{"id":2136,"date":"2023-04-02T10:05:46","date_gmt":"2023-04-02T02:05:46","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2136"},"modified":"2023-04-05T20:00:08","modified_gmt":"2023-04-05T12:00:08","slug":"foundation-of-advanced-mathematics-calculation-and-gradient-of-directional-derivative","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/02\/foundation-of-advanced-mathematics-calculation-and-gradient-of-directional-derivative\/","title":{"rendered":"\u9ad8\u7b49\u6570\u5b66\u57fa\u7840\uff1a\u65b9\u5411\u5bfc\u6570\u7684\u8ba1\u7b97\u4e0e\u68af\u5ea6"},"content":{"rendered":"

\u65b9\u5411\u5bfc\u6570<\/h2>\n

\u65b9\u5411\u5bfc\u6570\u5b9a\u4e49<\/h3>\n

\u5b9a\u7406\uff1a\u82e5\u51fd\u6570$f(x,y,z)$\u5728\u70b9$P(x,y,z)$\u5904\u53ef\u5fae\uff0c\u6cbf\u4efb\u610f\u65b9\u5411$l$\u7684\u65b9\u5411\u5bfc\u6570<\/p>\n

$$
\n\\frac{\\partial f}{\\partial l} = \\frac{\\partial f}{\\partial x} \\cos \\alpha + \\frac{\\partial f}{\\partial y} \\cos \\beta + \\frac{\\partial f}{\\partial z} \\cos \\gamma
\n$$<\/p>\n

\u5176\u4e2d$\\alpha,\\beta,\\gamma$\u4e3a$l$\u7684\u65b9\u5411\u89d2<\/p>\n

<\/p>\n

\u8bc1\u660e\uff1a\u53c8\u51fd\u6570$f(x,y,z)$\u5728\u70b9$P$\u53ef\u5fae<\/p>\n

$$
\n\\begin{align}
\n\\Delta f &= \\frac{\\partial f}{\\partial x} \\Delta x + \\frac{\\partial f}{\\partial y} \\Delta y + \\frac{\\partial f}{\\partial z} \\Delta z + o(\\rho) \\
\n&= \\rho(\\frac{\\partial f}{\\partial x} \\cos \\alpha + \\frac{\\partial f}{\\partial y} \\cos \\beta + \\frac{\\partial f}{\\partial z} \\cos \\gamma) + o(\\rho)
\n\\end{align}
\n$$<\/p>\n

$$
\n\\frac{\\partial f}{\\partial l} = \\lim \\limits_{\\rho \\to 0} \\frac{\\Delta f}{\\rho} = \\rho(\\frac{\\partial f}{\\partial x} \\cos \\alpha + \\frac{\\partial f}{\\partial y} \\cos \\beta + \\frac{\\partial f}{\\partial z} \\cos \\gamma)
\n$$<\/p>\n

\u5bf9\u4e8e\u4e8c\u5143\u51fd\u6570$f(x,y)$\uff0c\u5728\u70b9$P(x,y)$\u5904\u6cbf\u65b9\u5411$l$\uff08\u65b9\u5411\u89d2\u4e3a$\\alpha,\\beta$\uff09\u7684\u65b9\u5411\u5bfc\u6570\u4e3a<\/p>\n

$$
\n\\begin{align}
\n\\frac{\\partial f}{\\partial l} &= \\lim \\limits_{\\rho \\to 0} \\frac{f(x+\\Delta x,y+\\Delta y) – f(x,y)}{\\rho} \\
\n&= f_x(x,y) \\cos \\alpha + f_y(x,y) \\cos \\beta
\n\\end{align}
\n$$<\/p>\n

$$
\n\\rho = \\sqrt{(\\Delta x)^2 + (\\Delta y)^2}, \\, \\Delta x = \\rho \\cos \\alpha, \\, \\Delta y = \\rho \\cos \\beta
\n$$<\/p>\n

\u7279\u522b<\/p>\n

$l$\u4e0e$x$\u8f74\u540c\u5411$\\alpha=0,\\beta=\\frac{\\pi}{2}$\u65f6\uff0c\u6709$\\frac{\\partial f}{\\partial l}=\\frac{\\partial f}{\\partial x}$
\n$l$\u4e0e$x$\u8f74\u53cd\u5411$\\alpha=\\pi,\\beta=\\frac{\\pi}{2}$\u65f6\uff0c\u6709$\\frac{\\partial f}{\\partial l}=-\\frac{\\partial f}{\\partial x}$<\/p>\n

\u65b9\u5411\u5bfc\u6570\u610f\u4e49<\/h3>\n

\u65b9\u5411\u5bfc\u6570(Directional Derivative)<\/strong>\uff1a\u6709\u65f6\u4e0d\u4ec5\u4ec5\u9700\u8981\u77e5\u9053\u51fd\u6570\u5728\u5750\u6807\u8f74\u4e0a\u7684\u53d8\u5316\u7387\uff08\u5373\u504f\u5bfc\u6570\uff09\uff0c\u800c\u4e14\u8fd8\u9700\u8981\u8bbe\u6cd5\u6c42\u5f97\u51fd\u6570\u5728\u5176\u4ed6\u7279\u5b9a\u65b9\u5411\u4e0a\u7684\u53d8\u5316\u7387\uff0c\u800c\u65b9\u5411\u5bfc\u6570\u5c31\u662f\u51fd\u6570\u5728\u5176\u5b83\u7279\u5b9a\u65b9\u5411\u4e0a\u7684\u53d8\u5316\u7387\u3002<\/p>\n

\u5982\u679c\u51fd\u6570$z=f(x,y)$\u5728\u70b9$P(x,y)$\u662f\u53ef\u5fae\u5206\u7684\uff0c\u90a3\u4e48\u51fd\u6570\u5728\u8be5\u70b9\u6cbf\u7740\u4efb\u4e00\u65b9\u5411$L$\u7684\u65b9\u5411\u5bfc\u6570\u90fd\u5b58\u5728\uff0c\u4e14\u8ba1\u7b97\u516c\u5f0f\u4e3a\uff1a<\/p>\n

$$
\n\\frac{\\partial f}{\\partial l} = \\frac{\\partial f}{\\partial x} \\cos \\alpha + \\frac{\\partial f}{\\partial y} \\cos \\beta
\n$$<\/p>\n

\u65b9\u5411\u5bfc\u6570\u4f8b\u9898<\/h3>\n

\u4f8b\uff1a\u6c42\u51fd\u6570$u=x^2yz$\u5728\u70b9$P(1,1,1)$\u6cbf\u5411\u91cf$\\vec{l}=(2,-1,3)$\u7684\u65b9\u5411\u5bfc\u6570<\/p>\n

\u89e3\uff1a\u5411\u91cf$\\vec{l}$\u7684\u65b9\u5411\u4f59\u5f26\u4e3a<\/p>\n

$$
\n\\cos \\alpha = \\frac{2}{\\sqrt{14}},\\,\\cos \\beta = \\frac{-1}{\\sqrt{14}},\\,\\cos \\gamma = \\frac{3}{\\sqrt{14}}
\n$$<\/p>\n

$$
\n\\frac{\\partial u}{\\partial l} \\left|p = \\left(2xyz\u00b7\\frac{2}{\\sqrt{14}} – x^2z\u00b7\\frac{1}{\\sqrt{14}} + x^2y\u00b7\\frac{3}{\\sqrt{14}}\\right) \\right|<\/em>{(1,1,1)} = \\frac{6}{\\sqrt{14}}
\n$$<\/p>\n

\u68af\u5ea6<\/h2>\n

\u68af\u5ea6(gradient)\u7684\u6982\u5ff5\u53ca\u8ba1\u7b97<\/h3>\n

\u5728\u7a7a\u95f4\u7684\u6bcf\u4e00\u4e2a\u70b9\u90fd\u53ef\u4ee5\u786e\u5b9a\u65e0\u9650\u591a\u4e2a\u65b9\u5411\uff0c\u56e0\u6b64\uff0c\u4e00\u4e2a\u591a\u5143\u51fd\u6570\u5728\u67d0\u4e2a\u70b9\u4e5f\u5fc5\u7136\u6709\u65e0\u9650\u591a\u4e2a\u65b9\u5411\u5bfc\u6570\u3002\u5728\u8fd9\u65e0\u9650\u591a\u4e2a\u65b9\u5411\u5bfc\u6570\u4e2d\uff0c\u63cf\u8ff0\u6700\u5927\u65b9\u5411\u5bfc\u6570\u53ca\u5176\u6240\u6cbf\u65b9\u5411\u7684\u77e2\u91cf\uff0c\u5c31\u662f\u68af\u5ea6\u3002\u68af\u5ea6\u662f\u573a\u8bba\u91cc\u7684\u4e00\u4e2a\u57fa\u672c\u6982\u5ff5\u3002<\/p>\n

\u65b9\u5411\u5bfc\u6570<\/p>\n

$$
\n\\frac{\\partial f}{\\partial l} = \\lim \\limits_{\\lambda \\to 0} \\frac{\\Delta f}{\\rho} = \\rho(\\frac{\\partial f}{\\partial x} \\cos \\alpha + \\frac{\\partial f}{\\partial y} \\cos \\beta + \\frac{\\partial f}{\\partial z} \\cos \\gamma)
\n$$<\/p>\n

\u4ee4\u5411\u91cf<\/p>\n

$$
\n\\vec{G} = \\left( \\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z} \\right)
\n$$<\/p>\n

$$
\n\\vec{l^o} = ( \\cos \\alpha, \\cos \\beta, \\cos \\gamma )
\n$$<\/p>\n

$$
\n\\frac{\\partial f}{\\partial l} = \\vec{G}\u00b7\\vec{l^o} = |\\vec{G}|\\cos(\\vec{G},\\vec{l^o}) \\,\\,\\,\\, (|\\vec{l^o}|=1)
\n$$<\/p>\n

\u5f53$\\vec{l^o}$\u4e0e$\\vec{G}$\u65b9\u5411\u4e00\u81f4\u65f6\uff0c\u65b9\u5411\u5bfc\u6570\u53d6\u5f97\u6700\u5927\u503c<\/p>\n

$$
\n\\operatorname{max}(\\frac{\\partial f}{\\partial l}) = |\\vec{G}|
\n$$<\/p>\n

$$
\n\\vec{G}:\\left \\{
\n\\begin{array} {c}
\n\u65b9\u5411: f\u53d8\u5316\u7387\u6700\u5927\u7684\u65b9\u5411 \\
\n\u6a21: f\u7684\u6700\u5927\u53d8\u5316\u7387\u4e4b\u503c
\n\\end{array}
\n\\right.
\n$$<\/p>\n

\u68af\u5ea6\u5b9a\u4e49<\/h3>\n

\u5411\u91cf$\\vec{G}$\uff1a\u79f0\u4e3a\u51fd\u6570$f(P)$\u5728\u70b9$P$\u5904\u7684\u68af\u5ea6(gradient)\uff0c\u8bb0\u4f5c$\\operatorname{grad} f$\uff0c\u5373<\/p>\n

$$
\n\\operatorname{grad} f = \\left( \\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z} \\right) = \\left( \\frac{\\partial f}{\\partial x}\\vec{i} + \\frac{\\partial f}{\\partial y}\\vec{j}, \\frac{\\partial f}{\\partial z}\\vec{k} \\right)
\n$$<\/p>\n

\u540c\u6837\u53ef\u5b9a\u4e49\u4e8c\u5143\u51fd\u6570$f(x,y)$\u5728\u70b9$P(x,y)$\u5904\u7684\u68af\u5ea6<\/p>\n

$$
\n\\operatorname{grad} f = \\frac{\\partial f}{\\partial x}\\vec{i} + \\frac{\\partial f}{\\partial y}\\vec{j} = \\left( \\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y} \\right)
\n$$<\/p>\n

\n

\u8bf4\u660e\uff1a\u51fd\u6570\u7684\u65b9\u5411\u5bfc\u6570\u4e3a\u68af\u5ea6\u5728\u8be5\u65b9\u5411\u4e0a\u7684\u6295\u5f71<\/p>\n<\/blockquote>\n

$\\nabla = \\left( \\frac{\\partial}{\\partial x},\\frac{\\partial}{\\partial y} \\right)$\u5f15\u7528\u8bb0\u53f7\uff0c\u79f0\u4e3a\u5948\u5e03\u62c9(Nebla)\u7b97\u7b26\uff0c\u6216\u79f0\u4e3a\u5411\u91cf\u5fae\u5206\u7b97\u5b50\uff0c\u6216\u54c8\u5bc6\u987f(W.R.Hamilton)\u7b97\u5b50\u3002\u5219\u68af\u5ea6\u53ef\u8bb0\u4e3a<\/p>\n

$$
\n\\operatorname{grad} f = \\left( \\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y} \\right) = \\nabla f
\n$$<\/p>\n

\u51fd\u6570$f$\u6cbf\u68af\u5ea6$\\operatorname{grad} f$\u65b9\u5411\uff0c\u589e\u52a0\uff08\u4e0a\u5347\uff09\u6700\u5feb
\n\u51fd\u6570$f$\u6cbf\u8d1f\u68af\u5ea6$-\\operatorname{grad} f$\u65b9\u5411\uff0c\u51cf\u5c0f\uff08\u4e0b\u964d\uff09\u6700\u5feb<\/p>\n

\u4ee5\u4e09\u5143\u51fd\u6570\u4e3a\u4f8b\uff0c\u8bbe$u=f(x,y,z)$\u5728\u70b9$P(x,y,z)$\u5904\u53ef\u5fae\u5206\uff0c\u5219\u51fd\u6570\u5728\u8be5\u70b9\u7684\u68af\u5ea6\u4e3a<\/p>\n

$$
\n\\operatorname{grad} f = \\nabla f = \\frac{\\partial f}{\\partial x}\\vec{i} + \\frac{\\partial f}{\\partial y}\\vec{j} + \\frac{\\partial f}{\\partial z}\\vec{k} = \\left( \\frac{\\partial f}{\\partial x},\\frac{\\partial f}{\\partial y},\\frac{\\partial f}{\\partial z} \\right) = \\left( \\frac{\\partial f}{\\partial (x,y,z)} \\right)
\n$$<\/p>\n

\u68af\u5ea6\u662f\u51fd\u6570$u=f(x,y,z)$\u5728\u70b9$P$\u5904\u53d6\u5f97\u6700\u5927\u65b9\u5411\u5bfc\u6570\u7684\u65b9\u5411\uff0c\u6700\u5927\u65b9\u5411\u5bfc\u6570\u4e3a<\/p>\n

$$
\n|\\operatorname{grad} f| = \\sqrt{(\\frac{\\partial f}{\\partial x})^2 + (\\frac{\\partial f}{\\partial y})^2 + (\\frac{\\partial f}{\\partial z})^2}
\n$$<\/p>\n

\u51fd\u6570$u=f(x,y,z)$\u5728\u70b9$P$\u5904\u6cbf\u65b9\u5411$\\vec{l}$\u7684\u65b9\u5411\u5bfc\u6570<\/p>\n

$$
\n\\frac{\\partial f}{\\partial l} = \\operatorname{grad} f \u00b7 \\vec{l^o} = \\nabla f \u00b7 \\vec{l^o}
\n$$<\/p>\n

\u68af\u5ea6\u793a\u4f8b<\/h3>\n

\u4f8b1\uff1a\u6c42$\\operatorname{grad} = \\frac{1}{x^2+y^2}$<\/p>\n

\u89e3\uff1a\u8fd9\u91cc$f(x,y)=\\frac{1}{x^2+y^2}$<\/p>\n

\u56e0\u4e3a $\\frac{\\partial f}{\\partial x}=-\\frac{2x}{(x^2+y^2)^2},\\frac{\\partial f}{\\partial y}=-\\frac{2y}{(x^2+y^2)^2}$<\/p>\n

\u6240\u4ee5 $\\operatorname{grad} \\frac{1}{x^2+y^2} = -\\frac{2x}{(x^2+y^2)^2} \\vec{i} -\\frac{2y}{(x^2+y^2)^2} \\vec{j}$<\/p>\n

\u4f8b2\uff1a\u8bbe$f(x,y,z)=x^3-xy^2-z$\uff0c$f(x,y,z)$\u5728\u70b9$P(1,1,0)$\u5904\u6cbf\u4ec0\u4e48\u65b9\u5411\u53d8\u5316\u6700\u5feb\uff0c\u5728\u8fd9\u65b9\u5411\u7684\u53d8\u5316\u7387\u662f\u591a\u5c11\uff1f<\/p>\n

\u89e3\uff1a$\\nabla f = f_xi+f_yj+f_zk = (3x^2-y^2)i-2xyj-k$<\/p>\n

$\\nabla f(1,1,0) = 2i-2j-k$<\/p>\n

\u6cbf$\\nabla f(1,1,0)$\u65b9\u5411\uff0c\u589e\u52a0\u6700\u5feb\uff08\u4e0a\u5347\uff09\uff0c$-\\nabla f(1,1,0)$\u65b9\u5411\uff0c\u51cf\u5c0f\u6700\u5feb\uff08\u4e0b\u964d\uff09<\/p>\n

$
\n\\operatorname{max} \\left\\{\\frac{\\partial f}{\\partial l}\\left|_p\\right.\\right\\} = |\\operatorname{grad} f| = |\\operatorname{grad} f(1,1,0)| = 3
\n$<\/p>\n

$
\n\\operatorname{min} \\left\\{\\frac{\\partial f}{\\partial l}\\left|_p\\right.\\right\\} = -|\\operatorname{grad} f| = -|\\operatorname{grad} f(1,1,0)| = -3
\n$<\/p>\n","protected":false},"excerpt":{"rendered":"

\u65b9\u5411\u5bfc\u6570 \u65b9\u5411\u5bfc\u6570\u5b9a\u4e49 \u5b9a\u7406\uff1a\u82e5\u51fd\u6570$f(x,y,z)$\u5728\u70b9$P(x,y,z)$\u5904\u53ef\u5fae\uff0c\u6cbf\u4efb\u610f\u65b9\u5411$l$\u7684\u65b9\u5411 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[509],"tags":[],"class_list":["post-2136","post","type-post","status-publish","format-standard","hentry","category-mathematics-fundamentals"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2136","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=2136"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2136\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2136"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2136"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2136"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}