{"id":61053,"date":"2023-11-11T18:39:25","date_gmt":"2023-11-11T10:39:25","guid":{"rendered":"https:\/\/docs.pingcode.com\/?p=61053"},"modified":"2023-11-11T18:39:28","modified_gmt":"2023-11-11T10:39:28","slug":"%e6%9b%b2%e9%9d%a2%e6%b3%95%e5%90%91%e9%87%8f%e5%92%8c%e6%9b%b2%e7%ba%bf%e5%88%87%e5%90%91%e9%87%8f%e6%9c%89%e4%bb%80%e4%b9%88%e5%8c%ba%e5%88%ab","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/61053.html","title":{"rendered":"\u66f2\u9762\u6cd5\u5411\u91cf\u548c\u66f2\u7ebf\u5207\u5411\u91cf\u6709\u4ec0\u4e48\u533a\u522b"},"content":{"rendered":"\n<blockquote class=\"wp-block-quote\">\n<p>\u66f2\u9762\u6cd5\u5411\u91cf\u548c\u66f2\u7ebf\u5207\u5411\u91cf\u7684\u533a\u522b\uff1a1.\u66f2\u9762\u6cd5\u5411\u91cf\u7684\u5b9a\u4e49\u548c\u8ba1\u7b97\uff1b2.\u66f2\u7ebf\u5207\u5411\u91cf\u7684\u5b9a\u4e49\u548c\u8ba1\u7b97\uff1b3.\u66f2\u9762\u6cd5\u5411\u91cf\u548c\u66f2\u7ebf\u5207\u5411\u91cf\u7684\u7269\u7406\u610f\u4e49\uff1b4.\u5e94\u7528\u9886\u57df\u7684\u5dee\u5f02\u3002\u66f2\u7ebf\u5207\u5411\u91cf\u5219\u6307\u6cbf\u66f2\u7ebf\u8fd0\u52a8\u7684\u77ac\u65f6\u65b9\u5411\uff0c\u5f3a\u8c03\u7684\u662f\u66f2\u7ebf\u5728\u8be5\u70b9\u7684\u8fd0\u52a8\u8d8b\u52bf\u3002<\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\">1.\u66f2\u9762\u6cd5\u5411\u91cf\u7684\u5b9a\u4e49\u548c\u8ba1\u7b97<\/h2>\n\n\n\n<p>\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u66f2\u9762\u53ef\u4ee5\u901a\u8fc7\u4e00\u4e2a\u4e8c\u5143\u51fd\u6570\u6765\u63cf\u8ff0\uff0c\u6bd4\u5982 f(x, y, z) = 0<em>f<\/em>(<em>x<\/em>,<em>y<\/em>,<em>z<\/em>)=0\u3002\u5728\u8fd9\u4e2a\u66f2\u9762\u4e0a\u7684\u4efb\u610f\u4e00\u70b9\uff0c\u90fd\u6709\u4e00\u4e2a\u552f\u4e00\u786e\u5b9a\u7684\u6cd5\u5411\u91cf\uff0c\u5b83\u5782\u76f4\u4e8e\u901a\u8fc7\u8be5\u70b9\u7684\u66f2\u9762\u7684\u5207\u5e73\u9762\u3002\u5728\u6570\u5b66\u4e0a\uff0c\u8fd9\u4e2a\u6cd5\u5411\u91cf\u53ef\u4ee5\u901a\u8fc7\u66f2\u9762\u4e0a\u7684\u68af\u5ea6\u6765\u8ba1\u7b97\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5982\u679c\u66f2\u9762\u7531\u9690\u51fd\u6570 f(x, y, z) = 0<em>f<\/em>(<em>x<\/em>,<em>y<\/em>,<em>z<\/em>)=0 \u7ed9\u51fa\uff0c\u90a3\u4e48\u6cd5\u5411\u91cf \\mathbf{n}<strong>n<\/strong> \u5728\u70b9 (x_0, y_0, z_0)(<em>x<\/em>0\u200b,<em>y<\/em>0\u200b,<em>z<\/em>0\u200b) \u53ef\u4ee5\u8868\u793a\u4e3a\u68af\u5ea6 \\nabla f\u2207<em>f<\/em> \u5728\u8be5\u70b9\u7684\u503c\uff0c\u5373\uff1a \\mathbf{n} = \\nabla f(x_0, y_0, z_0) = \\left(\\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z}\\right) \\Bigg|_{(x_0, y_0, z_0)}<strong>n<\/strong>=\u2207<em>f<\/em>(<em>x<\/em>0\u200b,<em>y<\/em>0\u200b,<em>z<\/em>0\u200b)=(\u2202<em>x<\/em>\u2202<em>f<\/em>\u200b,\u2202<em>y<\/em>\u2202<em>f<\/em>\u200b,\u2202<em>z<\/em>\u2202<em>f<\/em>\u200b)\u2223\u2223\u200b(<em>x<\/em>0\u200b,<em>y<\/em>0\u200b,<em>z<\/em>0\u200b)\u200b<br>\u5bf9\u4e8e\u53c2\u6570\u5316\u66f2\u9762\uff0c\u4f8b\u5982\u7531\u5411\u91cf\u503c\u51fd\u6570 \\mathbf{r}(u, v)<strong>r<\/strong>(<em>u<\/em>,<em>v<\/em>) \u63cf\u8ff0\u7684\u66f2\u9762\uff0c\u6cd5\u5411\u91cf\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u4e24\u4e2a\u504f\u5bfc\u6570\u5411\u91cf\u7684\u53c9\u79ef\u6765\u83b7\u5f97\uff1a \\mathbf{n} = \\frac{\\partial \\mathbf{r}}{\\partial u} \\times \\frac{\\partial \\mathbf{r}}{\\partial v}<strong>n<\/strong>=\u2202<em>u<\/em>\u2202<strong>r<\/strong>\u200b\u00d7\u2202<em>v<\/em>\u2202<strong>r<\/strong>\u200b<br><br><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">2.\u66f2\u7ebf\u5207\u5411\u91cf\u7684\u5b9a\u4e49\u548c\u8ba1\u7b97<\/h2>\n\n\n\n<p>\u4e0e\u66f2\u9762\u6cd5\u5411\u91cf\u4e0d\u540c\uff0c\u66f2\u7ebf\u5207\u5411\u91cf\u63cf\u8ff0\u7684\u662f\u66f2\u7ebf\u4e0a\u4e00\u70b9\u7684\u5207\u7ebf\u65b9\u5411\u3002\u5bf9\u4e8e\u53c2\u6570\u5316\u8868\u793a\u7684\u66f2\u7ebf \\mathbf{r}(t)<strong>r<\/strong>(<em>t<\/em>)\uff0c\u5728\u53c2\u6570 t<em>t<\/em> \u7684\u67d0\u4e00\u503c\u5904\uff0c\u66f2\u7ebf\u7684\u5207\u5411\u91cf\u53ef\u4ee5\u901a\u8fc7\u6c42\u5bfc\u6570\u83b7\u5f97\uff1a \\mathbf{T} = \\frac{d\\mathbf{r}}{dt} \\Bigg|_{t_0}<strong>T<\/strong>=<em>dtd<\/em><strong>r<\/strong>\u200b\u2223\u2223\u200b<em>t<\/em>0\u200b\u200b<\/p>\n\n\n\n<p>\u5207\u5411\u91cf\u63d0\u4f9b\u4e86\u66f2\u7ebf\u5728\u70b9 \\mathbf{r}(t_0)<strong>r<\/strong>(<em>t<\/em>0\u200b) \u5904\u7684\u77ac\u65f6\u8fd0\u52a8\u65b9\u5411\uff0c\u662f\u7814\u7a76\u66f2\u7ebf\u5c40\u90e8\u6027\u8d28\u7684\u91cd\u8981\u5de5\u5177\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">3.\u66f2\u9762\u6cd5\u5411\u91cf\u548c\u66f2\u7ebf\u5207\u5411\u91cf\u7684\u7269\u7406\u610f\u4e49<\/h2>\n\n\n\n<p>\u5728\u7269\u7406\u5b66\u4e2d\uff0c\u6cd5\u5411\u91cf\u548c\u5207\u5411\u91cf\u6709\u7740\u76f4\u89c2\u7684\u7269\u7406\u610f\u4e49\u3002\u4f8b\u5982\uff0c\u5728\u5149\u6ed1\u8868\u9762\u4e0a\u7684\u53cd\u5c04\u548c\u6298\u5c04\u73b0\u8c61\u4e2d\uff0c\u6cd5\u5411\u91cf\u7528\u4e8e\u8ba1\u7b97\u5165\u5c04\u5149\u7ebf\u548c\u53cd\u5c04\u5149\u7ebf\u7684\u65b9\u5411\uff1b\u800c\u5728\u52a8\u529b\u5b66\u4e2d\uff0c\u5207\u5411\u91cf\u5219\u7528\u6765\u63cf\u8ff0\u7269\u4f53\u6cbf\u66f2\u7ebf\u8def\u5f84\u7684\u901f\u5ea6\u65b9\u5411\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">4.\u5e94\u7528\u9886\u57df\u7684\u5dee\u5f02<\/h2>\n\n\n\n<p>\u867d\u7136\u6cd5\u5411\u91cf\u548c\u5207\u5411\u91cf\u90fd\u662f\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u63cf\u8ff0\u65b9\u5411\u7684\u5de5\u5177\uff0c\u4f46\u5b83\u4eec\u5728\u5404\u81ea\u7684\u5e94\u7528\u9886\u57df\u5185\u6709\u7740\u4e0d\u540c\u7684\u91cd\u70b9\u3002\u4f8b\u5982\uff0c\u5728\u8ba1\u7b97\u6d41\u4f53\u529b\u5b66\u4e2d\uff0c\u6cd5\u5411\u91cf\u7528\u4e8e\u786e\u5b9a\u6d41\u4f53\u4e0e\u56fa\u4f53\u8fb9\u754c\u7684\u4ea4\u4e92\u4f5c\u7528\uff0c\u800c\u5728\u9053\u8def\u8bbe\u8ba1\u4e2d\uff0c\u5207\u5411\u91cf\u5219\u7528\u4e8e\u786e\u5b9a\u9053\u8def\u7684\u5f2f\u66f2\u7a0b\u5ea6\u548c\u8f66\u8f86\u7684\u8fd0\u52a8\u8f68\u8ff9\u3002<\/p>\n\n\n\n<p>\u7ed3\u8bba:\u66f2\u9762\u6cd5\u5411\u91cf\u548c\u66f2\u7ebf\u5207\u5411\u91cf\u867d\u7136\u6709\u5171\u540c\u70b9\uff0c\u4f46\u5728\u5b9a\u4e49\u3001\u8ba1\u7b97\u548c\u5e94\u7528\u4e0a\u6709\u7740\u672c\u8d28\u7684\u4e0d\u540c\u3002\u901a\u8fc7\u5bf9\u8fd9\u4e24\u4e2a\u6982\u5ff5\u7684\u8be6\u7ec6\u5256\u6790\uff0c\u6211\u4eec\u4e0d\u4ec5\u80fd\u66f4\u597d\u5730\u7406\u89e3\u5b83\u4eec\u7684\u6570\u5b66\u672c\u8d28\uff0c\u8fd8\u80fd\u6df1\u5165\u8ba4\u8bc6\u5230\u5b83\u4eec\u5728\u73b0\u5b9e\u4e16\u754c\u4e2d\u7684\u91cd\u8981\u4f5c\u7528\u3002\u5728\u4efb\u4f55\u6d89\u53ca\u5230\u66f2\u9762\u548c\u66f2\u7ebf\u63cf\u8ff0\u7684\u9886\u57df\uff0c\u6b63\u786e\u5730\u4f7f\u7528\u6cd5\u5411\u91cf\u548c\u5207\u5411\u91cf\u5bf9\u4e8e\u95ee\u9898\u7684\u89e3\u51b3\u90fd\u662f\u81f3\u5173\u91cd\u8981\u7684\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img fetchpriority=\"high\" decoding=\"async\" width=\"484\" height=\"300\" src=\"https:\/\/cdn-docs.pingcode.com\/wp-content\/uploads\/2023\/11\/\u5982\u4f55\u5728\u9e92\u9e9f\u64cd\u4f5c\u7cfb\u7edf\u4e0a\u5b89\u88c5\u548c\u7ba1\u7406\u5b57\u4f53-55.png?x-oss-process=image\/auto-orient,1\/format,webp\" alt=\"\" class=\"wp-image-61056\" \/><\/figure><\/div>\n\n\n<p><strong>\u5e38\u89c1\u95ee\u7b54\uff1a<\/strong><\/p>\n\n\n\n<ul>\n<li>\u95ee\uff1a\u66f2\u9762\u6cd5\u5411\u91cf\u548c\u66f2\u7ebf\u5207\u5411\u91cf\u5728\u51e0\u4f55\u4e0a\u6709\u4ec0\u4e48\u672c\u8d28\u533a\u522b\uff1f<\/li>\n\n\n\n<li>\u7b54\uff1a\u51e0\u4f55\u4e0a\uff0c\u66f2\u9762\u6cd5\u5411\u91cf\u662f\u5782\u76f4\u4e8e\u66f2\u9762\u5728\u67d0\u4e00\u70b9\u7684\u5411\u91cf\uff0c\u4ee3\u8868\u66f2\u9762\u5728\u8be5\u70b9\u7684\u5782\u76f4\u65b9\u5411\uff0c\u800c\u66f2\u7ebf\u5207\u5411\u91cf\u662f\u8d34\u4e8e\u66f2\u7ebf\u5728\u67d0\u4e00\u70b9\u7684\u5207\u7ebf\u65b9\u5411\u7684\u5411\u91cf\uff0c\u4ee3\u8868\u66f2\u7ebf\u5728\u8be5\u70b9\u7684\u65b9\u5411\u6216\u8fd0\u52a8\u8d8b\u52bf\u3002<\/li>\n<\/ul>\n\n\n\n<ul>\n<li>\u95ee\uff1a\u5982\u4f55\u8ba1\u7b97\u66f2\u9762\u4e0a\u67d0\u4e00\u70b9\u7684\u6cd5\u5411\u91cf\uff1f<\/li>\n\n\n\n<li>\u7b54\uff1a\u66f2\u9762\u4e0a\u67d0\u70b9\u7684\u6cd5\u5411\u91cf\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u8be5\u70b9\u7684\u68af\u5ea6\u6765\u5f97\u5230\uff0c\u5982\u679c\u66f2\u9762\u7531\u9690\u51fd\u6570 f(x, y, z) = 0f(x,y,z)=0 \u7ed9\u51fa\uff0c\u5219\u6cd5\u5411\u91cf\u662f\u8be5\u51fd\u6570\u68af\u5ea6\u7684\u65b9\u5411\uff1b\u5982\u679c\u66f2\u9762\u662f\u7531\u53c2\u6570\u65b9\u7a0b\u7ed9\u51fa\uff0c\u90a3\u4e48\u6cd5\u5411\u91cf\u53ef\u4ee5\u901a\u8fc7\u8ba1\u7b97\u4e24\u4e2a\u53c2\u6570\u504f\u5bfc\u6570\u5411\u91cf\u7684\u53c9\u79ef\u5f97\u5230\u3002<\/li>\n<\/ul>\n\n\n\n<ul>\n<li>\u95ee\uff1a\u5728\u7269\u7406\u5b66\u4e2d\uff0c\u66f2\u9762\u6cd5\u5411\u91cf\u6709\u54ea\u4e9b\u5e94\u7528\uff1f<\/li>\n\n\n\n<li>\u7b54\uff1a\u5728\u7269\u7406\u5b66\u4e2d\uff0c\u66f2\u9762\u6cd5\u5411\u91cf\u5e38\u7528\u4e8e\u63cf\u8ff0\u529b\u7684\u65b9\u5411\uff0c\u5982\u8ba1\u7b97\u5149\u7684\u53cd\u5c04\u548c\u6298\u5c04\u3001\u786e\u5b9a\u7535\u573a\u548c\u78c1\u573a\u4e2d\u7684\u529b\u7ebf\u65b9\u5411\uff0c\u4ee5\u53ca\u6d41\u4f53\u52a8\u529b\u5b66\u4e2d\u786e\u5b9a\u6d41\u4f53\u4e0e\u56fa\u4f53\u8fb9\u754c\u7684\u4ea4\u4e92\u4f5c\u7528\u7b49\u3002<\/li>\n<\/ul>\n\n\n\n<ul>\n<li>\u95ee\uff1a\u4e3a\u4ec0\u4e48\u66f2\u7ebf\u5207\u5411\u91cf\u91cd\u8981\uff1f<\/li>\n\n\n\n<li>\u7b54\uff1a\u66f2\u7ebf\u5207\u5411\u91cf\u5bf9\u4e8e\u63cf\u8ff0\u7269\u4f53\u6cbf\u7740\u66f2\u7ebf\u8def\u5f84\u7684\u52a8\u6001\u7279\u6027\u81f3\u5173\u91cd\u8981\uff0c\u6bd4\u5982\u5728\u8fd0\u52a8\u5b66\u4e2d\u5206\u6790\u7269\u4f53\u7684\u901f\u5ea6\u548c\u52a0\u901f\u5ea6\u65b9\u5411\uff0c\u5728\u9053\u8def\u8bbe\u8ba1\u4e2d\u786e\u5b9a\u8f66\u8f86\u7684\u9884\u671f\u8f68\u8ff9\uff0c\u4ee5\u53ca\u5728\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u4e2d\u751f\u6210\u5e73\u6ed1\u7684\u52a8\u753b\u8def\u5f84\u3002<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"\u66f2\u9762\u6cd5\u5411\u91cf\u548c\u66f2\u7ebf\u5207\u5411\u91cf\u7684\u533a\u522b\uff1a1.\u66f2\u9762\u6cd5\u5411\u91cf\u7684\u5b9a\u4e49\u548c\u8ba1\u7b97\uff1b2.\u66f2\u7ebf\u5207\u5411\u91cf\u7684\u5b9a\u4e49\u548c\u8ba1\u7b97\uff1b3.\u66f2\u9762\u6cd5\u5411\u91cf\u548c\u66f2\u7ebf\u5207\u5411 [&hellip;]","protected":false},"author":7,"featured_media":61056,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[37],"tags":[],"acf":[],"_links":{"self":[{"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/posts\/61053"}],"collection":[{"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/comments?post=61053"}],"version-history":[{"count":0,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/posts\/61053\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/media\/61056"}],"wp:attachment":[{"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/media?parent=61053"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/categories?post=61053"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/docs.pingcode.com\/wp-json\/wp\/v2\/tags?post=61053"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}