{"id":1652,"date":"2022-03-27T21:09:44","date_gmt":"2022-03-27T13:09:44","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1652"},"modified":"2025-02-26T11:04:06","modified_gmt":"2025-02-26T03:04:06","slug":"degree_mapping","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/03\/27\/degree_mapping\/","title":{"rendered":"\u6620\u5c04\u5ea6"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u8fd9\u5468\u662f\u4f9d\u65e7\u5b85\u5bb6\u7684\u4e00\u5468, \u4e0b\u5468\u4fbf\u662f\u6e05\u660e\u5047\u671f\u4e86, \u53bb\u5e74\u53bb\u4e86\u5357\u4eac\u73a9, \u4eca\u5e74\u7531\u4e8e\u75ab\u60c5\u4f30\u8ba1\u54ea\u4e5f\u53bb\u4e0d\u4e86\u4e86, \u5c31\u4e56\u4e56\u5446\u5728\u5b81\u6ce2\u73a9\u53ed~ \u6620\u5c04\u5ea6\u662f\u5728\u5b66\u4e60\u57fa\u672c\u7fa4\u7684\u5e94\u7528\u65f6\u770b\u5230\u7684\u4e00\u4e2a\u6982\u5ff5, \u53ef\u4ee5\u6781\u5927\u5730\u7b80\u5316\u5f88\u591a\u5b9a\u7406\u7684\u8bc1\u660e\u8fc7\u7a0b, \u5982Brouwer\u4e0d\u52a8\u70b9\u5b9a\u7406\u4e0e\u4ee3\u6570\u57fa\u672c\u5b9a\u7406.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/159012125\">\u4ece\u73af\u7ed5\u6570\u5230\u6620\u5c04\u5ea6\uff08\u4e00\uff09<\/a><br \/>\n2. <a href=\"https:\/\/www.zhihu.com\/question\/56105919\/answer\/147788473\">\u4ec0\u4e48\u662f\u8986\u53e0\u6620\u5c04(covering map)\uff1f\u8986\u53e0\u6620\u5c04\u6709\u54ea\u4e9b\u6027\u8d28\uff1f<\/a><\/p>\n<p><strong>\u5b9a\u4e49<\/strong> \u8bbe\u8fde\u7eed\u6620\u5c04$f: S^1 \\to S^1$\u6ee1\u8db3$$f_\\pi(\\left \\langle a_{1, 0} \\right \\rangle) = \\left \\langle a_{1, \\theta} \\right \\rangle  ^n\uff0c$$\u5219\u79f0$n$\u4e3a$f$\u7684\u6620\u5c04\u5ea6, \u8bb0\u4e3a$deg\\ f$. \u5176\u4e2d, $a_{n, \\theta}$\u662f\u4ee5$x_0 = (cos\\theta, sin\\theta) $$ \\in S^1$\u4e3a\u57fa\u70b9\u7684\u7ed5\u5706\u5468$n$\u5708\u7684\u95ed\u9053\u8def:$$a_{n, \\theta} : [0, 1] \\to S^1, t \\mapsto (cos(2\\pi nt + \\theta), sin(2\\pi nt + \\theta)).$$\u6b64\u5916, $\\left \\langle a_{n, \\theta} \\right \\rangle = \\left \\langle a_{1, \\theta} \\right \\rangle ^n$.<\/p>\n<p>\u4ece\u6620\u5c04\u5ea6\u7684\u5b9a\u4e49\u51fa\u53d1, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u5341\u5206\u76f4\u89c2\u7684\u51e0\u4f55\u89e3\u91ca: \u8fde\u7eed\u6620\u5c04$f$\u53ef\u4ee5\u770b\u4f5c\u5706\u4e0a\u7684\u4e00\u4e2a\u5c01\u95ed\u8def\u5f84, \u82e5$deg\\ f = n$, \u5219\u8bf4\u660e\u6cbf\u7740\u8def\u5f84\u8d70\u5b8c\u56de\u5230\u51fa\u53d1\u70b9\u540e, \u7ed5&#8221;\u4e2d\u5fc3&#8221;\u8f6c\u4e86$n$\u5708.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8fd9\u5468\u662f\u4f9d\u65e7\u5b85\u5bb6\u7684\u4e00\u5468, \u4e0b\u5468\u4fbf\u662f\u6e05\u660e\u5047\u671f\u4e86, \u53bb\u5e74\u53bb\u4e86\u5357\u4eac\u73a9, \u4eca\u5e74\u7531\u4e8e\u75ab\u60c5\u4f30\u8ba1\u54ea\u4e5f\u53bb\u4e0d\u4e86\u4e86, \u5c31\u4e56\u4e56\u5446\u5728\u5b81\u6ce2 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/03\/27\/degree_mapping\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u6620\u5c04\u5ea6<\/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":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1652"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1652"}],"version-history":[{"count":11,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1652\/revisions"}],"predecessor-version":[{"id":3610,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1652\/revisions\/3610"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1652"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1652"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1652"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}