\u5728\u4e00\u4e2a\u591a\u53d8\u91cf\u7684\u51fd\u6570\u4e2d\uff0c\u504f\u5bfc\u6570\u5c31\u662f\u5173\u4e8e\u5176\u4e2d\u4e00\u4e2a\u53d8\u91cf\u7684\u5bfc\u6570\u800c\u4fdd\u6301\u5176\u5b83\u53d8\u91cf\u6052\u5b9a\u4e0d\u53d8\u3002\u5047\u5b9a\u4e8c\u5143\u51fd\u6570$z=f(x,y)$\uff0c\u70b9$(x_0,y_0)$\u662f\u5176\u5b9a\u4e49\u57df\u5185\u7684\u4e00\u4e2a\u70b9\uff0c\u5c06$y$\u56fa\u5b9a\u5728$y_0$\u4e0a\uff0c\u800c$x$\u5728$x_0$\u4e0a\u589e\u91cf$\\Delta x$\uff0c\u76f8\u5e94\u7684\u51fd\u6570$z$\u6709\u589e\u91cf$\\Delta z = f(x_0+\\Delta x, y_0) – f(x_0, y_0)$\u3002$\\Delta z$\u548c$\\Delta x$\u7684\u6bd4\u503c\u5f53$\\Delta x$\u7684\u503c\u8d8b\u8fd1\u4e8e$0$\u7684\u65f6\u5019\uff0c\u5982\u679c\u6781\u9650\u5b58\u5728\uff0c\u90a3\u4e48\u6b64\u6781\u9650\u503c\u79f0\u4e3a\u51fd\u6570$z = f(x,y)$\u5728\u70b9$(x_0,y_0)$\u5904\u5bf9$x$\u7684\u504f\u5bfc\u6570(partial derivative)\uff0c\u8bb0\u4f5c\uff1a$f’_x(x_0,y_0)$<\/p>\n
<\/p>\n
\u5bf9$x$\u7684\u504f\u5bfc\u6570\uff1a<\/p>\n
$$
\n\\left. \\frac{\\partial f}{\\partial x}\\right|{x=x_0\\y=y_0}
\n$$<\/p>\n
\u5bf9$y$\u7684\u504f\u5bfc\u6570\uff1a<\/p>\n
$$
\n\\left. \\frac{\\partial f}{\\partial y}\\right|{x=x_0\\y=y_0}
\n$$<\/p>\n
\u63a8\u5e7f\uff1a\u5bf9\u4e8e\u4e09\u5143\u51fd\u6570$u=f(x,y,z)$\u53ef\u7c7b\u4f3c<\/p>\n
\u5b9a\u4e49$u$\u5728\u70b9$P_0(x_0,y_0,z_0)$\u5206\u522b\u5bf9$x,y,x$\u7684\u504f\u5bfc\u6570<\/p>\n
$$ $$ $$ \u504f\u5bfc\u6570\u662f\u591a\u5143\u51fd\u6570\u5bf9\u5176\u4e2d\u67d0\u4e00\u4e2a\u81ea\u53d8\u91cf\uff08\u5176\u4f59\u81ea\u53d8\u91cf\u89c6\u4e3a\u5e38\u91cf\uff09\u7684\u53d8\u5316\u7387\u3002<\/p>\n \u4f8b\uff1a\u6c42$z=x^2+3xy+y^2$\u5728\u70b9$(1,2)$\u5904\u7684\u504f\u5bfc\u6570<\/p>\n $ $ \u51fd\u6570$z=f(x,y)$\u7684\u4e8c\u9636\u504f\u5bfc\u6570\u4e3a<\/p>\n $$ $$ \u7c7b\u4f3c\u53ef\u4ee5\u5b9a\u4e49\u66f4\u9ad8\u9636\u7684\u504f\u5bfc\u6570\uff0c\u4f8b\u5982<\/p>\n $z=f(x,y)$\u5173\u4e8e$x$\u76843\u9636\u504f\u5bfc\u6570\u4e3a<\/p>\n $$ $z=f(x,y)$\u5173\u4e8e$x$\u7684$n-1$\u9636\u504f\u5bfc\u6570\uff0c\u518d\u5173\u4e8e$y$\u7684\u4e00\u9636\u504f\u5bfc\u6570\u4e3a<\/p>\n $$ \u5b9a\u4e49\uff1a\u4e8c\u9636\u53ca\u4e8c\u9636\u4ee5\u4e0a\u7684\u504f\u5bfc\u6570\u7edf\u79f0\u4e3a\u9ad8\u9636\u504f\u5bfc\u6570<\/p>\n \u5411\u91cf\uff1a\u662f\u6307\u5177\u6709$n$\u4e2a\u4e92\u76f8\u72ec\u7acb\u6027\u8d28(\u7eac\u5ea6)\u7684\u5bf9\u8c61\u7684\u8868\u793a\uff0c\u5411\u91cf\u5e38\u4f7f\u7528\u5b57\u6bcd+\u7bad\u5934\u7684\u5f62\u5f0f\u8fdb\u884c\u6807\u793a\uff0c\u4e5f\u53ef\u4ee5\u4f7f\u7528\u51e0\u4f55\u5750\u6807\u6765\u8868\u793a\u5411\u91cf\uff0c\u6bd4\u5982$\\overrightarrow{a} = \\overrightarrow{OP} = xi + yj + zk$\uff0c\u53ef\u4ee5\u7528\u5750\u6807$(i,j,k)$\u8868\u793a\u5411\u91cf$\\overrightarrow{a}$<\/p>\n \u5411\u91cf\u7684\u6a21\uff1a\u5411\u91cf\u7684\u5927\u5c0f\uff0c\u4e5f\u5c31\u662f\u5411\u91cf\u7684\u957f\u5ea6\uff0c\u5411\u91cf\u5750\u6807\u5230\u539f\u70b9\u7684\u8ddd\u79bb\uff0c\u5e38\u8bb0\u4f5c$|\\overrightarrow{a}|$<\/p>\n \u5355\u4f4d\u5411\u91cf\uff1a\u957f\u5ea6\u4e3a\u4e00\u4e2a\u5355\u4f4d\uff08\u5373\u6a21\u4e3a1\uff09\u7684\u5411\u91cf\u5c31\u53eb\u505a\u5355\u4f4d\u5411\u91cf<\/p>\n \u8bbe\u4e24\u5411\u91cf\u4e3a\uff1a$\\overrightarrow{a}=(x_1,y_1),\\overrightarrow{b}=(x_2,y_2)$\uff0c\u5e76\u4e14$\\overrightarrow{a}$\u548c$\\overrightarrow{b}$\u4e4b\u95f4\u7684\u5939\u89d2\u4e3a$\\theta$<\/p>\n \u6570\u91cf\u79ef\uff1a\u4e24\u4e2a\u5411\u91cf\u7684\u6570\u91cf\u79ef\uff08\u5185\u79ef\u3001\u70b9\u79ef\uff09\u662f\u4e00\u4e2a\u6570\u91cf\/\u5b9e\u6570\uff0c\u8bb0\u4f5c$\\overrightarrow{a} \\cdot \\overrightarrow{b}$<\/p>\n $$ $$ \u5411\u91cf\u79ef\uff1a\u4e24\u4e2a\u5411\u91cf\u7684\u5411\u91cf\u79ef\uff08\u5916\u79ef\u3001\u53c9\u79ef\uff09\u662f\u4e00\u4e2a\u5411\u91cf\uff0c\u8bb0\u4f5c$\\overrightarrow{a} \u00d7 \\overrightarrow{b}$\uff0c\u5411\u91cf\u79ef\u5373\u4e24\u4e2a\u4e0d\u5171\u7ebf\u975e\u96f6\u5411\u91cf\u6240\u5728\u5e73\u9762\u7684\u4e00\u7ec4\u6cd5\u5411\u91cf<\/p>\n $$ $$ \u5982\u679c\u4e24\u4e2a\u5411\u91cf\u7684\u70b9\u79ef\u4e3a\u96f6\uff0c\u90a3\u4e48\u79f0\u8fd9\u4e24\u4e2a\u5411\u91cf\u4e92\u4e3a\u6b63\u4ea4\u5411\u91cf\u3002\u5728\u51e0\u4f55\u610f\u4e49\u4e0a\u6765\u8bb2\uff0c\u6b63\u4ea4\u5411\u91cf\u5728\u4e8c\u7ef4\/\u4e09\u7ef4\u7a7a\u95f4\u4e0a\u5176\u5b9e\u5c31\u662f\u4e24\u4e2a\u5411\u91cf\u5782\u76f4\u3002<\/p>\n \u5982\u679c\u4e24\u4e2a\u6216\u591a\u4e2a\u5411\u91cf\uff0c\u5b83\u4eec\u7684\u70b9\u79ef\u5747\u4e3a0\uff0c\u90a3\u4e48\u5b83\u4eec\u4e92\u76f8\u79f0\u4e3a\u6b63\u4ea4\u5411\u91cf\u3002<\/p>\n \u5b9a\u4e49\uff1a\u82e5\u51fd\u6570$f(x,y,z)$\u5728\u70b9$P(x,y,z)$\u5904\u6cbf\u65b9\u5411$l$(\u65b9\u5411\u89d2\u4e3a$\\alpha,\\beta,\\gamma$)\u5b58\u5728\u4e0b\u5217\u6781\u9650<\/p>\n $$ \u5176\u4e2d$\\rho = \\sqrt{(\\Delta x)^2 + (\\Delta y)^2 + (\\Delta z)^2}$\uff0c$\\Delta x = \\rho \\cos \\alpha,\\Delta y = \\rho \\cos \\beta,\\Delta z = \\rho \\cos \\gamma$\uff0c\u65b9\u5411\u89d2$\\cos^2 \\alpha + \\cos^2 \\beta + \\cos^2 \\gamma = 1$<\/p>\n \u5219\u79f0$\\frac{\\partial f}{\\partial l}$\u4e3a\u51fd\u6570\u5728\u70b9$P$\u5904\u6cbf\u65b9\u5411$l$\u7684\u65b9\u5411\u5bfc\u6570\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u591a\u5143\u51fd\u6570\u504f\u5bfc\u6570 \u5728\u4e00\u4e2a\u591a\u53d8\u91cf\u7684\u51fd\u6570\u4e2d\uff0c\u504f\u5bfc\u6570\u5c31\u662f\u5173\u4e8e\u5176\u4e2d\u4e00\u4e2a\u53d8\u91cf\u7684\u5bfc\u6570\u800c\u4fdd\u6301\u5176\u5b83\u53d8\u91cf\u6052\u5b9a\u4e0d\u53d8\u3002\u5047\u5b9a\u4e8c\u5143\u51fd\u6570$z […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2134","post","type-post","status-publish","format-standard","hentry","category-java"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2134","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2134"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2134\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2134"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2134"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2134"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nf_x(x_0,y_0,z0) = \\lim \\limits<\/em>{\\Delta x \\to 0} \\frac{f(x_0+\\Delta x,y_0,z_0)-f(x_0,y_0,z_0)}{\\Delta x}
\n$$<\/p>\n
\nf_y(x_0,y_0,z0) = \\lim \\limits<\/em>{\\Delta y \\to 0} \\frac{f(x_0,y_0+\\Delta y,z_0)-f(x_0,y_0,z_0)}{\\Delta y}
\n$$<\/p>\n
\nf_z(x_0,y_0,z0) = \\lim \\limits<\/em>{\\Delta z \\to 0} \\frac{f(x_0,y_0,z_0+\\Delta z)-f(x_0,y_0,z_0)}{\\Delta z}
\n$$<\/p>\n
\n\\frac{\\partial z}{\\partial x} = 2x + 3y \\,\\,\\,\\,\\,\\,\\,\\, \\frac{\\partial z}{\\partial y} = 3x + 2y
\n$<\/p>\n
\n\\left. \\frac{\\partial z}{\\partial x}\\right|{(1,2)} = 2\u00b71+3\u00b72 = 8 \\,\\,\\,\\,\\,\\,\\,\\, \\left. \\frac{\\partial z}{\\partial x}\\right|<\/em>{(1,2)} = 3\u00b71+2\u00b72 = 7
\n$<\/p>\n\u9ad8\u9636\u504f\u5bfc\u6570<\/h2>\n
\n\\left.
\n\\begin{array} {r}
\n\\frac{\\partial}{\\partial x}(\\frac{\\partial z}{\\partial x}) = \\frac{\\partial^2 z}{\\partial x^2} = f”{xx} = f”<\/em>{11} \\
\n\\frac{\\partial}{\\partial y}(\\frac{\\partial z}{\\partial y}) = \\frac{\\partial^2 z}{\\partial y^2} = f”{yy} = f”<\/em>{22} \\
\n\\end{array}
\n\\right\\}\u7eaf\u504f\u5bfc
\n$$<\/p>\n
\n\\left.
\n\\begin{array} {r}
\n\\frac{\\partial}{\\partial y}(\\frac{\\partial z}{\\partial x}) = \\frac{\\partial^2 z}{\\partial x \\partial y} = f”{xy} = f”<\/em>{12} \\
\n\\frac{\\partial}{\\partial x}(\\frac{\\partial z}{\\partial y}) = \\frac{\\partial^2 z}{\\partial y \\partial x} = f”{yx} = f”<\/em>{21} \\
\n\\end{array}
\n\\right\\}\u6df7\u5408\u504f\u5bfc
\n$$<\/p>\n
\n\\frac{\\partial^3 z}{\\partial x^3} = \\frac{\\partial}{\\partial x}(\\frac{\\partial^2 z}{\\partial x^2})
\n$$<\/p>\n
\n\\frac{\\partial^n z}{\\partial x^{n-1} \\partial y} = \\frac{\\partial}{\\partial y}(\\frac{\\partial^{n-1} z}{\\partial x^{n-1}})
\n$$<\/p>\n\u5411\u91cf<\/h2>\n
\u5411\u91cf\u7684\u5b9a\u4e49<\/h3>\n
\u5411\u91cf\u7684\u8fd0\u7b97<\/h3>\n
\n\\overrightarrow{a} \\cdot \\overrightarrow{b} = |\\overrightarrow{a}| \\cdot |\\overrightarrow{b}| \\cdot \\cos \\theta
\n$$<\/p>\n
\n\\cos = \\frac{\\overrightarrow{a} \\cdot \\overrightarrow{b}}{|\\overrightarrow{a}||\\overrightarrow{b}|} = \\frac{x_1x_2+y_1y_2}{\\sqrt{x_1^2+x_2^2} \\cdot \\sqrt{y_1^2+y_2^2}}
\n$$<\/p>\n
\n\\overrightarrow{a} \u00d7 \\overrightarrow{b} = |\\overrightarrow{a}| \\cdot |\\overrightarrow{b}| \\cdot \\sin \\theta
\n$$<\/p>\n
\nS = \\frac{1}{2}|\\overrightarrow{a}||\\overrightarrow{b}|\\sin \\theta
\n$$<\/p>\n\u6b63\u4ea4\u5411\u91cf<\/h3>\n
\u65b9\u5411\u5bfc\u6570<\/h2>\n
\n\\lim \\limits{\\rho \\to 0} \\frac{\\Delta f}{\\rho} = \\lim \\limits<\/em>{\\rho \\to 0} \\frac{f(x+\\Delta x,y+\\Delta y,z+\\Delta z)-f(x,y,z)}{\\rho} = \\frac{\\partial f}{\\partial l}
\n$$<\/p>\n