{"id":425,"date":"2017-08-02T14:26:12","date_gmt":"2017-08-02T14:26:12","guid":{"rendered":"https:\/\/blog.vanabel.cn\/?p=425"},"modified":"2017-08-02T14:33:27","modified_gmt":"2017-08-02T14:33:27","slug":"jizuobiaoxiaqiexianggongbianweifendeyigefenbujifengongshi","status":"publish","type":"post","link":"https:\/\/blog.vanabel.cn\/?p=425","title":{"rendered":"\u6781\u5750\u6807\u4e0b\u5207\u5411\u5171\u53d8\u5fae\u5206\u7684\u4e00\u4e2a\u5206\u90e8\u79ef\u5206\u516c\u5f0f"},"content":{"rendered":"

\u5047\u8bbe$A$\u662f$G$-\u4e3b\u4e1b$P$\u7684\u4e00\u4e2a\u8054\u7edc. \u7279\u522b\u5730, \u6211\u4eec\u5047\u8bbe$G$\u662f\u7d27\u674e\u7fa4, \u8fd9\u6837\u4ed6\u53ef\u89c6\u4e3a\u67d0\u4e2a\u6b63\u4ea4\u7fa4\u7684\u5b50\u7fa4, \u8fdb\u800c\u5c40\u90e8\u5730, \u5c06$A$\u89c6\u4e3a$\\mathfrak{g}$-\u503c\u76841-\u5f62\u5f0f\u65f6, $A$\u662f\u53cd\u5bf9\u79f0\u77e9\u9635. \u73b0\u5728\u5047\u8bbe$u,v$\u662f$P$\u7684\u5c40\u90e8\u622a\u9762, \u5219\u5728\u6781\u5750\u6807\u7cfb\u4e0b, $A=A_rdr+A_\\theta d\\theta$ \u4ee5\u53ca
\n$$
\n\t\\nabla_A u=(d+A)u=u_{|r}dr+u_{|\\theta}d\\theta,\\quad u_{|r}=\\partial_r u+A_ru,\\quad u_{|\\theta}=\\partial_\\theta u+A_\\theta u.
\n$$
\n\u73b0\u5728, \u6211\u4eec\u5173\u4e8e$u_{|\\theta}v_{|\\theta}$\u6709\u5982\u4e0b\u7684\u5206\u90e8\u79ef\u5206\u516c\u5f0f.
\n$$
\n\\int_{S^1}u_{|\\theta}\\cdot v_{|\\theta}=-\\int_{S^1}u_{|\\theta\\theta}\\cdot v.
\n$$
\n<\/p>\n

\u4e8b\u5b9e\u4e0a, \u76f4\u63a5\u8ba1\u7b97\u53ef\u77e5
\n\\begin{align*}
\n\tu_{|\\theta}v_{|\\theta}&=(u_\\theta+A_\\theta u)\\cdot(v_\\theta+A_\\theta v)\\\\
\n\t&=\\partial_\\theta(u_\\theta\\cdot v)-u_{\\theta\\theta}\\cdot v+A_\\theta u\\cdot v_\\theta+u_\\theta\\cdot A_\\theta v+A_\\theta u\\cdot A_\\theta v\\\\
\n\t&=\\partial_\\theta(u_\\theta \\cdot v)-u_{\\theta\\theta}\\cdot v+\\partial_\\theta(A_\\theta u\\cdot v)-\\partial_\\theta(A_\\theta u)\\cdot v\\\\
\n &\\qquad+u_\\theta\\cdot A_\\theta v+A_\\theta u\\cdot A_\\theta v\\\\
\n\tu_{|\\theta\\theta}\\cdot v&=\\partial_\\theta(u_\\theta+A_\\theta u)\\cdot v+A_\\theta(u_\\theta+A_\\theta u)\\cdot v\\\\
\n\t&=u_{\\theta\\theta}\\cdot v+\\partial_\\theta(A_\\theta u)\\cdot v+A_\\theta u_\\theta\\cdot v+A_\\theta^2u\\cdot v\\\\
\n\t&=u_{\\theta\\theta}\\cdot v+\\partial_\\theta(A_\\theta u)\\cdot v-u_\\theta\\cdot A_\\theta v-A_\\theta u\\cdot A_\\theta v
\n\\end{align*}
\n\u6700\u540e\u4e00\u4e2a\u7b49\u53f7\u5904, \u6211\u4eec\u7528\u4e86$A_\\theta=-A_\\theta^T$.<\/p>\n

\u8fd9\u6837, \u6211\u4eec\u5f97\u5230
\n$$
\nu_{|\\theta}\\cdot v_{|\\theta}=\\partial_\\theta(u_\\theta\\cdot v+A_\\theta u\\cdot v)-u_{|\\theta\\theta}\\cdot v,
\n$$
\n\u7531\u6b64, \u6613\u89c1\u7ed3\u8bba\u6210\u7acb.<\/p>\n

\u6700\u540e, \u6211\u4eec
\n

\u6ce8\u8bb0 1<\/span>.<\/span> \u4e00\u822c\u800c\u8a00, \u4e0d\u4e00\u5b9a\u6709
\n$$
\n\\partial_ru_{|\\theta}=u_{r|\\theta}.
\n$$
\n<\/div>
\n\u4e8b\u5b9e\u4e0a,
\n$$
\n\\partial_ru_{|\\theta}=\\partial_r(u_\\theta+A_\\theta u)=u_{r\\theta}+\\partial_rA_\\theta u+A_\\theta u_r=u_{r|\\theta}+\\partial_rA_\\theta u.
\n$$<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5047\u8bbe$A$\u662f$G$-\u4e3b\u4e1b$P$\u7684\u4e00\u4e2a\u8054\u7edc. \u7279\u522b\u5730, \u6211\u4eec\u5047\u8bbe$G$\u662f\u7d27\u674e\u7fa4, \u8fd9\u6837\u4ed6\u53ef\u89c6\u4e3a\u67d0\u4e2a\u6b63\u4ea4\u7fa4\u7684\u5b50\u7fa4,… \u7ee7\u7eed\u9605\u8bfb\u6781\u5750\u6807\u4e0b\u5207\u5411\u5171\u53d8\u5fae\u5206\u7684\u4e00\u4e2a\u5206\u90e8\u79ef\u5206\u516c\u5f0f<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[87,88,38],"class_list":["post-425","post","type-post","status-publish","format-standard","hentry","category-math","tag-gongbianweifen","tag-fenbujifen","tag-38","entry"],"_links":{"self":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/425","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=425"}],"version-history":[{"count":5,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/425\/revisions"}],"predecessor-version":[{"id":430,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/425\/revisions\/430"}],"wp:attachment":[{"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=425"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=425"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=425"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}