{"id":2409,"date":"2011-10-25T21:18:53","date_gmt":"2011-10-25T13:18:53","guid":{"rendered":"http:\/\/vanabel.sinaapp.com\/?p=102"},"modified":"2011-10-25T21:18:53","modified_gmt":"2011-10-25T13:18:53","slug":"%e4%bc%aa%e7%90%83%e9%9d%a2%e5%8f%8a%e5%85%b6%e6%b5%8b%e5%9c%b0%e7%ba%bf","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=2409","title":{"rendered":"\u4f2a\u7403\u9762\u53ca\u5176\u6d4b\u5730\u7ebf"},"content":{"rendered":"

\u4f2a\u7403\u9762\u7684\u53c2\u6570\u65b9\u7a0b\u53ca\u5176\u9ad8\u65af\u66f2\u7387<\/h4>\n

\u4f2a\u7403\u9762(\u5982\u4e0b\u56fe)
\n\"\"<\/a>
\n\"\u62fd\u7269\u7ebf\"<\/a><\/p>\n

\u5b83\u662f\u7531\u62fd\u7269\u7ebf:
\n\\[
\nf(x)=\\rm{ArcCosh}\\left(
\n\\frac{1}{x}\\right) – \\sqrt{1 – x^2},\\quad x\\in[0,1],
\n\\]
\n\u7ed5$z$\u8f74\u65cb\u8f6c\u800c\u6765.<\/p>\n

(\u5982\u53f3\u56fe)\u800c\u6240\u8c13\u62fd\u7269\u7ebf\u662f\u6ee1\u8db3\u5982\u4e0b”\u62fd\u7269\u65b9\u7a0b”\u7684\u66f2\u7ebf:
\n\\[
\ny’=-\\frac{\\sqrt{1-x^2}}{x},
\n\\]
\n\u5373\u5176\u4e0a\u6bcf\u4e00\u70b9\u7684\u5207\u7ebf\u4e0e $y$ \u8f74\u7684\u4ea4\u70b9\u5230\u8be5\u70b9\u7684\u8ddd\u79bb\u4e3a\u5e38\u65701. (\u5982\u56fe\u4e2d\u7c97\u9ed1\u7ebf\u6240\u793a)<\/p>\n

\u8fd9\u6837\u4f2a\u7403\u9762\u7684\u53c2\u6570\u65b9\u7a0b\u53ef\u5199\u4e3a:
\n\\[
\nr(x,u)=\\left\\{x\\cos u,x\\sin u,\\rm{ArcCosh}\\left(\\frac{1}{x}\\right)- \\sqrt{1 – x^2}\\right\\}.
\n\\]
\n\u6839\u636e\u65cb\u8f6c\u66f2\u9762$\\left\\{x\\cos u,x\\sin u, f(x)\\right\\}$\u7684\u9ad8\u65af\u66f2\u7387\u8ba1\u7b97\u516c\u5f0f
\n\\[
\n\\frac{f’f”}{x\\left(1+(f’)^2\\right)^2},
\n\\]
\n\u53ef\u4ee5\u5f97\u5230\u4f2a\u7403\u9762\u7684\u9ad8\u65af\u66f2\u7387\u4e3a\u5e38\u6570-1.<\/strong>
\n\u5b83\u662f\u5e38\u8d1f\u66f2\u7387\u7684\u4e00\u4e2a\u6a21\u578b, \u6ce8\u610f\u5230\u5b83\u662f\u4e0d\u5b8c\u5907\u7684, \u4e8b\u5b9e\u4e0aHilbert\u8bc1\u660e\u4e86$\\mathbf{R}^3$\u4e2d\u4e0d\u5b58\u5728\u5b8c\u5907\u7684\u5d4c\u5165\u5e38\u8d1f\u9ad8\u65af\u66f2\u7387\u66f2\u9762. <\/strong><\/p>\n

\u4f2a\u7403\u9762\u7684\u6d4b\u5730\u7ebf<\/h4>\n

\u76f4\u63a5\u8ba1\u7b97\u4f2a\u7403\u9762\u7684\u6cd5\u5411$\\mathbf{n}$(\u975e\u5355\u4f4d)\u4e3a:
\n\\[
\n\\left\\{
\n-\\sqrt{1-x^2}\\cos u,-\\sqrt{1-x^2}\\sin u,-x\\right\\}.
\n\\]
\n\u8fd9\u6837, \u82e5\u4f2a\u7403\u9762\u4e0a\u7684\u66f2\u7ebf $r(t)=X(x(t),u(t))$\u4e3a\u6d4b\u5730\u7ebf, \u90a3\u4e48\u5b83\u5e94\u6ee1\u8db3\u5982\u4e0b\u7684\u6d4b\u5730\u7ebf\u65b9\u7a0b
\n\\[
\n\\frac{\\partial^2r(t)}{\\partial t^2}.\\mathbf n=0
\n\\]
\n\u5c06\u66f2\u7ebf\u65b9\u7a0b\u4ee3\u5165\u5e76\u6574\u7406\u5f97\u5230
\n\\[
\nx(t)^2(x(t)^2-1)u'(t)^2+x'(t)^2=0.\\tag{1}
\n\\]
\n\u7279\u522b\u5730, \u4ee4 $x(t)=t, t\\in[0,1]$, \u5219\u53ef\u89e3\u5f97
\n\\[
\nu(t)=\\mathbf{Re}\\left(
\n\\rm{ArcCoth\\sqrt{1-t^2}}\\right)
\n=\\frac{1}{2}\\ln\\frac{-t^2+2\\sqrt{1-t^2}+2}{t^2}.
\n\\]
\n\u4e8e\u662f, \u6211\u4eec\u53ef\u4ee5\u5177\u4f53\u7684\u6c42\u51fa\u8fd9\u6761\u6d4b\u5730\u7ebf. \u4f5c\u56fe\u5982\u4e0b:
\n\"\"<\/a>
\n\u4e8b\u5b9e\u4e0a, \u4ece\u6d4b\u5730\u7ebf$X(t)$\u5e94\u6ee1\u8db3\u7684\u65b9\u7a0b(1)\u8fd8\u53ef\u770b\u51fa, \u4f2a\u7403\u9762\u4e0a\u7684\u7ecf\u7ebf\u548c\u7eac\u7ebf\u90fd\u4e0d\u662f\u6d4b\u5730\u7ebf<\/strong>.<\/p>\n","protected":false},"excerpt":{"rendered":"

\u4f2a\u7403\u9762\u7684\u53c2\u6570\u65b9\u7a0b\u53ca\u5176\u9ad8\u65af\u66f2\u7387 \u4f2a\u7403\u9762(\u5982\u4e0b\u56fe)<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[166,131,800,769,402],"class_list":["post-2409","post","type-post","status-publish","format-standard","hentry","category-mathnotes","tag-166","tag-131","tag-lecture-notes","tag-769","tag-402"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2409","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2409"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/2409\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2409"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2409"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2409"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}