{"id":2071,"date":"2022-08-21T13:50:53","date_gmt":"2022-08-21T05:50:53","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=2071"},"modified":"2025-02-26T11:16:01","modified_gmt":"2025-02-26T03:16:01","slug":"winding_number_non-closed_curve","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/08\/21\/winding_number_non-closed_curve\/","title":{"rendered":"\u975e\u95ed\u66f2\u7ebf\u4e0a\u7684\u5708\u6570"},"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>\u4e66\u4e0aP241\u5bf9\u4e8e\u975e\u95ed\u66f2\u7ebf\u4e0a\u7684\u5708\u6570\u5b9a\u4e49\u4e00\u5e26\u800c\u8fc7, \u672c\u6587\u4e88\u4ee5\u8865\u5145.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/math.stackexchange.com\/questions\/3575488\/winding-number-on-non-closed-curve\">Winding Number on non-closed curve<\/a><br \/>\n2. <a href=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/08\/The-Winding-Number.pdf\">The Winding Number<\/a><br \/>\n3. <a href=\"https:\/\/www.hellocharlie.top\/2020\/04\/20\/%E5%8D%95%E4%BD%8D%E9%80%9F%E5%BA%A6%E5%8F%82%E6%95%B0%E5%8C%96\/\">\u5355\u4f4d\u901f\u5ea6\u53c2\u6570\u5316<\/a><\/p>\n<p><strong>1. \u95ed\u66f2\u7ebf\u4e0a\u7684\u5708\u6570<\/strong><\/p>\n<p><strong>\u5b9a\u4e491<\/strong> $C^*$\u4e0a\u7684\u5e45\u89d2\u51fd\u6570Arg\u4e3a\u4e00\u4e2a\u591a\u503c\u51fd\u6570, \u5176\u5b9a\u4e49\u4e3a$$Arg \\ z = \\{ \\theta \\in R | e^{i \\theta} = \\frac{z}{|z|} \\}.$$<strong>\u5b9a\u4e492<\/strong> \u4ee4$a \\in C$, $\\gamma$\u4e3a$C \\backslash \\{ a \\}$\u4e0a\u7684\u4e00\u6761\u9053\u8def, $\\theta_0 \\in R$\u4e3a$Arg(\\gamma(0) &#8211; $$ a)$\u4e2d\u7684\u4e00\u4e2a\u503c:$$\\theta_0 \\in Arg(\\gamma(0) &#8211; a).$$\u5219\u5b58\u5728\u4e00\u4e2a\u552f\u4e00\u7684\u8fde\u7eed\u51fd\u6570$\\theta: [0, 1] \\mapsto R$\u4f7f\u5f97$\\theta(0) = \\theta_0$, \u6211\u4eec\u79f0$\\theta$\u4e3a$z \\mapsto $$ Arg(z &#8211; a)$\u5173\u4e8e$\\gamma$\u7684Continuous Choice:$$\\forall t \\in [0, 1], \\theta(t) \\in Arg(\\gamma(t) &#8211; a).$$<\/p>\n<p><strong>\u5b9a\u4e493<\/strong> \u4ee4$a \\in C$, $\\gamma$\u4e3a$C \\backslash \\{ a \\}$\u4e0a\u7684\u4e00\u6761\u9053\u8def, \u5219$z \\mapsto Arg(z &#8211; a)$\u5173\u4e8e$\\gamma$\u7684\u53d8\u5206\u7684\u5b9a\u4e49\u4e3a$$[z \\mapsto Arg(z &#8211; a)]_\\gamma = \\theta(1) &#8211; \\theta(0),$$\u5176\u4e2d, $\\theta$\u4e3a$z \\mapsto $$ Arg(z &#8211; a)$\u5173\u4e8e$\\gamma$\u7684Continuous Choice.<\/p>\n<p><strong>\u5b9a\u4e494<\/strong> \u4ee4$a \\in C$, $\\gamma$\u4e3a$C \\backslash \\{ a \\}$\u4e0a\u7684\u4e00\u6761\u9053\u8def, \u5219$\\gamma$\u7ed5$a$\u7684\u5708\u6570\u4e3a\u4e00\u6574\u6570, \u5176\u5b9a\u4e49\u4e3a$$ind(\\gamma, a) = \\frac{1}{2\\pi}[z \\mapsto Arg(z &#8211; a)]_\\gamma.$$<\/p>\n<p><strong>2. \u975e\u95ed\u66f2\u7ebf\u4e0a\u7684\u5708\u6570<\/strong><\/p>\n<p>\u53c2\u8003\u6750\u65991\u4e2d\u63d0\u5230\u4e86\u4e00\u79cd\u975e\u95ed\u66f2\u7ebf\u4e0a\u7684\u4e0d\u4f9d\u8d56\u4e8e\u8d77\u70b9\u9009\u53d6\u7684\u5708\u6570\u5b9a\u4e49, \u4ece\u76f4\u89c2\u4e0a\u6765\u770b, \u5b83\u662f\u4e00\u4e2a\u4eba\u6cbf\u7740\u66f2\u7ebf\u884c\u8d70\u7684\u65cb\u8f6c\u6570, \u4e14\u5728\u4eba\u4ece\u66f2\u7ebf\u7684\u4e00\u7aef\u884c\u8d70\u5230\u53e6\u5916\u4e00\u7aef\u7684\u8fc7\u7a0b\u4e2d, \u4eba\u7684\u671d\u5411\u4e0e\u5176\u8fd0\u52a8\u65b9\u5411\u603b\u662f\u4fdd\u6301\u4e00\u81f4\u7684. \u6570\u5b66\u4e0a\u7684\u5b9a\u4e49\u5982\u4e0b\u6240\u793a.<\/p>\n<p><strong>\u5b9a\u4e495<\/strong> \u5bf9\u4e8e\u7b1b\u5361\u5c14\u5750\u6807\u7cfb$(x, y)$, \u7ed9\u5b9a\u4e00\u6761\u66f2\u7ebf$a$\u7684\u5355\u4f4d\u901f\u5ea6\u53c2\u6570\u5316$r(t) = (x( $$ t), y(t))$, \u5176\u4e2d, $r(0)$\u4e0e$r(1)$\u5206\u522b\u8868\u793a\u66f2\u7ebf$a$\u7684\u4e24\u4e2a\u7aef\u70b9. \u8003\u8651\u901f\u5ea6\u5411\u91cf$\\dot{r} = (\\dot{x}( $$ t), \\dot{y}(t))$, \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u8fde\u7eed\u51fd\u6570$\\alpha(t)$, \u5b83\u63cf\u8ff0\u4e86\u5728\u65f6\u523b$t$\u5904\u7684\u8fd0\u52a8\u65b9\u5411, \u4f7f\u5f97$\\dot{r}  = (cos \\alpha (t), sin \\alpha (t))$. \u7531\u4e8e\u8fd0\u52a8\u65b9\u5411\u968f\u65f6\u95f4$dt$\u53d8\u5316\u7684\u89d2\u5ea6$d\\alpha$\u4e0e\u5750\u6807\u7cfb\u7684\u9009\u62e9\u65e0\u5173, \u6545\u5176\u5bfc\u6570$\\frac{d\\alpha}{dt}$\u662f\u552f\u4e00\u7684, \u4ece\u800c\u4e0d\u540c\u5f62\u5f0f\u7684$\\alpha(t)$\u4e4b\u95f4\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570. \u7531\u6b64, \u6211\u4eec\u7ed9\u51fa\u66f2\u7ebf$a$\u4e0a\u7684\u5708\u6570\u5b9a\u4e49:$$\\frac{1}{2\\pi} \\int_{0}^{T} \\frac{d\\alpha}{dt} = \\frac{\\alpha(1) &#8211; \\alpha(0)}{2\\pi}.$$\u8be5\u5708\u6570\u5b9a\u4e49\u5728\u4efb\u4f55\u66f2\u7ebf(\u65e0\u8bba\u662f\u5f00\u66f2\u7ebf\u8fd8\u662f\u95ed\u66f2\u7ebf) \u4e0a\u5747\u662fWell-Defined\u7684, \u5e76\u4e14\u4e0d\u4f9d\u8d56\u4e8e\u5750\u6807\u7cfb\u6216\u8005\u539f\u70b9\u7684\u9009\u53d6. \u5728\u8fd9\u79cd\u5708\u6570\u7684\u5b9a\u4e49\u4e0b, \u95ed\u66f2\u7ebf\u7684\u5708\u6570\u603b\u4e3a\u4e00\u4e2a\u6574\u6570, \u800c\u5f00\u66f2\u7ebf\u7684\u5708\u6570\u5219\u4e0d\u4e00\u5b9a\u4e3a\u4e00\u4e2a\u6574\u6570. \u4f8b\u5982, \u6781\u5750\u6807\u7cfb$(r, \\theta)$\u4e0b\u7531$r = \\theta$\u63cf\u8ff0\u7684\u5f00\u66f2\u7ebf(\u87ba\u7ebf) \u7684\u5708\u6570\u4e3a:<br \/>\n$\\cdot$ 1.5, $2\\pi \\le \\theta \\le 5\\pi$.<br \/>\n$\\cdot$ 2, $2\\pi \\le \\theta \\le 6\\pi$.<\/p>\n<p><strong>3. \u4ece\u590d\u8fed\u6620\u5c04\u7684\u89d2\u5ea6\u7406\u89e3\u5708\u6570<\/strong><\/p>\n<p>\u8003\u8651\u4e00\u4e2a\u628a\u76f4\u7ebf\u65e0\u7a77\u5730&#8221;\u7f20\u7ed5&#8221; \u5728\u66f2\u7ebf$a$\u4e0a\u7684\u590d\u8fed\u6620\u5c04$$p: E^1 \\to a, t \\mapsto (x(t), y(t)).$$\u5728\u8be5\u590d\u8fed\u6620\u5c04$p$\u4e0b, $a^\\uparrow = \\frac{\\alpha}{2\\pi}$, \u5219\u4e0a\u8ff0\u5708\u6570\u7684\u5b9a\u4e49\u5373\u4e3a$a^\\uparrow(1) &#8211; a^\\uparrow(0)$. \u7531\u4e8e\u4e0d\u540c\u5f62\u5f0f\u7684$\\alpha(t)$\u4e4b\u95f4\u4ec5\u76f8\u5dee\u4e00\u4e2a\u5e38\u6570, \u6545\u5f53\u9009\u53d6\u4e0d\u540c\u7684\u63d0\u5347\u8d77\u70b9$a^\\uparrow(0)$(\u53ef\u89c6\u4e3a\u76f8\u5dee\u4e00\u4e2a\u5e73\u79fb) \u65f6, \u76f8\u5e94\u7684\u63d0\u5347\u7ec8\u70b9$a^\\uparrow(1)$\u4e5f\u76f8\u5dee\u4e00\u4e2a\u76f8\u540c\u5e73\u79fb\u91cf\u7684\u5e73\u79fb.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e66\u4e0aP241\u5bf9\u4e8e\u975e\u95ed\u66f2\u7ebf\u4e0a\u7684\u5708\u6570\u5b9a\u4e49\u4e00\u5e26\u800c\u8fc7, \u672c\u6587\u4e88\u4ee5\u8865\u5145.<\/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\/2071"}],"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=2071"}],"version-history":[{"count":42,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2071\/revisions"}],"predecessor-version":[{"id":3624,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/2071\/revisions\/3624"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2071"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2071"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2071"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}