{"id":19753,"date":"2025-05-10T12:55:18","date_gmt":"2025-05-10T03:55:18","guid":{"rendered":"https:\/\/123210.net\/wordpress\/tatenaoshi\/?p=19753"},"modified":"2025-05-10T12:55:18","modified_gmt":"2025-05-10T03:55:18","slug":"%e3%83%9b%e3%83%83%e3%82%b8%e5%88%86%e8%a7%a3%ef%bc%88hodge-decomposition%ef%bc%89%e3%81%af%e3%80%81%e3%83%98%e3%83%ab%e3%83%a0%e3%83%9b%e3%83%ab%e3%83%84%e5%88%86%e8%a7%a3%e3%82%92%e4%b8%80%e8%88%ac","status":"publish","type":"post","link":"https:\/\/123210.net\/wordpress\/tatenaoshi\/%e3%83%9b%e3%83%83%e3%82%b8%e5%88%86%e8%a7%a3%ef%bc%88hodge-decomposition%ef%bc%89%e3%81%af%e3%80%81%e3%83%98%e3%83%ab%e3%83%a0%e3%83%9b%e3%83%ab%e3%83%84%e5%88%86%e8%a7%a3%e3%82%92%e4%b8%80%e8%88%ac\/","title":{"rendered":"\u30db\u30c3\u30b8\u5206\u89e3\uff08Hodge decomposition\uff09\u306f\u3001\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u5206\u89e3\u3092\u4e00\u822c\u5316\u3057\u305f\u6570\u5b66\u7684\u67a0\u7d44\u307f\u3067\u3001\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u4e0a\u306e\u5fae\u5206\u5f62\u5f0f\u3084\u30d9\u30af\u30c8\u30eb\u5834\u3092\u7279\u5b9a\u306e\u6210\u5206\u306b\u5206\u89e3\u3059\u308b\u624b\u6cd5\u3067\u3059\u3002\u9ad8\u6b21\u5143\u7a7a\u9593\u3084\u975e\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u3067\u306e\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u5206\u89e3\u306e\u7406\u8ad6\u7684\u57fa\u790e\u3092\u63d0\u4f9b\u3057\u3001\u7269\u7406\u5b66\u3001\u5de5\u5b66\u3001\u5e7e\u4f55\u5b66\u306a\u3069\u3067\u5e83\u304f\u5fdc\u7528\u3055\u308c\u307e\u3059\u3002"},"content":{"rendered":"
\n
1. \u30db\u30c3\u30b8\u5206\u89e3\u306e\u6982\u8981<\/span><\/span><\/div>\n
\u30db\u30c3\u30b8\u5206\u89e3\u306f\u3001\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53 <\/span>(<\/span>M<\/span>)<\/span> \u4e0a\u306e\u5fae\u5206\u5f62\u5f0f\uff08\u307e\u305f\u306f\u30d9\u30af\u30c8\u30eb\u5834\uff09\u3092\u4ee5\u4e0b\u306e3\u3064\u306e\u6210\u5206\u306b\u5206\u89e3\u3059\u308b\u5b9a\u7406\u3067\u3059\uff1a <\/span><\/span><\/p>\n
\u03a9k(M)=d\u03a9k\u22121(M)\u2295\u03b4\u03a9k+1(M)\u2295Hk(M),\\Omega^k(M) = d\\Omega^{k-1}(M) \\oplus \\delta\\Omega^{k+1}(M) \\oplus \\mathcal{H}^k(M),<\/span>\\Omega^k(M) = d\\Omega^{k-1}(M) \\oplus \\delta\\Omega^{k+1}(M) \\oplus \\mathcal{H}^k(M),<\/code><\/div>\n

\u3053\u3053\u3067\uff1a<\/span><\/span><\/div>\n

    \n
  • \n
    \n
    \u03a9k(M)\\Omega^k(M)<\/span>\\Omega^k(M)<\/code><\/div>\n

    \uff1a<\/span><\/span>(<\/span><\/span>M<\/span><\/span>)<\/span><\/span> \u4e0a\u306e <\/span><\/span>(<\/span><\/span>k<\/span><\/span>)<\/span><\/span>-\u5f62\u5f0f\u306e\u7a7a\u9593\u3002<\/span><\/span><\/div>\n<\/li>\n

  • \n
    \n
    d\u03a9k\u22121(M)d\\Omega^{k-1}(M)<\/span>d\\Omega^{k-1}(M)<\/code><\/div>\n

    \uff1a<\/span><\/span><\/p>\n

    (k\u22121)(k-1)<\/span>(k-1)<\/code><\/div>\n

    -\u5f62\u5f0f\u306e\u5916\u5fae\u5206\uff08exact forms\uff1a\u6b63\u78ba\u5f62\u5f0f\uff09\u3002<\/span><\/span><\/div>\n<\/li>\n

  • \n
    \n
    \u03b4\u03a9k+1(M)\\delta\\Omega^{k+1}(M)<\/span>\\delta\\Omega^{k+1}(M)<\/code><\/div>\n

    \uff1a<\/span><\/span><\/p>\n

    (k+1)(k+1)<\/span>(k+1)<\/code><\/div>\n

    -\u5f62\u5f0f\u306e\u4f59\u5fae\u5206\uff08coexact forms\uff1a\u4f59\u6b63\u78ba\u5f62\u5f0f\uff09\u3002<\/span><\/span><\/div>\n<\/li>\n

  • \n
    \n
    Hk(M)\\mathcal{H}^k(M)<\/span>\\mathcal{H}^k(M)<\/code><\/div>\n

    \uff1a\u8abf\u548c\u5f62\u5f0f\u306e\u7a7a\u9593\uff08<\/span><\/span><\/p>\n

    \u0394\u03c9=0\\Delta \\omega = 0<\/span>\\Delta \\omega = 0<\/code><\/div>\n

    , \u3064\u307e\u308a <\/span><\/span><\/p>\n

    d\u03c9=0d\\omega = 0<\/span>d\\omega = 0<\/code><\/div>\n

    \u304b\u3064 <\/span><\/span><\/p>\n

    \u03b4\u03c9=0\\delta\\omega = 0<\/span>\\delta\\omega = 0<\/code><\/div>\n

    \uff09\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n

    \u76f4\u611f\u7684\u89e3\u91c8<\/span><\/span>\uff1a<\/span><\/span><\/div>\n
      \n
    • \n
      \u4efb\u610f\u306e <\/span><\/span>(<\/span><\/span>k<\/span><\/span>)<\/span><\/span>-\u5f62\u5f0f <\/span><\/span><\/p>\n
      \u03c9\\omega<\/span>\\omega<\/code><\/div>\n

      \u306f\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u66f8\u3051\u308b\uff1a<\/span><\/span><\/div>\n

      \u03c9=d\u03b1+\u03b4\u03b2+h,\\omega = d\\alpha + \\delta\\beta + h,<\/span>\\omega = d\\alpha + \\delta\\beta + h,<\/code><\/div>\n
      \u3053\u3053\u3067 <\/span><\/span><\/p>\n
      \u03b1\\alpha<\/span>\\alpha<\/code><\/div>\n

      \u306f <\/span><\/span><\/p>\n

      (k\u22121)(k-1)<\/span>(k-1)<\/code><\/div>\n

      -\u5f62\u5f0f\u3001<\/span><\/span><\/p>\n

      \u03b2\\beta<\/span>\\beta<\/code><\/div>\n

      \u306f <\/span><\/span><\/p>\n

      (k+1)(k+1)<\/span>(k+1)<\/code><\/div>\n

      -\u5f62\u5f0f\u3001<\/span><\/span>(<\/span><\/span>h<\/span><\/span>)<\/span><\/span> \u306f\u8abf\u548c\u5f62\u5f0f\u3002<\/span><\/span><\/div>\n<\/li>\n

    • \n
      \u3053\u306e\u5206\u89e3\u306f\u76f4\u4ea4\u5206\u89e3\u3067\u3042\u308a\u3001\u5404\u6210\u5206\u306f <\/span><\/span><\/p>\n
      L2L^2<\/span>L^2<\/code><\/div>\n

      \u30ce\u30eb\u30e0\u306b\u95a2\u3057\u3066\u4e92\u3044\u306b\u76f4\u4ea4\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n

      \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593 <\/span><\/span><\/span><\/p>\n
      Rn\\mathbb{R}^n<\/span>\\mathbb{R}^n<\/code><\/div>\n

      \u306e\u5834\u5408<\/span><\/span>\uff1a<\/span><\/span><\/div>\n

        \n
      • \n
        3\u6b21\u5143\u3067\u306f\u3001\u30d9\u30af\u30c8\u30eb\u5834 <\/span><\/span><\/p>\n
        F\\mathbf{F}<\/span>\\mathbf{F}<\/code><\/div>\n

        \u306b\u5bfe\u3057\u3066\u30d8\u30eb\u30e0\u30db\u30eb\u30c4\u5206\u89e3\uff08<\/span><\/span><\/p>\n

        F=\u2207\u03d5+\u2207\u00d7A\\mathbf{F} = \\nabla \\phi + \\nabla \\times \\mathbf{A}<\/span>\\mathbf{F} = \\nabla \\phi + \\nabla \\times \\mathbf{A}<\/code><\/div>\n

        \uff09\u304c\u30db\u30c3\u30b8\u5206\u89e3\u306e\u7279\u5225\u306a\u5834\u5408\u3002<\/span><\/span><\/div>\n<\/li>\n

      • \n
        \u9ad8\u6b21\u5143\u3067\u306f\u3001\u30d9\u30af\u30c8\u30eb\u5834\u3092\u52fe\u914d\u5834\uff08<\/span><\/span><\/p>\n
        \u2207\u03d5\\nabla \\phi<\/span>\\nabla \\phi<\/code><\/div>\n

        \uff09\u3001\u767a\u6563\u30bc\u30ed\u5834\uff08\u4e00\u822c\u5316\u3055\u308c\u305f\u300c\u56de\u8ee2\u300d\u6210\u5206\uff09\u3001\u304a\u3088\u3073\u8abf\u548c\u5834\uff08<\/span><\/span><\/p>\n

        \u2207\u22c5H=0,\u2207\u00d7H=0\\nabla \\cdot \\mathbf{H} = 0, \\nabla \\times \\mathbf{H} = 0<\/span>\\nabla \\cdot \\mathbf{H} = 0, \\nabla \\times \\mathbf{H} = 0<\/code><\/div>\n

        \uff09\u306b\u5206\u89e3\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n

        \n
        2. \u6570\u5b66\u7684\u80cc\u666f<\/span><\/span><\/div>\n
        \u30db\u30c3\u30b8\u5206\u89e3\u3092\u7406\u89e3\u3059\u308b\u306b\u306f\u3001\u4ee5\u4e0b\u306e\u6982\u5ff5\u304c\u91cd\u8981\u3067\u3059\uff1a<\/span><\/span><\/div>\n
        (1) \u5fae\u5206\u5f62\u5f0f\u3068\u5916\u5fae\u5206<\/span><\/span><\/div>\n
          \n
        • \n
          (<\/span><\/span>k<\/span><\/span>)<\/span><\/span>-\u5f62\u5f0f\u306f\u3001\u591a\u69d8\u4f53\u4e0a\u306e\u300c\u30d9\u30af\u30c8\u30eb\u5834\u306e\u4e00\u822c\u5316\u300d\u3067\u3001\u4ea4\u4ee3\u30c6\u30f3\u30bd\u30eb\u3068\u3057\u3066\u632f\u308b\u821e\u3046\u3002<\/span><\/span><\/div>\n<\/li>\n
        • \n
          \u5916\u5fae\u5206 <\/span><\/span><\/p>\n
          d:\u03a9k(M)\u2192\u03a9k+1(M)d: \\Omega^k(M) \\to \\Omega^{k+1}(M)<\/span>d: \\Omega^k(M) \\to \\Omega^{k+1}(M)<\/code><\/div>\n

          \u306f\u3001\u5fae\u5206\u5f62\u5f0f\u306e\u300c\u52fe\u914d\u300d\u3084\u300c\u56de\u8ee2\u300d\u3092\u4e00\u822c\u5316\u3002<\/span><\/span><\/div>\n

            \n
          • \n
            \u6027\u8cea\uff1a<\/span><\/span><\/p>\n
            d\u2218d=0d \\circ d = 0<\/span>d \\circ d = 0<\/code><\/div>\n

            \uff08\u4f8b\uff1a<\/span><\/span><\/p>\n

            \u2207\u00d7(\u2207\u03d5)=0\\nabla \\times (\\nabla \\phi) = 0<\/span>\\nabla \\times (\\nabla \\phi) = 0<\/code><\/div>\n

            \u306e\u9ad8\u6b21\u5143\u7248\uff09\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

            (2) \u4f59\u5fae\u5206<\/span><\/span><\/div>\n
              \n
            • \n
              \u4f59\u5fae\u5206 <\/span><\/span><\/p>\n
              \u03b4:\u03a9k(M)\u2192\u03a9k\u22121(M)\\delta: \\Omega^k(M) \\to \\Omega^{k-1}(M)<\/span>\\delta: \\Omega^k(M) \\to \\Omega^{k-1}(M)<\/code><\/div>\n

              \u306f\u3001\u5916\u5fae\u5206\u306e\u53cc\u5bfe\u6f14\u7b97\u5b50\u3002<\/span><\/span><\/div>\n<\/li>\n

            • \n
              \u30ea\u30fc\u30de\u30f3\u8a08\u91cf <\/span><\/span>(<\/span><\/span>g<\/span><\/span>)<\/span><\/span> \u3092\u7528\u3044\u3066\u5b9a\u7fa9\u3055\u308c\u3001<\/span><\/span><\/p>\n
              \u03b4=(\u22121)n(k+1)+1\u2217d\u2217\\delta = (-1)^{n(k+1)+1} * d *<\/span>\\delta = (-1)^{n(k+1)+1} * d *<\/code><\/div>\n

              \uff08\u3053\u3053\u3067 <\/span><\/span><\/p>\n

              \u2217*<\/span>*<\/code><\/div>\n

              \u306f\u30db\u30c3\u30b8\u30b9\u30bf\u30fc\u6f14\u7b97\u5b50\uff09\u3002<\/span><\/span><\/div>\n<\/li>\n

            • \n
              \u4f8b\uff1a3\u6b21\u5143\u3067\u30d9\u30af\u30c8\u30eb\u5834 <\/span><\/span><\/p>\n
              F\\mathbf{F}<\/span>\\mathbf{F}<\/code><\/div>\n

              \u306b\u5bfe\u3057\u3001<\/span><\/span><\/p>\n

              \u03b4F\u223c\u2207\u22c5F\\delta \\mathbf{F} \\sim \\nabla \\cdot \\mathbf{F}<\/span>\\delta \\mathbf{F} \\sim \\nabla \\cdot \\mathbf{F}<\/code><\/div>\n

              \u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n

              (3) \u30db\u30c3\u30b8\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3<\/span><\/span><\/div>\n
                \n
              • \n
                \u30db\u30c3\u30b8\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306f <\/span><\/span><\/p>\n
                \u0394=d\u03b4+\u03b4d\\Delta = d\\delta + \\delta d<\/span>\\Delta = d\\delta + \\delta d<\/code><\/div>\n

                \u3067\u5b9a\u7fa9\u3002<\/span><\/span><\/div>\n<\/li>\n

              • \n
                \u8abf\u548c\u5f62\u5f0f <\/span><\/span>(<\/span><\/span>h<\/span><\/span>)<\/span><\/span> \u306f <\/span><\/span><\/p>\n
                \u0394h=0\\Delta h = 0<\/span>\\Delta h = 0<\/code><\/div>\n

                \u3092\u6e80\u305f\u3059\uff08\u3064\u307e\u308a\u3001<\/span><\/span><\/p>\n

                dh=0,\u03b4h=0dh = 0, \\delta h = 0<\/span>dh = 0, \\delta h = 0<\/code><\/div>\n

                \uff09\u3002<\/span><\/span><\/div>\n<\/li>\n

              • \n
                \u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u3067\u306f\u3001<\/span><\/span><\/p>\n
                \u0394\\Delta<\/span>\\Delta<\/code><\/div>\n

                \u306f\u901a\u5e38\u306e\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3 <\/span><\/span><\/p>\n

                \u22072\\nabla^2<\/span>\\nabla^2<\/code><\/div>\n

                \u306b\u9084\u5143\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n

                (4) <\/span><\/span><\/p>\n
                L2L^2<\/span>L^2<\/code><\/div>\n

                \u7a7a\u9593\u3068\u76f4\u4ea4\u6027<\/span><\/span><\/div>\n

                  \n
                • \n
                  \u30db\u30c3\u30b8\u5206\u89e3\u306f\u3001<\/span><\/span><\/p>\n
                  L2L^2<\/span>L^2<\/code><\/div>\n

                  \u30ce\u30eb\u30e0\uff08<\/span><\/span><\/p>\n

                  \u27e8\u03c9,\u03b7\u27e9=\u222bM\u03c9\u2227\u2217\u03b7\\langle \\omega, \\eta \\rangle = \\int_M \\omega \\wedge *\\eta<\/span>\\langle \\omega, \\eta \\rangle = \\int_M \\omega \\wedge *\\eta<\/code><\/div>\n

                  \uff09\u306b\u304a\u3044\u3066\u76f4\u4ea4\u5206\u89e3\u3002<\/span><\/span><\/div>\n<\/li>\n

                • \n
                  \n
                  d\u03b1d\\alpha<\/span>d\\alpha<\/code><\/div>\n

                  , <\/span><\/span><\/p>\n

                  \u03b4\u03b2\\delta\\beta<\/span>\\delta\\beta<\/code><\/div>\n

                  , <\/span><\/span>(<\/span><\/span>h<\/span><\/span>)<\/span><\/span> \u306f\u4e92\u3044\u306b\u76f4\u4ea4\uff1a<\/span><\/span><\/div>\n

                  \u27e8d\u03b1,\u03b4\u03b2\u27e9=0,\u27e8d\u03b1,h\u27e9=0,\u27e8\u03b4\u03b2,h\u27e9=0.\\langle d\\alpha, \\delta\\beta \\rangle = 0, \\quad \\langle d\\alpha, h \\rangle = 0, \\quad \\langle \\delta\\beta, h \\rangle = 0.<\/span>\\langle d\\alpha, \\delta\\beta \\rangle = 0, \\quad \\langle d\\alpha, h \\rangle = 0, \\quad \\langle \\delta\\beta, h \\rangle = 0.<\/code><\/div>\n<\/li>\n<\/ul>\n
                  \n
                  3. \u30db\u30c3\u30b8\u5206\u89e3\u5b9a\u7406<\/span><\/span><\/div>\n
                  \u5b9a\u7406<\/span><\/span>\uff08\u30b3\u30f3\u30d1\u30af\u30c8\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u306e\u5834\u5408\uff09\uff1a \u30b3\u30f3\u30d1\u30af\u30c8\u3067\u5411\u304d\u4ed8\u3051\u53ef\u80fd\u306a\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53 <\/span>(<\/span>M<\/span>)<\/span> \u4e0a\u306e\u4efb\u610f\u306e <\/span>(<\/span>k<\/span>)<\/span>-\u5f62\u5f0f <\/span><\/span><\/p>\n
                  \u03c9\u2208\u03a9k(M)\\omega \\in \\Omega^k(M)<\/span>\\omega \\in \\Omega^k(M)<\/code><\/div>\n

                  \u306f\u3001\u4e00\u610f\u306b\u6b21\u306e\u3088\u3046\u306b\u5206\u89e3\u3055\u308c\u308b\uff1a <\/span><\/span><\/p>\n

                  \u03c9=d\u03b1+\u03b4\u03b2+h,\\omega = d\\alpha + \\delta\\beta + h,<\/span>\\omega = d\\alpha + \\delta\\beta + h,<\/code><\/div>\n<\/div>\n
                    \n
                  • \n
                    \n
                    \u03b1\u2208\u03a9k\u22121(M)\\alpha \\in \\Omega^{k-1}(M)<\/span>\\alpha \\in \\Omega^{k-1}(M)<\/code><\/div>\n

                    , <\/span><\/span><\/p>\n

                    \u03b2\u2208\u03a9k+1(M)\\beta \\in \\Omega^{k+1}(M)<\/span>\\beta \\in \\Omega^{k+1}(M)<\/code><\/div>\n

                    , <\/span><\/span><\/p>\n

                    h\u2208Hk(M)h \\in \\mathcal{H}^k(M)<\/span>h \\in \\mathcal{H}^k(M)<\/code><\/div>\n

                    \u3002<\/span><\/span><\/div>\n<\/li>\n

                  • \n
                    \u5206\u89e3\u306f <\/span><\/span><\/p>\n
                    L2L^2<\/span>L^2<\/code><\/div>\n

                    \u7a7a\u9593\u3067\u76f4\u4ea4\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n

                    \u975e\u30b3\u30f3\u30d1\u30af\u30c8\u306e\u5834\u5408<\/span><\/span>\uff08\u4f8b\uff1a<\/span><\/span><\/p>\n
                    Rn\\mathbb{R}^n<\/span>\\mathbb{R}^n<\/code><\/div>\n

                    \uff09\uff1a<\/span><\/span><\/div>\n

                      \n
                    • \n
                      \n
                      Rn\\mathbb{R}^n<\/span>\\mathbb{R}^n<\/code><\/div>\n

                      \u3067\u306f\u3001\u7121\u9650\u9060\u3067\u306e\u6e1b\u8870\u6761\u4ef6\uff08\u4f8b\uff1a<\/span><\/span><\/p>\n

                      \u03c9\\omega<\/span>\\omega<\/code><\/div>\n

                      \u304c <\/span><\/span><\/p>\n

                      L2L^2<\/span>L^2<\/code><\/div>\n

                      \u3084\u30bd\u30dc\u30ec\u30d5\u7a7a\u9593\u306b\u5c5e\u3059\u308b\uff09\u3092\u4eee\u5b9a\u3002<\/span><\/span><\/div>\n<\/li>\n

                    • \n
                      \u8abf\u548c\u6210\u5206 <\/span><\/span><\/p>\n
                      Hk(Rn)\\mathcal{H}^k(\\mathbb{R}^n)<\/span>\\mathcal{H}^k(\\mathbb{R}^n)<\/code><\/div>\n

                      \u306f\u3001\u30c8\u30dd\u30ed\u30b8\u30fc\u306b\u3088\u308a\u30bc\u30ed\u306b\u306a\u308b\u5834\u5408\u3082\uff08\u4f8b\uff1a<\/span><\/span><\/p>\n

                      Rn\\mathbb{R}^n<\/span>\\mathbb{R}^n<\/code><\/div>\n

                      \u306f\u5358\u9023\u7d50\u306a\u306e\u3067 <\/span><\/span><\/p>\n

                      H1(Rn)=0\\mathcal{H}^1(\\mathbb{R}^n) = 0<\/span>\\mathcal{H}^1(\\mathbb{R}^n) = 0<\/code><\/div>\n

                      \uff09\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n

                      \n
                      4. \u8a3c\u660e\u306e\u6982\u8981<\/span><\/span><\/div>\n
                      \u30db\u30c3\u30b8\u5206\u89e3\u306e\u8a3c\u660e\u306f\u3001\u6955\u5186\u578b\u504f\u5fae\u5206\u65b9\u7a0b\u5f0f\u306e\u7406\u8ad6\u3068\u51fd\u6570\u89e3\u6790\u306b\u57fa\u3065\u304d\u307e\u3059\u3002\u4ee5\u4e0b\u306f\u6982\u8981\uff1a<\/span><\/span><\/div>\n
                        \n
                      1. \n
                        \u30db\u30c3\u30b8\u30e9\u30d7\u30e9\u30b7\u30a2\u30f3\u306e\u6027\u8cea<\/span><\/span><\/span>\uff1a<\/span><\/span><\/div>\n
                          \n
                        • \n
                          \n
                          \u0394\\Delta<\/span>\\Delta<\/code><\/div>\n

                          \u306f\u81ea\u5df1\u5171\u5f79\u3067\u6955\u5186\u578b\u6f14\u7b97\u5b50\u3002<\/span><\/span><\/div>\n<\/li>\n

                        • \n
                          \u30b3\u30f3\u30d1\u30af\u30c8\u591a\u69d8\u4f53\u3067\u306f\u3001<\/span><\/span><\/p>\n
                          \u0394\\Delta<\/span>\\Delta<\/code><\/div>\n

                          \u306e\u30b9\u30da\u30af\u30c8\u30eb\u306f\u96e2\u6563\u7684\u3067\u3001\u56fa\u6709\u5024\u306f\u975e\u8ca0\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n<\/li>\n

                        • \n
                          \u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u3068\u30dd\u30a2\u30bd\u30f3\u65b9\u7a0b\u5f0f<\/span><\/span><\/span>\uff1a<\/span><\/span><\/div>\n
                            \n
                          • \n
                            \n
                            \u03c9\\omega<\/span>\\omega<\/code><\/div>\n

                            \u306e\u6b63\u78ba\u6210\u5206 <\/span><\/span><\/p>\n

                            d\u03b1d\\alpha<\/span>d\\alpha<\/code><\/div>\n

                            \u306f\u3001<\/span><\/span><\/p>\n

                            \u0394\u03b1=\u03b4\u03c9\\Delta \\alpha = \\delta \\omega<\/span>\\Delta \\alpha = \\delta \\omega<\/code><\/div>\n

                            \u3092\u89e3\u304f\u3053\u3068\u3067\u5f97\u3089\u308c\u308b\u3002<\/span><\/span><\/div>\n<\/li>\n

                          • \n
                            \u4f59\u6b63\u78ba\u6210\u5206 <\/span><\/span><\/p>\n
                            \u03b4\u03b2\\delta\\beta<\/span>\\delta\\beta<\/code><\/div>\n

                            \u306f\u3001<\/span><\/span><\/p>\n

                            \u0394\u03b2=d\u03c9\\Delta \\beta = d\\omega<\/span>\\Delta \\beta = d\\omega<\/code><\/div>\n

                            \u3092\u89e3\u304f\u3002<\/span><\/span><\/div>\n<\/li>\n

                          • \n
                            \u8abf\u548c\u6210\u5206 <\/span><\/span>(<\/span><\/span>h<\/span><\/span>)<\/span><\/span> \u306f\u3001<\/span><\/span><\/p>\n
                            \u0394h=0\\Delta h = 0<\/span>\\Delta h = 0<\/code><\/div>\n

                            \u304b\u3064 <\/span><\/span><\/p>\n

                            \u03c9\u2212d\u03b1\u2212\u03b4\u03b2\\omega – d\\alpha – \\delta\\beta<\/span>\\omega - d\\alpha - \\delta\\beta<\/code><\/div>\n

                            \u304b\u3089\u8a08\u7b97\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n<\/li>\n

                          • \n
                            \u76f4\u4ea4\u6027\u306e\u4fdd\u8a3c<\/span><\/span><\/span>\uff1a<\/span><\/span><\/div>\n
                              \n
                            • \n
                              \n
                              L2L^2<\/span>L^2<\/code><\/div>\n

                              \u5185\u7a4d\u3092\u7528\u3044\u3066\u3001<\/span><\/span><\/p>\n

                              d\u03a9k\u22121d\\Omega^{k-1}<\/span>d\\Omega^{k-1}<\/code><\/div>\n

                              , <\/span><\/span><\/p>\n

                              \u03b4\u03a9k+1\\delta\\Omega^{k+1}<\/span>\\delta\\Omega^{k+1}<\/code><\/div>\n

                              , <\/span><\/span><\/p>\n

                              Hk\\mathcal{H}^k<\/span>\\mathcal{H}^k<\/code><\/div>\n

                              \u304c\u76f4\u4ea4\u3059\u308b\u3002<\/span><\/span><\/div>\n<\/li>\n

                            • \n
                              \u30b3\u30f3\u30d1\u30af\u30c8\u591a\u69d8\u4f53\u3067\u306f\u3001<\/span><\/span><\/p>\n
                              Hk\\mathcal{H}^k<\/span>\\mathcal{H}^k<\/code><\/div>\n

                              \u306e\u6b21\u5143\u306f\u6709\u9650\u3067\u3001\u30d9\u30c3\u30c1\u6570\uff08\u30b3\u30db\u30e2\u30ed\u30b8\u30fc\u7fa4\u306e\u6b21\u5143\uff09\u306b\u7b49\u3057\u3044\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n<\/li>\n

                            • \n
                              \u975e\u30b3\u30f3\u30d1\u30af\u30c8\u306e\u5834\u5408<\/span><\/span><\/span>\uff1a<\/span><\/span><\/div>\n
                                \n
                              • \n
                                \n
                                Rn\\mathbb{R}^n<\/span>\\mathbb{R}^n<\/code><\/div>\n

                                \u3067\u306f\u3001\u30b0\u30ea\u30fc\u30f3\u95a2\u6570\u3084\u30d5\u30fc\u30ea\u30a8\u89e3\u6790\u3092\u7528\u3044\u3066\u5206\u89e3\u3092\u69cb\u7bc9\u3002<\/span><\/span><\/div>\n<\/li>\n

                              • \n
                                \u9069\u5207\u306a\u5883\u754c\u6761\u4ef6\uff08\u4f8b\uff1a\u6e1b\u8870\u6761\u4ef6\uff09\u3067\u4e00\u610f\u6027\u3092\u4fdd\u8a3c\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n
                                \n
                                5. \u30db\u30c3\u30b8\u5206\u89e3\u306e\u5177\u4f53\u4f8b<\/span><\/span><\/div>\n
                                (1) <\/span><\/span><\/p>\n
                                R3\\mathbb{R}^3<\/span>\\mathbb{R}^3<\/code><\/div>\n

                                \u306e\u30d9\u30af\u30c8\u30eb\u5834<\/span><\/span><\/div>\n

                                  \n
                                • \n
                                  \u30d9\u30af\u30c8\u30eb\u5834 <\/span><\/span><\/p>\n
                                  F\\mathbf{F}<\/span>\\mathbf{F}<\/code><\/div>\n

                                  \u3092 <\/span><\/span><\/p>\n

                                  F=\u2207\u03d5+\u2207\u00d7A\\mathbf{F} = \\nabla \\phi + \\nabla \\times \\mathbf{A}<\/span>\\mathbf{F} = \\nabla \\phi + \\nabla \\times \\mathbf{A}<\/code><\/div>\n

                                  \u306b\u5206\u89e3\u3002<\/span><\/span><\/div>\n<\/li>\n

                                • \n
                                  \u3053\u3053\u3067\uff1a<\/span><\/span><\/div>\n
                                    \n
                                  • \n
                                    \n
                                    \u03d5\\phi<\/span>\\phi<\/code><\/div>\n

                                    \u306f <\/span><\/span><\/p>\n

                                    \u0394\u03d5=\u2207\u22c5F\\Delta \\phi = \\nabla \\cdot \\mathbf{F}<\/span>\\Delta \\phi = \\nabla \\cdot \\mathbf{F}<\/code><\/div>\n

                                    \u3092\u89e3\u3044\u3066\u5f97\u3089\u308c\u308b\u3002<\/span><\/span><\/div>\n<\/li>\n

                                  • \n
                                    \n
                                    A\\mathbf{A}<\/span>\\mathbf{A}<\/code><\/div>\n

                                    \u306f <\/span><\/span><\/p>\n

                                    \u0394A=\u2212\u2207\u00d7F\\Delta \\mathbf{A} = -\\nabla \\times \\mathbf{F}<\/span>\\Delta \\mathbf{A} = -\\nabla \\times \\mathbf{F}<\/code><\/div>\n

                                    \u3092\u89e3\u304d\u3001<\/span><\/span><\/p>\n

                                    \u2207\u22c5A=0\\nabla \\cdot \\mathbf{A} = 0<\/span>\\nabla \\cdot \\mathbf{A} = 0<\/code><\/div>\n

                                    \uff08Coulomb gauge\uff09\u3092\u8ab2\u3059\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n<\/li>\n

                                  • \n
                                    \u8abf\u548c\u6210\u5206\u306f <\/span><\/span><\/p>\n
                                    R3\\mathbb{R}^3<\/span>\\mathbb{R}^3<\/code><\/div>\n

                                    \u3067\u306f\u901a\u5e38\u30bc\u30ed\uff08\u5358\u9023\u7d50\u3067\u6e1b\u8870\u6761\u4ef6\u3092\u4eee\u5b9a\uff09\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n

                                    (2) 2\u6b21\u5143\u30c8\u30fc\u30e9\u30b9 <\/span><\/span><\/p>\n
                                    T2T^2<\/span>T^2<\/code><\/div>\n<\/div>\n
                                      \n
                                    • \n
                                      \u30c8\u30fc\u30e9\u30b9\u4e0a\u306e1-\u5f62\u5f0f <\/span><\/span><\/p>\n
                                      \u03c9\\omega<\/span>\\omega<\/code><\/div>\n

                                      \u3092\u5206\u89e3\u3002<\/span><\/span><\/div>\n<\/li>\n

                                    • \n
                                      \n
                                      H1(T2)\u2245R2\\mathcal{H}^1(T^2) \\cong \\mathbb{R}^2<\/span>\\mathcal{H}^1(T^2) \\cong \\mathbb{R}^2<\/code><\/div>\n

                                      \uff08\u30c8\u30fc\u30e9\u30b9\u306e1\u6b21\u30d9\u30c3\u30c1\u6570\uff1d2\uff09\u3002<\/span><\/span><\/div>\n<\/li>\n

                                    • \n
                                      \n
                                      \u03c9=d\u03b1+\u03b4\u03b2+h\\omega = d\\alpha + \\delta\\beta + h<\/span>\\omega = d\\alpha + \\delta\\beta + h<\/code><\/div>\n

                                      , \u3053\u3053\u3067 <\/span><\/span>(<\/span><\/span>h<\/span><\/span>)<\/span><\/span> \u306f\u30c8\u30fc\u30e9\u30b9\u306e\u975e\u81ea\u660e\u306a1\u5468\u671f\u306b\u5bfe\u5fdc\u3059\u308b\u8abf\u548c\u5f62\u5f0f\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n

                                      (3) \u9ad8\u6b21\u5143 <\/span><\/span><\/p>\n
                                      Rn\\mathbb{R}^n<\/span>\\mathbb{R}^n<\/code><\/div>\n<\/div>\n
                                        \n
                                      • \n
                                        \u30d9\u30af\u30c8\u30eb\u5834 <\/span><\/span><\/p>\n
                                        F:Rn\u2192Rn\\mathbf{F}: \\mathbb{R}^n \\to \\mathbb{R}^n<\/span>\\mathbf{F}: \\mathbb{R}^n \\to \\mathbb{R}^n<\/code><\/div>\n

                                        \u3092 <\/span><\/span><\/p>\n

                                        F=\u2207\u03d5+G\\mathbf{F} = \\nabla \\phi + \\mathbf{G}<\/span>\\mathbf{F} = \\nabla \\phi + \\mathbf{G}<\/code><\/div>\n

                                        \uff08<\/span><\/span><\/p>\n

                                        \u2207\u22c5G=0\\nabla \\cdot \\mathbf{G} = 0<\/span>\\nabla \\cdot \\mathbf{G} = 0<\/code><\/div>\n

                                        \uff09\u306b\u5206\u89e3\u3002<\/span><\/span><\/div>\n<\/li>\n

                                      • \n
                                        \n
                                        \u03d5\\phi<\/span>\\phi<\/code><\/div>\n

                                        \u306f <\/span><\/span><\/p>\n

                                        \u0394\u03d5=\u2207\u22c5F\\Delta \\phi = \\nabla \\cdot \\mathbf{F}<\/span>\\Delta \\phi = \\nabla \\cdot \\mathbf{F}<\/code><\/div>\n

                                        \u304b\u3089\u3001<\/span><\/span><\/p>\n

                                        G\\mathbf{G}<\/span>\\mathbf{G}<\/code><\/div>\n

                                        \u306f\u6b8b\u5dee\u3068\u3057\u3066\u8a08\u7b97\u3002<\/span><\/span><\/div>\n<\/li>\n<\/ul>\n

                                        \n
                                        6. \u5fdc\u7528\u4f8b\uff08\u8a73\u7d30\uff09<\/span><\/span><\/div>\n
                                        \u30db\u30c3\u30b8\u5206\u89e3\u306f\u3001\u524d\u8ff0\u306e\u9ad8\u6b21\u5143\u5fdc\u7528\uff08\u8d85\u5f26\u7406\u8ad6\u3001\u6d41\u4f53\u529b\u5b66\u3001\u30c7\u30fc\u30bf\u89e3\u6790\u306a\u3069\uff09\u306e\u57fa\u76e4\u3067\u3059\u3002\u4ee5\u4e0b\u306b\u5177\u4f53\u7684\u306a\u8a73\u7d30\u3092\u88dc\u8db3\uff1a<\/span><\/span><\/div>\n
                                          \n
                                        1. \n
                                          \u8d85\u5f26\u7406\u8ad6<\/span><\/span><\/span>\uff1a<\/span><\/span><\/div>\n