\u672c\u7ae0\u8ba8\u8bba\u8981\u70b9\uff1a<\/p>\n\n\n\n
\u5f53\u6211\u4eec\u5728\u8fdb\u5165\u9690\u51fd\u6570\u6c42\u5bfc\u8fd9\u4e00\u7ae0\u8282\u65f6\uff0c\u6211\u4eec\u4f1a\u9047\u5230\u4e00\u79cd\u5168\u65b0\u7684\u64cd\u4f5c\u65b9\u5f0f\uff1a<\/p>\n\n\n\n
$$ \\underbrace{\\frac{d}{dx}(x^2) = 2x}_{\\text{\u53d8\u91cf\u5339\u914d\uff1a\u76f4\u63a5\u6c42\u5bfc}} \\qquad \\qquad \\underbrace{\\frac{d}{dx}(y^2) = 2y \\cdot y'}_{\\text{\u53d8\u91cf\u4e0d\u540c\uff1a\u94fe\u5f0f\u6cd5\u5219}} $$<\/p>\n\n\n\n
\u5f88\u591a\u53c2\u8003\u4e66\u7531\u4e8e\u4fa7\u91cd\u8ba1\u7b97\uff0c\u4ec5\u4ec5\u91c7\u7528\u300c\u5728\u65b9\u7a0b\u4e24\u8fb9\u540c\u65f6\u5bf9 $x$ \u6c42\u5bfc\u300d\u7684\u63d0\u793a\u4fbf\u8349\u8349\u7565\u8fc7\uff0c\u672c\u6587\u5c06\u5c31\u6b64\u5c55\u5f00\u8ba8\u8bba\u3002<\/p>\n\n\n\n
\u5728\u4e00\u4e9b\u51fd\u6570\u4e2d\uff0c\u53d8\u91cf $y$ \u7531\u4e00\u4e2a\u53ea\u542b\u53d8\u91cf $x$ \u7684\u5f0f\u5b50\u8868\u793a\u3002<\/p>\n\n\n\n
\u663e\u51fd\u6570\uff1a<\/strong> $y = x + 1$ , $y = 3x$ , $y = 2x^2 + 3x + 7$ , $y = \\tan x$<\/p>\n\n\n\n \u8fd9\u4e9b\u51fd\u6570\u4e2d\uff0c$y$ \u4e0e $x$ \u7684\u5173\u7cfb\u662f\u660e\u786e\u7684\uff0c\u6211\u4eec\u79f0 $y$ \u662f $x$ \u7684\u663e\u51fd\u6570\u3002<\/p>\n\n\n\n \u9690\u51fd\u6570\uff1a<\/strong> $x^2 + y^2 = 1$, $\\sqrt{x} + \\sqrt{y} = 1$, $x^3 + y^3 - 3xy = 1$<\/p>\n\n\n\n \u5728\u8fd9\u4e9b\u51fd\u6570\u4e2d\uff0c\u6211\u4eec\u5f88\u96be\u751a\u81f3\u65e0\u6cd5\u7528 $x$ \u8868\u793a $y$\uff0c\u8fd9\u4e9b\u51fd\u6570\u88ab\u79f0\u4e3a\u9690\u51fd\u6570\u3002<\/p>\n\n\n\n \u4e25\u683c\u6765\u8bf4\uff0c\u5bf9\u4e8e $x^2+y^2=1$ \uff0c\u5b83\u662f\u4e00\u4e2a\u300c\u7531\u65b9\u7a0b $F(x,y)=0$ \u6240\u786e\u5b9a\u7684\u51fd\u6570\u5173\u7cfb\u300d\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u51fd\u6570\u3002\u56e0\u4e3a\u6839\u636e\u51fd\u6570\u5b9a\u4e49\uff0c\u4e00\u4e2a $x$ \u5bf9\u5e94\u4e0d\u80fd\u4e24\u4e2a $y$ (\u5706\u7684\u65b9\u7a0b\u663e\u7136\u8fdd\u80cc\u4e86\u8fd9\u4e00\u70b9)\u3002<\/p>\n\n\n\n \u867d\u7136\u5706\u5728\u6574\u4f53\u4e0a\u4e0d\u662f\u51fd\u6570\uff0c\u4f46\u5728\u5c40\u90e8\u8303\u56f4\u5185\uff08\u6bd4\u5982\u4e0a\u534a\u5706\uff09\uff0c$y$ \u786e\u5b9e\u53d7\u5230 $x$ \u7684\u5236\u7ea6\uff0c\u8fd9\u79cd\u7531\u65b9\u7a0b\u300c\u6697\u4e2d\u300d\u786e\u5b9a\u7684\u51fd\u6570\u5173\u7cfb\uff0c\u88ab\u79f0\u4e3a\u9690\u51fd\u6570\u3002<\/p>\n<\/blockquote>\n\n\n\n \u9762\u5bf9\u4e00\u4e2a\u50cf $x^2 + y^2 = 1$ \u7684\u65b9\u7a0b\uff0c\u6211\u4eec\u60f3\u6c42 $\\frac{dy}{dx}$\uff0c\u6c42\u5bfc\u65b9\u5f0f\u4e3a\uff1a<\/p>\n\n\n\n $$ \\underbrace{\\frac{d}{dx}(x^2) = 2x}_{\\text{\u53d8\u91cf\u5339\u914d\uff1a\u76f4\u63a5\u6c42\u5bfc}} \\qquad \\qquad \\underbrace{\\frac{d}{dx}(y^2) = 2y \\cdot y'}_{\\text{\u53d8\u91cf\u4e0d\u540c\uff1a\u94fe\u5f0f\u6cd5\u5219}} $$<\/p>\n\n\n\n \u6700\u7ec8\u6c42\u5bfc\u7ed3\u679c\u4e3a\uff1a$2x + 2y \\cdot y' = 0$<\/p>\n\n\n\n \u53ef\u662f\uff0c\u6211\u4eec\u4e3a\u4ec0\u4e48\u80fd\u540c\u65f6\u5bf9\u7b49\u5f0f\u4e24\u8fb9\u8fdb\u884c\u6c42\u5bfc\uff1f\u4e3a\u4ec0\u4e48\u4e24\u8005\u7684\u6c42\u5bfc\u65b9\u5f0f\u4e0d\u4e00\u6837\uff1f<\/p>\n\n\n\n \u8981\u56de\u7b54\u521a\u624d\u7684\u5408\u6cd5\u6027\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u5148\u5206\u6790\u8fd9\u4e2a\u51fd\u6570\uff1a<\/p>\n\n\n\n $$ x^2 + y^2 = 1 $$<\/p>\n\n\n\n \u5f53\u6211\u4eec\u8bf4\u70b9 $(x,y)$ \u5728\u5706 $x^2 + y^2 = 1$ \u4e0a\u8fd0\u52a8\u65f6\uff0c\u5f53\u6211\u4eec\u53d8\u6362\u5750\u6807\u7684 $x$ \u503c\u65f6\uff0c\u5750\u6807\u7684 $y$ \u503c\u5982\u4f55\u51b3\u5b9a\uff1f\u8fd9\u91cc\u6709\u4e00\u4e2a\u663e\u7136\u7684\u4e8b\u5b9e\uff1a$y$ \u7684\u53d6\u503c\u53d7\u5230 $x$ \u53d6\u503c\u7684\u5f71\u54cd\uff0c\u8fd9\u610f\u5473\u7740 $y$ \u7684\u503c\u5e76\u4e0d\u662f\u968f\u610f\u9009\u53d6\u7684\uff0c\u800c\u662f\u968f\u7740 $x$ \u7684\u53d8\u5316\u800c\u53d8\u5316\u7684\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5c40\u90e8\u8303\u56f4\u5185\uff0c$y$ \u662f $x$ \u7684\u51fd\u6570\u3002<\/p>\n\n\n\n \u4e3a\u4e86\u89e3\u91ca\u8fd9\u4e00\u70b9\uff0c\u6211\u4eec\u5148\u6682\u65f6\u629b\u5f00 $y$ \u8fd9\u4e2a\u9762\u5177\uff0c\u7ed9\u5b83\u4e00\u4e2a\u65b0\u7684\u540d\u5b57\uff1a\u300c$g(x)$\u300d\u3002\u6b64\u65f6\uff0c\u539f\u65b9\u7a0b\u5c31\u53d8\u6210\u4e86\u4e00\u4e2a\u5173\u4e8e $x$ \u7684\u6052\u7b49\u5f0f\uff1a<\/p>\n\n\n\n $$ x^2 + [g(x)]^2 = 1 $$<\/p>\n\n\n\n Question\uff1a<\/strong> \u6b64\u5904\u4e3a\u4ec0\u4e48\u4e00\u5b9a\u662f\u6052\u7b49\u5f0f\uff1f<\/p>\n\n\n\n \u76f4\u89c2\u611f\u53d7\u4e0a\u662f\uff1a\u6211\u4eec\u8ba8\u8bba\u7684\u524d\u63d0\u662f\u70b9 $(x, y)$ \u59cb\u7ec8\u5728\u66f2\u7ebf\u4e0a\u8fd0\u52a8\u3002<\/p>\n\n\n\n \u5f53\u6211\u4eec\u628a $y$ \u770b\u4f5c $g(x)$ \u65f6\uff0c\u6211\u4eec\u5b9e\u9645\u4e0a\u662f\u9650\u5236\u4e86 $y$ \u7684\u53d6\u503c\u5fc5\u987b\u6ee1\u8db3\u8fd9\u4e2a\u65b9\u7a0b\u3002\u65e2\u7136\u70b9\u5728\u66f2\u7ebf\u4e0a\uff0c\u90a3\u4e48\u628a\u5750\u6807\u4ee3\u5165\u65b9\u7a0b\uff0c\u7b49\u5f0f\u5de6\u8fb9\u7684\u8ba1\u7b97\u7ed3\u679c\u5c31\u6c38\u8fdc\u7b49\u4e8e\u53f3\u8fb9\u7684\u5e38\u6570 1\u3002<\/p>\n\n\n\n \u65e2\u7136\u5de6\u53f3\u4e24\u8fb9\u59cb\u7ec8\u76f8\u7b49\uff08\u5373\u4e3a\u6052\u7b49\u5f0f\uff09\uff0c\u90a3\u4e48\u5b83\u4eec\u968f $x$ \u7684\u53d8\u5316\u7387\uff08\u5bfc\u6570\uff09\u4e5f\u81ea\u7136\u5fc5\u987b\u76f8\u7b49\uff08\u56e0\u4e3a\u5982\u679c\u53d8\u5316\u7387\u4e0d\u76f8\u7b49\uff0c\u90a3\u5c31\u6ca1\u529e\u6cd5\u6210\u4e3a\u300c\u6052\u7b49\u5f0f\u300d\uff0c\u5c31\u50cf\u4e24\u4e2a\u51fd\u6570\uff0c\u5982\u679c\u53d8\u5316\u7387\u4e0d\u76f8\u7b49\uff0c\u90a3\u4ed6\u4eec\u7684\u6570\u503c\u6709\u53ef\u80fd\u5904\u5904\u76f8\u7b49\u5417\uff1f\uff09\u3002<\/p>\n<\/blockquote>\n\n\n\n \u65e2\u7136\u7b49\u53f7\u5de6\u53f3\u4e24\u8fb9\u59cb\u7ec8\u76f8\u7b49\uff0c\u90a3\u4e48\uff0c\u4ed6\u4eec\u968f\u7740 $x$ \u7684\u53d8\u5316\u7387\u4e5f\u5fc5\u7136\u4fdd\u6301\u4e00\u81f4\uff0c\u8fd9\u6b63\u662f\u6211\u4eec\u53ef\u4ee5\u300c\u4e24\u8fb9\u540c\u65f6\u6c42\u5bfc\u300d\u7684\u4f9d\u636e\u3002<\/p>\n\n\n\n \u66f4\u8fdb\u4e00\u6b65\uff0c\u7531\u4e8e\u94fe\u5f0f\u6cd5\u5219\uff1a<\/p>\n\n\n\n $$ [g(x)^2]' = 2g(x) \\cdot g'(x) $$<\/p>\n\n\n\n \u8fd9\u65f6\u5019\u6211\u4eec\u5c31\u80fd\u89e3\u91ca\u4e3a\u4ec0\u4e48 $(y^2)' = 2y \\cdot y'$ \u4e86\uff01\u5b83\u7684\u6838\u5fc3\u601d\u60f3\u662f\u300c\u628a $y$ \u89c6\u4e3a $x$ \u7684\u51fd\u6570\uff0c\u7136\u540e\u5229\u7528\u94fe\u5f0f\u6cd5\u5219\u300d\u3002<\/p>\n\n\n\n \u540e\u7eed\u5185\u5bb9\u4f5c\u8005\u6b63\u5728\u70e7\u8111\u7684\u601d\u8003\u600e\u4e48\u5199\u6bd4\u8f83\u5bb9\u6613\u88ab\u5927\u5bb6\u6240\u63a5\u53d7\uff0c\u6682\u65f6\u5148\u653e\u4e00\u4e9b\u5b9a\u7406\uff0c\u4f5c\u8005\u5c06\u5c3d\u5feb\u5b8c\u6210<\/p>\n<\/blockquote>\n\n\n\n \u2160.\u6216\u9690\u51fd\u6570\u5b58\u5728\u5b9a\u7406\uff1a<\/strong><\/p>\n\n\n\n \u8b66\u544a\uff1a\u6b64\u540e\u7684\u8bc1\u660e\u524d\u7f6e\u77e5\u8bc6\uff1a\u7b49\u9ad8\u7ebf\/\u5168\u5fae\u5206\/\u68af\u5ea6\u3002\u7b14\u8005\u5c06\u5c1d\u8bd5\u5c3d\u53ef\u80fd\u7ed9\u51fa\u8bb2\u89e3\uff0c\u4f46\u662f\u53d7\u9650\u4e8e\u6c34\u5e73\uff0c\u53ef\u80fd\u4f1a\u6709\u7eb0\u6f0f\uff0c\u5e0c\u671b\u5404\u4f4d\u8c05\u89e3\u3002<\/em><\/p>\n<\/blockquote>\n\n\n\n \u5728\u524d\u6587\u4e2d\uff0c\u6211\u4eec\u901a\u8fc7\u201c\u4e24\u8fb9\u6c42\u5bfc\u201d\u7684\u7b97\u672f\u6280\u5de7\u5f97\u5230\u4e86\u7ed3\u679c\u3002\u4f46\u6570\u5b66\u7684\u4e25\u8c28\u6027\u8981\u6c42\u6211\u4eec\u4e0d\u80fd\u53ea\u6ee1\u8db3\u4e8e\u201c\u7b97\u51fa\u4e86\u4ec0\u4e48\u201d\uff0c\u800c\u5fc5\u987b\u7406\u89e3\u201c\u610f\u5473\u7740\u4ec0\u4e48\u201d\u3002<\/p>\n\n\n\n \u8981\u771f\u6b63\u4ece\u5e95\u5c42\u7406\u89e3\u9690\u51fd\u6570\u6c42\u5bfc\uff0c\u6211\u4eec\u9700\u8981\u5f15\u5165\u4e09\u4e2a\u66f4\u4e3a\u672c\u8d28\u7684\u6570\u5b66\u5de5\u5177\uff1a\u7b49\u9ad8\u7ebf\uff08Level Set\uff09<\/strong>\u3001\u5168\u5fae\u5206\uff08Total Differential\uff09<\/strong> \u548c \u68af\u5ea6\uff08Gradient\uff09<\/strong>\u3002<\/p>\n\n\n\n 1. \u9690\u51fd\u6570\u5b58\u5728\u5b9a\u7406 1\uff1a\u7531\u65b9\u7a0b $F(x,y)=0$ \u786e\u5b9a\u7684\u4e00\u5143\u9690\u51fd\u6570 $y=f(x)$<\/strong><\/p>\n\n\n\n \u8bbe\u51fd\u6570 $F(x,y)$ \u5728\u70b9 $P(x_0, y_0)$ \u7684\u67d0\u4e00\u90bb\u57df\u5185\u5177\u6709\u8fde\u7eed\u504f\u5bfc\u6570\uff0c\u4e14 $F(x_0, y_0) = 0$\uff0c$F_y(x_0, y_0) \\neq 0$\uff0c\u5219\u65b9\u7a0b $F(x,y)=0$ \u5728\u70b9 $(x_0, y_0)$ \u7684\u67d0\u4e00\u90bb\u57df\u5185\u6052\u80fd\u552f\u4e00<\/strong>\u786e\u5b9a\u4e00\u4e2a\u8fde\u7eed\u4e14\u5177\u6709\u8fde\u7eed\u5bfc\u6570\u7684\u51fd\u6570 $y=f(x)$\uff0c\u5b83\u6ee1\u8db3\u6761\u4ef6 $y_0 = f(x_0)$\uff0c\u5e76\u6709\uff1a<\/p>\n\n\n\n $$ \\frac{dy}{dx} = -\\frac{F_x}{F_y} $$<\/p>\n\n\n\n 2. \u9690\u51fd\u6570\u5b58\u5728\u5b9a\u7406 2\uff1a\u7531\u65b9\u7a0b $F(x,y,z)=0$ \u786e\u5b9a\u7684\u4e8c\u5143\u9690\u51fd\u6570 $z=f(x,y)$<\/strong><\/p>\n\n\n\n \u8bbe\u51fd\u6570 $F(x,y,z)$ \u5728\u70b9 $P(x_0, y_0, z_0)$ \u7684\u67d0\u4e00\u90bb\u57df\u5185\u5177\u6709\u8fde\u7eed\u504f\u5bfc\u6570\uff0c\u4e14 $F(x_0, y_0, z_0) = 0$\uff0c$F_z(x_0, y_0, z_0) \\neq 0$\uff0c\u5219\u65b9\u7a0b $F(x,y,z)=0$ \u5728\u70b9 $(x_0, y_0, z_0)$ \u7684\u67d0\u4e00\u90bb\u57df\u5185\u6052\u80fd\u552f\u4e00<\/strong>\u786e\u5b9a\u4e00\u4e2a\u8fde\u7eed\u4e14\u5177\u6709\u8fde\u7eed\u504f\u5bfc\u6570\u7684\u51fd\u6570 $z=f(x,y)$\uff0c\u5b83\u6ee1\u8db3\u6761\u4ef6 $z_0 = f(x_0, y_0)$\uff0c\u5e76\u6709\uff1a<\/p>\n\n\n\n $$ \\frac{\\partial z}{\\partial x} = -\\frac{F_x}{F_z} \\qquad , \\qquad \\frac{\\partial z}{\\partial y} = -\\frac{F_y}{F_z} $$<\/p>\n\n\n\n \u6ce8\uff1a<\/strong> $z$ \u65e0\u8bba\u5bf9\u8c01\u6c42\u5bfc\uff0c\u6c42\u51e0\u9636\u5bfc\uff0c\u6c42\u5bfc\u540e\u7684\u65b0\u51fd\u6570\u548c $z$ \u5177\u6709\u76f8\u540c\u590d\u5408\u7ed3\u6784\u3002<\/p>\n<\/blockquote>\n","protected":false},"excerpt":{"rendered":" \u5fae\u79ef\u5206\u96be\u70b9\u89e3\u6790\uff08\u4e00\uff09 \u9690\u51fd\u6570\u6c42\u5bfc \u672c\u7ae0\u8ba8\u8bba\u8981\u70b9\uff1a \u5f53\u6211\u4eec\u5728\u8fdb\u5165\u9690\u51fd\u6570\u6c42\u5bfc\u8fd9\u4e00\u7ae0\u8282\u65f6\uff0c\u6211\u4eec\u4f1a\u9047\u5230\u4e00\u79cd\u5168\u65b0\u7684\u64cd\u4f5c\u65b9\u5f0f\uff1a $$ \\und …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"emotion":"","emotion_color":"","title_style":"","license":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-23","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/blog.spinoi.com\/index.php\/wp-json\/wp\/v2\/posts\/23","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.spinoi.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.spinoi.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.spinoi.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.spinoi.com\/index.php\/wp-json\/wp\/v2\/comments?post=23"}],"version-history":[{"count":2,"href":"https:\/\/blog.spinoi.com\/index.php\/wp-json\/wp\/v2\/posts\/23\/revisions"}],"predecessor-version":[{"id":25,"href":"https:\/\/blog.spinoi.com\/index.php\/wp-json\/wp\/v2\/posts\/23\/revisions\/25"}],"wp:attachment":[{"href":"https:\/\/blog.spinoi.com\/index.php\/wp-json\/wp\/v2\/media?parent=23"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.spinoi.com\/index.php\/wp-json\/wp\/v2\/categories?post=23"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.spinoi.com\/index.php\/wp-json\/wp\/v2\/tags?post=23"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\u4e8c\u3001\u9690\u51fd\u6570\u7684\u6c42\u5bfc\u6cd5\u5219<\/h4>\n\n\n\n
\u4e09\u3001\u9690\u51fd\u6570\u4e0e\u590d\u5408\u51fd\u6570<\/h4>\n\n\n\n
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\u56db\u3001\u4e25\u8c28\u7684\u6570\u5b66\u5206\u6790\uff1a<\/h4>\n\n\n\n
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