{\\left(x,y,z \\right)}\\\\\\frac{\\partial}{\\partial y} \\varphi{\\left(x,y,z \\right)}\\\\\\frac{\\partial}{\\partial z} \\varphi{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u52fe\u914d\u306e\u8a08\u7b97\u4f8b<\/h5>\n
\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb $$\\boldsymbol{r} = (x, y, z)$$ \u306e\u5927\u304d\u3055
\n$$r = |\\boldsymbol{r}| = \\sqrt{x^2 + y^2 + z^2}$$ \u306b\u3064\u3044\u3066\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
\n
\n
In\u00a0[45]:<\/div>\n
\n
\n
\n
# \u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u306e\u5b9a\u7fa9<\/span>\r\n\r\nrvec<\/span> =<\/span> Matrix<\/span>(<\/span>var<\/span>(<\/span>'x y z'<\/span>,<\/span> real<\/span> =<\/span> True<\/span>))<\/span>\r\ndisplay<\/span>(<\/span>rvec<\/span>)<\/span>\r\n\r\nr<\/span> =<\/span> rvec<\/span>.<\/span>norm<\/span>()<\/span>\r\ndisplay<\/span>(<\/span>r<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle \\left[\\begin{matrix}x\\\\y\\\\z\\end{matrix}\\right]$<\/div>\n<\/div>\n
\n
<\/div>\n
$\\displaystyle \\sqrt{x^{2} + y^{2} + z^{2}}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\nabla r$ \u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[46]:<\/div>\n
\n
\n
\n
grad<\/span>(<\/span>r<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[46]:<\/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
\n
<\/div>\n
\n
\n
\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<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[47]:<\/div>\n
\n
\n
\n
grad<\/span>(<\/span>r<\/span>)<\/span> ==<\/span> rvec<\/span>\/<\/span>r<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[47]:<\/div>\n
\n
True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\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
\n
\n
In\u00a0[48]:<\/div>\n
\n
\n
\n
grad<\/span>(<\/span>1<\/span>\/<\/span>r<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[48]:<\/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
\n
\n
In\u00a0[49]:<\/div>\n
\n
\n
\n
grad<\/span>(<\/span>1<\/span>\/<\/span>r<\/span>)<\/span> ==<\/span> -<\/span>rvec<\/span>\/<\/span>r<\/span>**<\/span>3<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[49]:<\/div>\n
\n
True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u5834\u306e\u767a\u6563 (div()<\/code> \u306e\u5b9a\u7fa9)<\/h4>\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
\n
\n
In\u00a0[50]:<\/div>\n
\n
\n
\n
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
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u5834 $\\boldsymbol{B}$ \u306e\u767a\u6563<\/p>\n
$$\\nabla\\cdot\\boldsymbol{B} = \\mbox{div}\\, \\boldsymbol{B} =
\n\\frac{\\partial B_x}{\\partial x} + \\frac{\\partial B_y}{\\partial y} +\\frac{\\partial B_x}{\\partial x}$$<\/p>\n
\u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[51]:<\/div>\n
\n
\n
\n
# \u30d9\u30af\u30c8\u30eb\u5834 B \u306e\u5b9a\u7fa9<\/span>\r\nB_x<\/span> =<\/span> Function<\/span>(<\/span>'B_x'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nB_y<\/span> =<\/span> Function<\/span>(<\/span>'B_y'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nB_z<\/span> =<\/span> Function<\/span>(<\/span>'B_z'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\n\r\nBvec<\/span> =<\/span> Matrix<\/span>([<\/span>B_x<\/span>,<\/span> B_y<\/span>,<\/span> B_z<\/span>])<\/span>\r\nBvec<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[51]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}B_{x}{\\left(x,y,z \\right)}\\\\B_{y}{\\left(x,y,z \\right)}\\\\B_{z}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[52]:<\/div>\n
\n
\n
\n
# B \u306e\u767a\u6563<\/span>\r\ndiv<\/span>(<\/span>Bvec<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[52]:<\/div>\n
$\\displaystyle \\frac{\\partial}{\\partial x} B_{x}{\\left(x,y,z \\right)} + \\frac{\\partial}{\\partial y} B_{y}{\\left(x,y,z \\right)} + \\frac{\\partial}{\\partial z} B_{z}{\\left(x,y,z \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u767a\u6563 \u306e\u8a08\u7b97\u4f8b<\/h5>\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
\n
\n
In\u00a0[53]:<\/div>\n
\n
\n
\n
# \u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u304c\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u308b\u3053\u3068\u3092\u78ba\u8a8d<\/span>\r\nrvec<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[53]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}x\\\\y\\\\z\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[54]:<\/div>\n
\n
\n
\n
div<\/span>(<\/span>rvec<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[54]:<\/div>\n
$\\displaystyle 3$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u5834\u306e\u56de\u8ee2 (rot()<\/code> \u306e\u5b9a\u7fa9)<\/h4>\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
\n
\n
In\u00a0[55]:<\/div>\n
\n
\n
\n
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
\n
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u5834 $\\boldsymbol{A}$ \u306e\u56de\u8ee2 $\\nabla\\times\\boldsymbol{A} = \\mbox{rot}\\, \\boldsymbol{A} $ \u3092\u78ba\u8a8d\u3057\u307e\u3059\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[56]:<\/div>\n
\n
\n
\n
# \u30d9\u30af\u30c8\u30eb\u5834 A \u306e\u5b9a\u7fa9<\/span>\r\nA_x<\/span> =<\/span> Function<\/span>(<\/span>'A_x'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nA_y<\/span> =<\/span> Function<\/span>(<\/span>'A_y'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nA_z<\/span> =<\/span> Function<\/span>(<\/span>'A_z'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\n\r\nAvec<\/span> =<\/span> Matrix<\/span>([<\/span>A_x<\/span>,<\/span> A_y<\/span>,<\/span> A_z<\/span>])<\/span>\r\nAvec<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[56]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}A_{x}{\\left(x,y,z \\right)}\\\\A_{y}{\\left(x,y,z \\right)}\\\\A_{z}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[57]:<\/div>\n
\n
\n
\n
rot<\/span>(<\/span>Avec<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[57]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}- \\frac{\\partial}{\\partial z} A_{y}{\\left(x,y,z \\right)} + \\frac{\\partial}{\\partial y} A_{z}{\\left(x,y,z \\right)}\\\\\\frac{\\partial}{\\partial z} A_{x}{\\left(x,y,z \\right)} – \\frac{\\partial}{\\partial x} A_{z}{\\left(x,y,z \\right)}\\\\- \\frac{\\partial}{\\partial y} A_{x}{\\left(x,y,z \\right)} + \\frac{\\partial}{\\partial x} A_{y}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u56de\u8ee2\u306e\u8a08\u7b97\u4f8b<\/h5>\n
$$\\nabla\\times \\left(\\frac{\\boldsymbol{r}}{r}\\right)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[58]:<\/div>\n
\n
\n
\n
rvec<\/span>\/<\/span>r<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[58]:<\/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
\n
\n
In\u00a0[59]:<\/div>\n
\n
\n
\n
rot<\/span>(<\/span>rvec<\/span>\/<\/span>r<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[59]:<\/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
\u25cb\u7df4\u7fd2\uff1a\u767a\u6563\u3084\u56de\u8ee2\u306e\u8a08\u7b97<\/h4>\n
$\\displaystyle 1. \\ \\nabla\\cdot\\left(\\frac{\\boldsymbol{r}}{r}\\right)$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\displaystyle 2. \\ \\nabla\\cdot\\left(\\frac{\\boldsymbol{r}}{r^n}\\right)$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\displaystyle 3. \\ \\nabla\\times\\left(\\frac{\\boldsymbol{r}}{r}\\right)$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\displaystyle 4. \\nabla\\times\\left(\\frac{\\boldsymbol{r}}{r^n}\\right)$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u52fe\u914d (grad) \u306e\u767a\u6563 (div)\uff1a\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306e\u5b9a\u7fa9<\/h4>\n
\u30b9\u30ab\u30e9\u30fc\u5834 $\\varphi$ \u306e\u52fe\u914d $\\nabla \\varphi$ \u306e\u767a\u6563 $$\\nabla\\cdot(\\nabla \\varphi) = \\nabla^2 \\varphi$$
\n\u3053\u3053\u3067<\/p>\n
$$\\nabla^2 \\equiv \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2} +\\frac{\\partial^2}{\\partial z^2}$$<\/p>\n
\u306f\u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\uff08\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\uff09\u3002<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[60]:<\/div>\n
\n
\n
\n
varphi<\/span> =<\/span> Function<\/span>(<\/span>'varphi'<\/span>)(<\/span>x<\/span>,<\/span> y<\/span>,<\/span> z<\/span>)<\/span>\r\nvarphi<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[60]:<\/div>\n
$\\displaystyle \\varphi{\\left(x,y,z \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[61]:<\/div>\n
\n
\n
\n
div<\/span>(<\/span>grad<\/span>(<\/span>varphi<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[61]:<\/div>\n
$\\displaystyle \\frac{\\partial^{2}}{\\partial x^{2}} \\varphi{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial y^{2}} \\varphi{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial z^{2}} \\varphi{\\left(x,y,z \\right)}$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u30b9\u30ab\u30e9\u30fc\u306b\u4f5c\u7528\u3059\u308b\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306e\u5b9a\u7fa9<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[62]:<\/div>\n
\n
\n
\n
# \u30b9\u30ab\u30e9\u30fc\u95a2\u6570\u306b\u4f5c\u7528\u3059\u308b\u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50\u306e\u5b9a\u7fa9<\/span>\r\ndef<\/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
\n
<\/div>\n
\n
\n
\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
\n
\n
In\u00a0[63]:<\/div>\n
\n
\n
\n
laplacian<\/span>(<\/span>1<\/span>\/<\/span>r<\/span>)<\/span>.<\/span>simplify<\/span>()<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[63]:<\/div>\n
$\\displaystyle 0$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u52fe\u914d (grad) \u306e\u56de\u8ee2 (rot)\uff1a\u6052\u7b49\u5f0f<\/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
\n
\n
In\u00a0[64]:<\/div>\n
\n
\n
\n
grad<\/span>(<\/span>varphi<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[64]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}\\frac{\\partial}{\\partial x} \\varphi{\\left(x,y,z \\right)}\\\\\\frac{\\partial}{\\partial y} \\varphi{\\left(x,y,z \\right)}\\\\\\frac{\\partial}{\\partial z} \\varphi{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[65]:<\/div>\n
\n
\n
\n
rot<\/span>(<\/span>grad<\/span>(<\/span>varphi<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[65]:<\/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)\uff1a\u6052\u7b49\u5f0f<\/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
\u307e\u305a\u306f\u30d9\u30af\u30c8\u30eb\u5834 $\\boldsymbol{a}$ \u306e\u5b9a\u7fa9\uff1a<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[66]:<\/div>\n
\n
\n
\n
# \u30d9\u30af\u30c8\u30eb\u5834 a \u306e\u5b9a\u7fa9<\/span>\r\na_x<\/span> =<\/span> Function<\/span>(<\/span>'a_x'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\na_y<\/span> =<\/span> Function<\/span>(<\/span>'a_y'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\na_z<\/span> =<\/span> Function<\/span>(<\/span>'a_z'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\n\r\navec<\/span> =<\/span> Matrix<\/span>([<\/span>a_x<\/span>,<\/span> a_y<\/span>,<\/span> a_z<\/span>])<\/span>\r\navec<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[66]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}a_{x}{\\left(x,y,z \\right)}\\\\a_{y}{\\left(x,y,z \\right)}\\\\a_{z}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\nabla\\times \\boldsymbol{a}$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[67]:<\/div>\n
\n
\n
\n
rot<\/span>(<\/span>avec<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[67]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}- \\frac{\\partial}{\\partial z} a_{y}{\\left(x,y,z \\right)} + \\frac{\\partial}{\\partial y} a_{z}{\\left(x,y,z \\right)}\\\\\\frac{\\partial}{\\partial z} a_{x}{\\left(x,y,z \\right)} – \\frac{\\partial}{\\partial x} a_{z}{\\left(x,y,z \\right)}\\\\- \\frac{\\partial}{\\partial y} a_{x}{\\left(x,y,z \\right)} + \\frac{\\partial}{\\partial x} a_{y}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
$\\nabla\\cdot\\left(\\nabla\\times \\boldsymbol{a}\\right)$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[68]:<\/div>\n
\n
\n
\n
div<\/span>(<\/span>rot<\/span>(<\/span>avec<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[68]:<\/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
<\/div>\n
\n
\n
\u30d9\u30af\u30c8\u30eb\u306b\u4f5c\u7528\u3059\u308b\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306e\u5b9a\u7fa9<\/h5>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[69]:<\/div>\n
\n
\n
\n
# \u30d9\u30af\u30c8\u30eb\u306b\u4f5c\u7528\u3059\u308b Laplacian \u3092\u5b9a\u7fa9<\/span>\r\ndef<\/span> Laplacian<\/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[70]:<\/div>\n
\n
\n
\n
rot<\/span>(<\/span>rot<\/span>(<\/span>avec<\/span>))<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[70]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}- \\frac{\\partial^{2}}{\\partial y^{2}} a_{x}{\\left(x,y,z \\right)} – \\frac{\\partial^{2}}{\\partial z^{2}} a_{x}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial y\\partial x} a_{y}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial z\\partial x} a_{z}{\\left(x,y,z \\right)}\\\\- \\frac{\\partial^{2}}{\\partial x^{2}} a_{y}{\\left(x,y,z \\right)} – \\frac{\\partial^{2}}{\\partial z^{2}} a_{y}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial y\\partial x} a_{x}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial z\\partial y} a_{z}{\\left(x,y,z \\right)}\\\\- \\frac{\\partial^{2}}{\\partial x^{2}} a_{z}{\\left(x,y,z \\right)} – \\frac{\\partial^{2}}{\\partial y^{2}} a_{z}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial z\\partial x} a_{x}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial z\\partial y} a_{y}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[71]:<\/div>\n
\n
\n
\n
grad<\/span>(<\/span>div<\/span>(<\/span>avec<\/span>))<\/span> -<\/span> Laplacian<\/span>(<\/span>avec<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[71]:<\/div>\n
$\\displaystyle \\left[\\begin{matrix}- \\frac{\\partial^{2}}{\\partial y^{2}} a_{x}{\\left(x,y,z \\right)} – \\frac{\\partial^{2}}{\\partial z^{2}} a_{x}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial y\\partial x} a_{y}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial z\\partial x} a_{z}{\\left(x,y,z \\right)}\\\\- \\frac{\\partial^{2}}{\\partial x^{2}} a_{y}{\\left(x,y,z \\right)} – \\frac{\\partial^{2}}{\\partial z^{2}} a_{y}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial y\\partial x} a_{x}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial z\\partial y} a_{z}{\\left(x,y,z \\right)}\\\\- \\frac{\\partial^{2}}{\\partial x^{2}} a_{z}{\\left(x,y,z \\right)} – \\frac{\\partial^{2}}{\\partial y^{2}} a_{z}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial z\\partial x} a_{x}{\\left(x,y,z \\right)} + \\frac{\\partial^{2}}{\\partial z\\partial y} a_{y}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[72]:<\/div>\n
\n
\n
\n
rot<\/span>(<\/span>rot<\/span>(<\/span>avec<\/span>))<\/span> ==<\/span> grad<\/span>(<\/span>div<\/span>(<\/span>avec<\/span>))<\/span> -<\/span> Laplacian<\/span>(<\/span>avec<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
Out[72]:<\/div>\n
\n
True<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u25cb\u7df4\u7fd2\uff1a\u30d9\u30af\u30c8\u30eb\u306e\u5916\u7a4d\u306e\u767a\u6563<\/h4>\n
\u4ee5\u4e0b\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b\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<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[73]:<\/div>\n
\n
\n
\n
E_x<\/span> =<\/span> Function<\/span>(<\/span>'E_x'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nE_y<\/span> =<\/span> Function<\/span>(<\/span>'E_y'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nE_z<\/span> =<\/span> Function<\/span>(<\/span>'E_z'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nB_x<\/span> =<\/span> Function<\/span>(<\/span>'B_x'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nB_y<\/span> =<\/span> Function<\/span>(<\/span>'B_y'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nB_z<\/span> =<\/span> Function<\/span>(<\/span>'B_z'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\n\r\nE<\/span> =<\/span> Matrix<\/span>([<\/span>E_x<\/span>,<\/span> E_y<\/span>,<\/span> E_z<\/span>])<\/span>\r\nB<\/span> =<\/span> Matrix<\/span>([<\/span>B_x<\/span>,<\/span> B_y<\/span>,<\/span> B_z<\/span>])<\/span>\r\n\r\ndisplay<\/span>(<\/span>E<\/span>)<\/span>\r\ndisplay<\/span>(<\/span>B<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle \\left[\\begin{matrix}E_{x}{\\left(x,y,z \\right)}\\\\E_{y}{\\left(x,y,z \\right)}\\\\E_{z}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n
\n
<\/div>\n
$\\displaystyle \\left[\\begin{matrix}B_{x}{\\left(x,y,z \\right)}\\\\B_{y}{\\left(x,y,z \\right)}\\\\B_{z}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
<\/div>\n
\n
\n
\u25cb\u7df4\u7fd2\uff1a\u30b9\u30ab\u30e9\u30fc\u3068\u30d9\u30af\u30c8\u30eb\u306e\u7a4d\u306e\u767a\u6563<\/h4>\n
\u4ee5\u4e0b\u306e\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b\u3002<\/p>\n
$$\\nabla\\cdot\\left( \\varphi\\,\\boldsymbol{D}\\right) =
\n\\left(\\nabla\\varphi\\right)\\cdot\\boldsymbol{D} +\\varphi\\left(\\nabla\\cdot\\boldsymbol{D}\\right)$$<\/p>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[74]:<\/div>\n
\n
\n
\n
varphi<\/span> =<\/span> Function<\/span>(<\/span>'varphi'<\/span>)(<\/span>x<\/span>,<\/span> y<\/span>,<\/span> z<\/span>)<\/span>\r\ndisplay<\/span>(<\/span>varphi<\/span>)<\/span>\r\n\r\nD_x<\/span> =<\/span> Function<\/span>(<\/span>'D_x'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nD_y<\/span> =<\/span> Function<\/span>(<\/span>'D_y'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\nD_z<\/span> =<\/span> Function<\/span>(<\/span>'D_z'<\/span>)(<\/span>x<\/span>,<\/span>y<\/span>,<\/span>z<\/span>)<\/span>\r\n\r\nD<\/span> =<\/span> Matrix<\/span>([<\/span>D_x<\/span>,<\/span> D_y<\/span>,<\/span> D_z<\/span>])<\/span>\r\ndisplay<\/span>(<\/span>D<\/span>)<\/span>\r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
\n
<\/div>\n
$\\displaystyle \\varphi{\\left(x,y,z \\right)}$<\/div>\n<\/div>\n
\n
<\/div>\n
$\\displaystyle \\left[\\begin{matrix}D_{x}{\\left(x,y,z \\right)}\\\\D_{y}{\\left(x,y,z \\right)}\\\\D_{z}{\\left(x,y,z \\right)}\\end{matrix}\\right]$<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n
\n
\n
In\u00a0[\u00a0]:<\/div>\n
\n
\n
\n
 \r\n<\/pre>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"
SymPy \u3067 Matrix() \u3092\u4f7f\u3063\u305f\u30d9\u30af\u30c8\u30eb\u30fb\u884c\u5217\u306e\u5b9a\u7fa9\u3068\u6f14\u7b97\uff0c\u304a\u3088\u3073\u52fe\u914d (grad)\uff0c\u767a\u6563 (div)\uff0c\u56de\u8ee2 (rot) \u3092\u5b9a\u7fa9\u3057\u3066\u30d9\u30af\u30c8\u30eb\u89e3\u6790\u3002<\/p>
\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":33,"featured_media":0,"parent":9502,"menu_order":50,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-9689","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\/9689","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=9689"}],"version-history":[{"count":3,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/9689\/revisions"}],"predecessor-version":[{"id":9693,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/9689\/revisions\/9693"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/9502"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=9689"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}