{"id":4885,"date":"2023-01-10T14:22:43","date_gmt":"2023-01-10T05:22:43","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=4885"},"modified":"2024-07-10T10:36:44","modified_gmt":"2024-07-10T01:36:44","slug":"%e5%8f%82%e8%80%83%ef%bc%9a%e6%a5%95%e5%86%86%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%82%922%e9%87%8d%e7%a9%8d%e5%88%86%e3%81%a7%e6%b1%82%e3%82%81%e3%82%8b","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6c\/%e5%a4%9a%e9%87%8d%e7%a9%8d%e5%88%86%ef%bc%9a%e5%a4%9a%e5%a4%89%e6%95%b0%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e5%88%86\/%e5%86%86%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%82%922%e9%87%8d%e7%a9%8d%e5%88%86%e3%81%a7%e6%b1%82%e3%82%81%e3%82%8b\/%e5%8f%82%e8%80%83%ef%bc%9a%e6%a5%95%e5%86%86%e3%81%ae%e9%9d%a2%e7%a9%8d%e3%82%922%e9%87%8d%e7%a9%8d%e5%88%86%e3%81%a7%e6%b1%82%e3%82%81%e3%82%8b\/","title":{"rendered":"\u53c2\u8003\uff1a\u6955\u5186\u306e\u9762\u7a4d\u30922\u91cd\u7a4d\u5206\u3067\u6c42\u3081\u308b"},"content":{"rendered":"

<\/p>\n

$$S = \\iint_D dx\\, dy, \\quad D: \\frac{x^2}{a^2} +\\frac{y^2}{b^2}\u00a0 \\leq 1$$<\/p>\n

\n

\\(\\displaystyle S = \\iint_D dx\\, dy\\) \u306f\u9818\u57df \\(D\\) \u306e\u9762\u7a4d\u3092\u8868\u3059\u306e\u3067\u3042\u3063\u305f\u3002 \\(\\displaystyle \\frac{x^2}{a^2} +\\frac{y^2}{b^2}\u00a0 \\leq 1\\) \u3064\u307e\u308a\uff0c\u9818\u57df \\(D\\) \u306f\u9577\u534a\u5f84\u30fb\u77ed\u534a\u5f84 \\(a, b\\) \u306e\u6955\u5186\u306e\u9762\u7a4d\u3067\u3042\u308b\u304b\u3089\uff0c\u516c\u5f0f\u3088\u308a \\(S = \\pi a b\\) \u3067\u3042\u308b\u3002\u3067\u306f\u5b9f\u969b\u306b \\(\\displaystyle \\iint_{D} dx\\, dy\\) \u3092\u3069\u3046\u3084\u3063\u3066\u8a08\u7b97\u3059\u308b\u304b\u3068\u3044\u3046\u8a71\u3002<\/p>\n

\u300c\u5186\u306e\u9762\u7a4d\u30922\u91cd\u7a4d\u5206\u3067\u6c42\u3081\u308b<\/a>\u300d\u3067\uff0c\u5186\u306e\u9762\u7a4d\u306f\u6c42\u3081\u3066\u3044\u308b\u306e\u3067\uff0c\u53c2\u8003\u307e\u3067\u306b\u6955\u5186\u306e\u9762\u7a4d\u306b\u3064\u3044\u3066\u3082\u307e\u3068\u3081\u3066\u304a\u304f\u3002\u6955\u5186\u306e\u9762\u7a4d\u306f\u5929\u6587\u5b66\u30fb\u5b87\u5b99\u7269\u7406\u5b66\u306e\u30b1\u30d7\u30e9\u30fc\u306e\u7b2c2\u6cd5\u5247\uff08\u9762\u7a4d\u901f\u5ea6\u4e00\u5b9a\u5247\uff09\u306e\u969b\u306b\u4f7f\u3046\u306e\u3067\uff0c\u5ff5\u306e\u305f\u3081\u306b Maxima \u3067\u6c42\u3081\u3066\u3044\u308b\uff08\u300cMaxima-Jupyter \u3067\u6955\u5186\u306e\u9762\u7a4d\u3092\u6c42\u3081\u308b<\/a>\u300d\uff09\u304c\uff0cMaxima \u306b\u983c\u3089\u305a\uff0c\u4eba\u529b\u3067\u3082\u7a4d\u5206\u3057\u305f\u304f\u306a\u308b\u3067\u3057\u3087\uff1f<\/p>\n

\u4ee5\u4e0b\u3067\u793a\u3059\u3088\u3046\u306b\uff0c\u7d20\u76f4\u306b\u6955\u5186\u306e\u4e2d\u5fc3\u3092\u539f\u70b9\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u3067\u7d2f\u6b21\u7a4d\u5206\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u9593\u9055\u3063\u3066\uff0c\u6955\u5186\u306e\u7126\u70b9\u3092\u539f\u70b9\u3068\u3057\u305f\u6975\u5ea7\u6a19\u3092\u4f7f\u3063\u30662\u91cd\u7a4d\u5206\u3057\u3088\u3046\u3068\u3059\u308b\u3082\u306e\u306a\u3089\uff0c\u75db\u3044\u76ee\u306b\u3042\u3044\u307e\u3059\u3002<\/p>\n

\u7d2f\u6b21\u7a4d\u5206\u3092\u4f7f\u3046<\/h3>\n

\"\"\u9818\u57df \\(D\\) \u306e\u6761\u4ef6\u5f0f\u304b\u3089\uff0c
$$D: \\frac{x^2}{a^2} +\\frac{y^2}{b^2}\u00a0 \\leq 1 \\quad \\Rightarrow \\quad y^2 \\leq b^2 \\left(1 – \\frac{x^2}{a^2} \\right)$$<\/p>\n

$$\\therefore -b\\sqrt{1 – \\frac{x^2}{a^2}} \\leq y \\leq\u00a0 b \\sqrt{1 – \\frac{x^2}{a^2}} $$<\/p>\n

\\begin{eqnarray}
S &=& \\iint_{D} dx \\, dy \\\\
&=& \\int_{-a}^{a}\u00a0 \\left\\{\\int_{-b\\sqrt{1 – \\frac{x^2}{a^2}}}^{b \\sqrt{1 – \\frac{x^2}{a^2}}} dy\\right\\} dx\\\\
&=& 2 \\int_{-a}^a b \\sqrt{1 – \\frac{x^2}{a^2}} dx
\\end{eqnarray}<\/p>\n

\u9ad8\u6821\u6570\u5b66\u3067\u306f\uff0c$y = f(x)$ \u3068 $y = g(x)$ \u304c\u533a\u9593 $x_1 \\leq x \\leq x_2$ \u3067 $f(x) \\geq g(x)$ \u306e\u3068\u304d\uff0c$y = f(x), \\ y = g(x), \\ x = x_1, \\ x = x_2$ \u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d $S$ \u304c<\/p>\n

$$S = \\int_{x_1}^{x_2} \\left( f(x) – g(x) \\right) \\, dx$$<\/p>\n

\u3068\u306a\u308b\u3068\u7fd2\u3063\u305f\u3068\u601d\u3046\u304c\uff0c\u3053\u306e\u5f0f\u304c\u4e0a\u8a18\u306e\u7d2f\u6b21\u7a4d\u5206\u3067\u5c0e\u304b\u308c\u305f\u3053\u3068\u306b\u306a\u308b\u3002<\/p>\n

\u3053\u3053\u3067\uff0c\\(x = a \\sin\\theta\\) \u3068\u5909\u6570\u5909\u63db\u3059\u308b\u3068\uff0c\\(\\displaystyle \\sqrt{1 – \\frac{x^2}{a^2}} = \\cos\\theta, dx = a \\cos\\theta d\\theta\\)\u3002
\\begin{eqnarray}
S &=& 2 \\int_{-a}^a b \\sqrt{1 – \\frac{x^2}{a^2}} dx \\\\
\n&=& 2 a b \\int_{-\\pi\/2}^{\\pi\/2} \\cos^2\\theta d\\theta\\\\
&=& a b \\int_{-\\pi\/2}^{\\pi\/2} (1 + \\cos 2\\theta) d\\theta \\\\
&=&a b \\left[ \\theta + \\frac{1}{2}\\sin 2\\theta\\right]_{-\\pi\/2}^{\\pi\/2} = \\pi a b
\\end{eqnarray}
$$\\therefore S = \\pi a b$$<\/p>\n

\u7126\u70b9\u3092\u539f\u70b9\u3068\u3057\u305f\u6975\u5ea7\u6a19\u306b\u3088\u308b\u7d2f\u6b21\u7a4d\u5206<\/h3>\n

\u7126\u70b9\u3092\u539f\u70b9\u3068\u3057\u305f\u6975\u5ea7\u6a19 $r, \\phi$ \u3067\u8868\u3057\u305f\u6955\u5186\u306e\u5f0f\u306f<\/p>\n

$$r(\\phi) = \\frac{a (1-e^2)}{1 + e \\cos\\phi}, \\quad b = a \\sqrt{1-e^2}$$<\/p>\n

\u3067\u3042\u3063\u305f\u3002\u3053\u306e\u6975\u5ea7\u6a19\u3067\u8868\u3059\u3068\uff0c\u9818\u57df \\(D\\) \u306f<\/p>\n

$$ 0 \\leq r \\leq r(\\phi) , \\quad 0 \\leq \\phi \\leq 2 \\pi$$<\/p>\n

\u3068\u306a\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u6955\u5186\u306e\u9762\u7a4d\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308b\u304c…<\/p>\n

\\begin{eqnarray} S &=& \\iint_{D} r \\,dr \\,d\\phi \\\\
\n&=&\u00a0 \\int_0^{2\\pi} d\\phi \\int_0^{r(\\phi)} r \\, dr \\\\
\n&=& \\frac{1}{2} \\int_0^{2\\pi} r^2(\\phi) \\,d\\phi \\\\
\n&=& a^2 (1-e^2)^2 \\int_0^{2\\pi} \\frac{1}{2 (1+e\\cos\\phi)^2}\\, d\\phi
\n\\end{eqnarray}<\/p>\n

$\\cos\\phi$ \u306e\u6709\u7406\u95a2\u6570 $\\displaystyle \\frac{1}{(1+e\\cos\\phi)^2}$ \u306e\u7a4d\u5206\u306b\u306a\u308a\uff0c\u30cf\u30de\u308a\u307e\u3059\u3002\u4ee5\u524d\u3084\u3063\u3066\u305f\u5929\u6587\u5b66\u306e\u6388\u696d\u3067\uff0c\u300c\u5929\u6587\u5b66\u3067\u306f\u6955\u5186\u306e\u4e2d\u5fc3\u3092\u539f\u70b9\u3068\u3057\u305f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19\u7cfb\u306a\u3069\u4f7f\u308f\u306a\u3044\u3002\u7126\u70b9\u3092\u539f\u70b9\u3068\u3057\u305f\u6975\u5ea7\u6a19\u3060\u3051\u304c\u51fa\u3066\u304f\u308b\u306e\u3060\u3002<\/strong><\/span>\u300d\u306a\u3069\u3068\u529b\u8aac\u3057\u305f\u624b\u524d\uff0c\u6955\u5186\u306e\u9762\u7a4d\u3082 $r(\\phi)$ \u3060\u3051\u3067\u6c42\u3081\u3088\u3046\u3068\u3057\u3066\uff0c\u5927\u30cf\u30de\u308a\u3057\u307e\u3057\u305f\u3002\u3053\u308c\u306f\u3084\u3063\u3066\u306f\u3044\u3051\u306a\u3044\u4f8b\u3068\u3057\u3066\u3002<\/p>\n

\u7b54\u3048\u306f\u65e2\u306b $ S = \\pi a b$ \u3068\u6c42\u307e\u3063\u3066\u3044\u308b\u306e\u3067\uff0c\u3053\u3093\u306a\u7169\u308f\u3057\u3044\u7a4d\u5206\u306b\u6642\u9593\u3092\u6f70\u3057\u305f\u304f\u306a\u3044\u3068\u601d\u3046\u4eba\u306f\uff0c\u4ee5\u4e0b\u306f\u8aad\u307f\u98db\u3070\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n

\u307e\u305a\uff0c\u6388\u696d\u3067\u3084\u3063\u305f\u300csin \ud835\udc65, cos \ud835\udc65 \u306e\u6709\u7406\u95a2\u6570\u306e\u7a4d\u5206<\/a>\u300d\u306e\u30bb\u30aa\u30ea\u30fc\u306b\u305d\u3063\u3066\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5909\u63db\u3092\u884c\u306a\u3063\u3066\u7f6e\u63db\u7a4d\u5206\u306e\u683c\u597d\u306b\u3059\u308b\u3002<\/p>\n

\\begin{eqnarray}
\n\\tan\\frac{\\phi}{2}&\\equiv& t\\\\
\nd\\phi &=& \\frac{2}{1 + t^2} dt \\\\
\n\\cos\\phi &=& \\frac{1-t^2}{1+t^2}
\n\\end{eqnarray}<\/p>\n

\u307e\u305a\u306f\u4e0d\u5b9a\u7a4d\u5206\u306e\u5f62\u3067\uff0c\u5909\u6570\u5909\u63db\u5f8c\uff0c\u90e8\u5206\u5206\u6570\u306b\u5206\u89e3\u3057\uff0c\u3055\u3089\u306b\u90e8\u5206\u7a4d\u5206\u3082\u99c6\u4f7f\u3057\u3066…\uff08\u3063\u3066\u3053\u3093\u306a\u306e\u79c1\u306e\u8155\u529b\u3067\u306f\u3067\u304d\u307e\u305b\u3093\u3002\u3053\u3053\u3060\u3051\u3053\u3063\u305d\u308a Maxima \u4f7f\u3044\u307e\u3057\u305f\u3002\uff09<\/p>\n

\\begin{eqnarray}&&
\n\\int \\frac{1}{2 (1+e\\cos\\phi)^2}\\, d\\phi \\\\
\n&=& \\int \\frac{1}{\\left(1+e\\frac{1-t^2}{1+t^2} \\right)^2}\\frac{dt}{1+t^2}\\\\
\n&=& \\int \\frac{1+t^2}{\\left\\{(1-e) t^2 + (1+e) \\right\\}^2}\\,dt \\\\
\n&=& \\int \\left\\{\\frac{1+e}{ (1-e^2)^{-\\frac{3}{2}}} \\frac{\\sqrt{\\frac{1-e}{1+e}}}{1 + \\frac{1-e}{1+e} t^2}
\n– \\frac{2 e}{ (1-e^2)^{-\\frac{3}{2}}}\\frac{\\sqrt{\\frac{1-e}{1+e}}}{\\left(1 + \\frac{1-e}{1+e} t^2\\right)^2}\\right\\} dt \\\\
\n&&\\qquad\\qquad \\left( x \\equiv \\sqrt{\\frac{1-e}{1+e}} t \\right)\\\\&=& \\frac{1+e}{ (1-e^2)^{-\\frac{3}{2}}}\\int \\frac{dx}{1 + x^2}
\n– \\frac{e}{ (1-e^2)^{-\\frac{3}{2}}}\\int \\frac{2 dx}{\\left(1 + x^2\\right)^2}\u00a0 \\\\
\n&=& \\frac{1}{ (1-e^2)^{-\\frac{3}{2}}}\\int \\frac{dx}{1 + x^2} – \\frac{e}{ (1-e^2)^{-\\frac{3}{2}}}\\frac{ x}{1+x^2} \\\\
\n&=& (1-e^2)^{-\\frac{3}{2}} \\tan^{-1} \\left(\\sqrt{\\frac{1-e}{1+e}} t \\right)
\n– \\frac{e t}{(1-e^2) \\left\\{ (1-e)t^2 + (1+e)\\right\\}}
\n\\end{eqnarray}<\/p>\n

(\u3053\u3053\u3067\u00a0 $ \\int \\frac{2 dx}{\\left(1 + x^2\\right)^2} = \\int \\frac{dx}{1+x^2} + \\frac{x}{1+x^2}$ \u3092\u4f7f\u3063\u305f\u3002)<\/p>\n

\u4e0a\u8a18\u7b2c2\u9805\u306f\uff0c$t = \\tan\\frac{\\phi}{2} $ \u304c $0$ \u306e\u3068\u304d\u3082 $\\pm\\infty$ \u306e\u3068\u304d\u3082\u30bc\u30ed\u306b\u306a\u308b\u306e\u3067\u5b9a\u7a4d\u5206\u306b\u306f\u5bc4\u4e0e\u3057\u306a\u3044\u3002\u3057\u305f\u304c\u3063\u3066\uff08\u7a4d\u5206\u533a\u9593\u5185\u3067 $\\tan\\frac{\\phi}{2}$ \u306e\u767a\u6563\u304c\u3042\u308b\u304b\u3089\u614e\u91cd\u306b\uff09<\/p>\n

\\begin{eqnarray}&&
\n\\int_0^{2\\pi} \\frac{1}{2 (1+e\\cos\\phi)^2}\\, d\\phi \\\\
\n&=& (1-e^2)^{-\\frac{3}{2}} \\Biggl[\\tan^{-1} \\left(\\sqrt{\\frac{1-e}{1+e} }\\tan\\frac{\\phi}{2} \\right)
\n\\Biggr]_0^{\\pi-0}\\\\
\n&& + (1-e^2)^{-\\frac{3}{2}} \\Biggl[\\tan^{-1} \\left(\\sqrt{\\frac{1-e}{1+e} }\\tan\\frac{\\phi}{2} \\right)
\n\\Biggr]_{\\pi+0}^{2\\pi} \\\\
\n&=& (1-e^2)^{-\\frac{3}{2}} \\left\\{ \\left(\\frac{\\pi}{2} – 0\\right) + \\left(0 – \\left(-\\frac{\\pi}{2}\\right) \\right)\\right\\}\\\\
\n&=& (1-e^2)^{-\\frac{3}{2}} \\,\\pi
\n\\end{eqnarray}<\/p>\n

\u6700\u7d42\u7684\u306b<\/p>\n

\\begin{eqnarray}
\nS &=&\u00a0 a^2 (1-e^2)^2 \\int_0^{2\\pi} \\frac{1}{2 (1+e\\cos\\phi)^2}\\, d\\phi\\\\
\n&=& a^2 (1-e^2)^2 \\times (1-e^2)^{-\\frac{3}{2}} \\,\\pi \\\\
\n&=& \\pi a^2 \\sqrt{1-e^2} \\\\
\n&=& \\pi a b
\n\\end{eqnarray}<\/p>\n

\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2\u3092\u4f7f\u3063\u305f\u7f6e\u63db\u7a4d\u5206<\/h3>\n

Memo\u300c\u771f\u8fd1\u70b9\u96e2\u89d2\u3068\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2\u3068\u306e\u95a2\u4fc2\u306b\u3064\u3044\u3066\u3082\u3046\u5c11\u3057<\/a>\u300d\u306b\u307e\u3068\u3081\u305f\u3088\u3046\u306b\uff0c\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u6570 $u$ \u3092\u4f7f\u3063\u305f\u7f6e\u63db\u7a4d\u5206\u306b\u3059\u308b\u3068\uff0c\u7c21\u5358\u3002$u$ \u306f\u696d\u754c\u7528\u8a9e\u3067\u300c\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2<\/strong><\/span>\u300d\u307e\u305f\u306f\u300c\u96e2\u5fc3\u8fd1\u70b9\u89d2 <\/strong><\/span>(Wikipedia)<\/a>\u300d\u3068\u547c\u3076\u3002\u3053\u3053\u3067\u306f\u6728\u4e0b\u5b99\u8457\u300c\u5929\u4f53\u3068\u8ecc\u9053\u306e\u529b\u5b66\u300d\u306b\u306a\u3089\u3044\uff0c\u300c\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2<\/strong><\/span>\u300d\u3067\u3002<\/p>\n

\u96e2\u5fc3\u8fd1\u70b9\u96e2\u89d2<\/strong><\/span>\u306e\u5e7e\u4f55\u5b66\u7684\u610f\u5473\u3065\u3051\u3082\u8208\u5473\u6df1\u3044\u304c\uff0c\u3053\u3053\u3067\u306f\u3042\u304f\u307e\u3067\u7f6e\u63db\u7a4d\u5206\u306e\u305f\u3081\u306e\u5909\u6570\u5909\u63db\u3067\u3042\u308b\u3068\u3044\u3046\u6570\u5b66\u7684\u9053\u5177\u3068\u3057\u3066\u306e\u6709\u52b9\u6027\u306e\u307f\u3092\u5f37\u8abf\u3059\u308b\u306e\u307f\u306b\u3068\u3069\u3081\u308b\u3002\u89d2\u5ea6\u5ea7\u6a19 $\\phi$ \uff08\u3053\u308c\u306f\u696d\u754c\u7528\u8a9e\u3067\u300c\u771f\u8fd1\u70b9\u96e2\u89d2<\/strong><\/span>\u300d\u3068\u3082\u547c\u3070\u308c\u308b\uff09\u3068 $u$ \u306e\u95a2\u4fc2\u306f\uff0c<\/p>\n

\\begin{eqnarray}
\n\\frac{1 -e^2}{1 + e \\cos\\phi} &\\equiv& 1 -e \\cos u \\\\
\nd\\phi &=& \\frac{\\sqrt{1 -e^2}}{1 -e \\cos u} \\,du
\n\\end{eqnarray}<\/p>\n

$u$ \u3092\u4f7f\u3063\u3066\u7f6e\u63db\u7a4d\u5206\u3059\u308b\u3068\uff0c\uff08\u7a4d\u5206\u7bc4\u56f2\u304c $[0, 2\\pi]$ \u306e\u307e\u307e\u306a\u306e\u3082\u4fbf\u5229\u306a\u3068\u3053\u308d\uff09<\/p>\n

\\begin{eqnarray}
\n\\int_0^{2\\pi} \\frac{1}{2 (1+e\\cos\\phi)^2}\\, d\\phi
\n&=& \\frac{1}{2 (1 -e^2)^2} \\int_0^{2 \\pi} (1 -e \\cos u)^2 \\cdot \\frac{\\sqrt{1 -e^2}}{1 -e \\cos u} \\,du \\\\
\n&=& \\frac{\\sqrt{1 -e^2}}{2 (1 -e^2)^2} \\int_0^{2 \\pi} (1 -e \\cos u)\\, du \\\\
\n&=& \\frac{\\sqrt{1 -e^2}}{2 (1 -e^2)^2} \\Bigl[ u -e \\sin u \\Bigr]_0^{2 \\pi} \\\\
\n&=& \\frac{ \\pi \\sqrt{1 -e^2}}{(1 -e^2)^2}
\n\\end{eqnarray}<\/p>\n

\u5ea7\u6a19\u5909\u63db\u3057\uff0c\u30e4\u30b3\u30d3\u30a2\u30f3\u3092\u8a08\u7b97\u3057\u3066\u7a4d\u5206<\/h3>\n

\u6955\u5186\u306e\u5185\u90e8\u3092\u8868\u3059\u9818\u57df \\(D\\) \u306f\u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19 $x, y$ \u3067<\/p>\n

$$ D: \\frac{x^2}{a^2} + \\frac{y^2}{a^2 (1 -e^2)} \\leq 1$$<\/p>\n

\u3068\u66f8\u3051\u305f\u3002\u3053\u3053\u3067\uff0c$a$ \u306f\u8ecc\u9053\u9577\u534a\u5f84\uff0c$e$ \u306f\u96e2\u5fc3\u7387\u3067\u3042\u308a\uff0c\u77ed\u534a\u5f84 $b$ \u3068\u306f $b = a \\sqrt{1 -e^2}$ \u306e\u95a2\u4fc2\u304c\u3042\u308b\u306e\u3067\u3042\u3063\u305f\u3002<\/p>\n

\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5ea7\u6a19\u5909\u63db\u3092\u304a\u3053\u306a\u3063\u3066\uff0c\u65b0\u3057\u3044\u5ea7\u6a19\u5909\u6570 $u, v$ \u306b\u3064\u3044\u3066\u306e\u7a4d\u5206\u306b\u306a\u304a\u3057\u3066\u307f\u3088\u3046\u3002<\/p>\n

\\begin{eqnarray}
\nx &=& u \\cos v \\\\
\ny &=& u \\sqrt{1-e^2} \\sin v \\\\
\n\\therefore\\ \\ \\frac{x^2}{a^2} + \\frac{y^2}{a^2 (1 -e^2)} &=& \\frac{u^2}{a^2} \\leq 1
\n\\end{eqnarray}<\/p>\n

\u3057\u305f\u304c\u3063\u3066\uff0c\u9818\u57df $D$ \u306f\uff0c$u, v$ \u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068<\/p>\n

$$D: \\ 0 \\leq u \\leq a, \\\u00a0 0, \\leq v \\leq 2 \\pi$$<\/p>\n

\u3068\u306a\u308b\u3002\u3053\u306e\u5ea7\u6a19\u5909\u63db\u306b\u3088\u3063\u3066\uff0c\u5fae\u5c0f\u9762\u7a4d\u8981\u7d20\u306f<\/p>\n

$$ dx\\, dy = \\frac{\\partial (x, y)}{\\partial (u, v)} \\, du\\, dv = \\cdots$$<\/p>\n

\u3068\u306a\u308b\u3002\u3053\u3053\u3067 $\\displaystyle \\frac{\\partial (x, y)}{\\partial (u, v)}$ \u306f\u5909\u63db\u306e\u30e4\u30b3\u30d3\u30a2\u30f3\u3002\u5b66\u751f\u8af8\u541b\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3061\u3083\u3093\u3068\u30e4\u30b3\u30d3\u30a2\u30f3\u3092\u8a08\u7b97\u3059\u308b\u3093\u3067\u3059\u3088\u3002<\/p>\n

$$
\n\\frac{\\partial(x,y)}{\\partial(u, v)} \\equiv
\n\\begin{vmatrix}
\n\\frac{\\partial x}{\\partial u} & \\frac{\\partial x}{\\partial v}\\\\
\n\\frac{\\partial y}{\\partial u} & \\frac{\\partial y}{\\partial v}\\\\
\n\\end{vmatrix}
\n= \\frac{\\partial x}{\\partial u} \\frac{\\partial y}{\\partial v} \u2013 \\frac{\\partial y}{\\partial u}\\frac{\\partial x}{\\partial v} = \\cdots
\n$$<\/p>\n

\u6700\u7d42\u7684\u306b\u9762\u7a4d\u306f\uff0c<\/p>\n

\\begin{eqnarray}
\nS &=& \\iint_D dx\\, dy\\\\
\n&=& \\iint_D \\frac{\\partial (x, y)}{\\partial (u, v)} \\, du\\, dv \\\\
\n&=& \\int_0^a du \\int_0^{2 \\pi} dv \\frac{\\partial(x,y)}{\\partial(u, v)} = \\cdots
\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2338,"menu_order":5,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-4885","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\/4885","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=4885"}],"version-history":[{"count":50,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/4885\/revisions"}],"predecessor-version":[{"id":9186,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/4885\/revisions\/9186"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2338"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=4885"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}