{"id":14512,"date":"2024-10-07T11:40:37","date_gmt":"2024-10-07T03:40:37","guid":{"rendered":"https:\/\/www.orczhou.com\/?p=14512"},"modified":"2026-01-16T11:12:40","modified_gmt":"2026-01-16T03:12:40","slug":"directional-derivative-partial-derivative","status":"publish","type":"post","link":"https:\/\/www.orczhou.com\/index.php\/2024\/10\/directional-derivative-partial-derivative\/","title":{"rendered":"\u4e8c\u5143\u51fd\u6570\u7684\u504f\u5bfc\u6570\u3001\u65b9\u5411\u5bfc\u6570\u3001\u68af\u5ea6"},"content":{"rendered":"\n\n\n\n

\u68af\u5ea6\u4e0b\u964d\u6cd5\uff08\u6216\u8005\u5176\u6539\u8fdb\u7b97\u6cd5\uff09\u662f\u673a\u5668\u5b66\u4e60\u7684\u57fa\u7840\u7b97\u6cd5\u4e4b\u4e00\u3002\u5728\u4e86\u89e3\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4f1a\u7ecf\u5e38\u770b\u5230\u4e00\u53e5\u8bdd\uff1a\u201c\u68af\u5ea6\u662f\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u53d8\u5316\u7387\u6700\u5927\u7684\u65b9\u5411\u201d\u3002\u672c\u6587\u4ece\u8f83\u4e3a\u4e25\u683c\u6570\u5b66\u8bc1\u660e\u7684\u89d2\u5ea6\u8bf4\u660e\u4e3a\u4ec0\u4e48\u662f\u8fd9\u6837\u3002\u7406\u89e3\u8fd9\u4e2a\u8bc1\u660e\u8fc7\u7a0b\uff0c\u53ef\u4ee5\u5f88\u597d\u7684\u7406\u89e3\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\uff0c\u53ca\u5176\u4f18\u5316\u7b97\u6cd5\u6216\u8005\u4f18\u5316\u65b9\u5411\u3002<\/p>\n\n\n\n

\u672c\u6587\u4e3b\u8981\u8003\u8651\u4e8c\u5143\u51fd\u6570\u573a\u666f\uff0c\u5373 \\(z=f(x,y) \\)\u3002\u539f\u56e0\u662f\u4e00\u5143\u51fd\u6570\u573a\u666f\u8fc7\u4e8e\u7b80\u5355\uff0c\u4e0d\u5177\u6709\u4ee3\u8868\u6027\uff0c\u53e6\u5916\uff0c\u4e8c\u5143\u573a\u666f\u5411\u591a\u5143\u573a\u666f\u63a8\u5e7f\u4e5f\u8fd8\u6bd4\u8f83\u597d\u7406\u89e3\u3002<\/p>\n\n\n\n

1. \u57fa\u672c\u6982\u5ff5<\/h3>\n\n\n\n

1.1 \u504f\u5bfc\u6570<\/h4>\n\n\n\n

\u504f\u5bfc\u6570\u7684\u5b9a\u4e49\u6bd4\u8f83\u597d\u7406\u89e3\uff0c\u5373\u56fa\u5b9a\u4e00\u4e2a\u53d8\u91cf\uff08\u5f53\u505a\u5e38\u6570\uff09\uff0c\u5bf9\u53e6\u4e00\u4e2a\u53d8\u91cf\u6c42\u5bfc\uff0c\u8bb0\u4f5c\uff1a<\/p>\n\n\n\n

$$ \\frac{\\partial z}{\\partial x} \\; , \\; \\frac{\\partial z}{\\partial y} $$<\/p>\n\n\n\n

1.2 \u68af\u5ea6\u5411\u91cf<\/h4>\n\n\n\n

\u7531\u5404\u4e2a\u504f\u5bfc\u6570\u7ec4\u6210\u7684\u5411\u91cf\uff0c\u5c31\u53eb\u68af\u5ea6\u5411\u91cf\uff0c\u901a\u5e38\u8bb0\u4f5c\uff1a\\( \\nabla \\)\uff0c\u6709\uff1a<\/p>\n\n\n

$$ \\nabla f = (\\frac{\\partial z}{\\partial x} , \\frac{\\partial z}{\\partial y} ) $$<\/p>\n\n\n\n

\u591a\u5143\/\u591a\u7ef4\u573a\u666f\uff0c\u5219\u5e38\u8bb0\u4f5c\uff1a<\/p>\n\n\n

$$ \\nabla f = (\\frac{\\partial f}{\\partial x_1} , \\frac{\\partial f}{\\partial x_2} … , \\frac{\\partial f}{\\partial x_n} ) $$<\/p>\n\n\n\n

1.3 \u65b9\u5411\u5bfc\u6570<\/h4>\n\n\n\n

\u591a\u5143\u51fd\u6570\u6ca1\u6709\u7b80\u5355\u7684\u201c\u5bfc\u6570\u201d\u7684\u6982\u5ff5\u3002\u4f46\u4e3a\u4e86\u7814\u7a76\u591a\u5143\u51fd\u6570\u5728\u67d0\u70b9\u7684\u53d8\u5316\u7387\uff0c\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u201c\u65b9\u5411\u5bfc\u6570\u201d\u3002<\/p>\n\n\n\n

\u5177\u4f53\u7684\uff0c\u8003\u8651\u51fd\u6570 \\( z = f(x,y) \\)\uff0c\u8be5\u51fd\u6570\u5b9a\u4e49\u57df\u4e3a\\( \\mathbb{R}^2 \\)\uff0c\u5176\u65b9\u5411\u5411\u91cf\u662f $$ \\{ u,v | u^2 +v^2 = 1 \\} $$\uff0c\u53d6\u5176\u4e2d\u7684\u4e00\u4e2a\u65b9\u5411 \\( l = (u_0,v_0) \\)\uff0c\u5e76\u5047\u8bbe\u8be5\u65b9\u5411\u4e0e\\( x \\)\u8f74\u6b63\u65b9\u5411\u5939\u89d2\u4e3a\\( \\theta \\)\u3002<\/p>\n\n\n\n

\u90a3\u4e48\uff0c\u51fd\u6570\\( z = f(x,y) \\)\u5728\u70b9\\( (x_0,y_0) \\)\u5904\uff0c\u5728\u65b9\u5411 \\( l = (u_0,v_0) \\)\u7684\u5bfc\u6570\u8bb0\u4f5c <\/p>\n\n\n\n

$$ \\frac{\\partial z}{\\partial l} |_{(x_0,y_0)} $$<\/p>\n\n\n\n

\n
\n

2. \u76f4\u89c2\u7406\u89e3\u65b9\u5411\u5bfc\u6570<\/h3>\n\n\n\n

\u56fe1\u662f\u4e00\u4e2a\u975e\u5e38\u6e05\u6670\u7684\u5173\u4e8e\u65b9\u5411\u5bfc\u6570\u7684\u56fe\u4f8b\u3002\u7eff\u8272\u66f2\u9762\u5373\u4e3a \\( z = f(x,y) \\)\uff0c\u5728\u70b9\\( A^\\prime \\)\u4e0a\u8003\u8651\u65b9\u5411\u4e3a\\( \\vec{h}\\)\u7684\u65b9\u5411\u5bfc\u6570\u3002\u8fc7\u70b9\\( A^\\prime \\)\u4e0e\u65b9\u5411\\( \\vec{h}\\)\uff0c\u4e0e\\( z \\)\u8f74\u5e73\u884c\uff0c\u5b58\u5728\u4e00\u4e2a\u5e73\u9762\uff0c\u5373\u56fe1\u4e2d\u7684\u534a\u900f\u660e\u7684\u5e73\u9762\uff0c\u8be5\u5e73\u9762\u4e0e \\( z = f(x,y) \\)\u76f8\u4ea4\u4e0e\u4e00\u6761\u66f2\u7ebf\uff0c\u5373\u56fe1\u4e2d\u7684\u9ec4\u8272\u66f2\u7ebf\u3002<\/p>\n\n\n\n

\u90a3\u4e48\uff0c\u8be5\u65b9\u5411\u5bfc\u6570\uff0c\u5373\u4e3a\u5728\u8be5\u9ec4\u8272\u66f2\u7ebf\u4e0a\uff0c\\( A^\\prime \\)\u4f4d\u7f6e\u7684\u5bfc\u6570\u3002\u8fd9\u5c31\u662f\u5173\u4e8e\u65b9\u5411\u5bfc\u6570\u7684\u76f4\u89c2\u7406\u89e3\u3002<\/p>\n\n\n\n

\u6240\u4ee5\uff0c\u504f\u5bfc\u6570\\( \\frac{\\partial z}{\\partial x} \\; , \\; \\frac{\\partial z}{\\partial y} \\)\u53ef\u4ee5\u7406\u89e3\u4e3a\u5728\\( (1,0) \\)\u548c\\( (0,1) \\)\u8fd9\u4e24\u4e2a\u65b9\u5411\u4e0a\u7684\u65b9\u5411\u5bfc\u6570\u3002<\/p>\n<\/div>\n\n\n\n

\n
<\/div>\n\n\n\n
\"\"
\u56fe1\uff1a\u6765\u81eaWikipedia: Directional derivative<\/figcaption><\/figure>\n<\/div>\n<\/div>\n\n\n\n

<\/p>\n\n\n\n

\n
\n

\u4e0e\u4e00\u822c\u7684\u5bfc\u6570\u5b9a\u4e49\u7c7b\u4f3c\u7684\uff0c\u53ef\u4ee5\u5b9a\u4e49\u65b9\u5411\u5bfc\u6570\uff1a<\/p>\n\n\n

$$ \\frac{\\partial z}{\\partial l} |_{(x_0,y_0)} = \\lim\\limits_{P \\to P_0} = \\frac{f(P) – f(P_0)}{||P-P_0||} = \\lim\\limits_{\\rho \\to 0} \\frac{\\Delta z}{ \\rho } $$<\/p>\n<\/div>\n\n\n\n

\n
\"\"
\u56fe2\uff1a\\( P \\) \u70b9\u5728\\( (u,v) \\)\u65b9\u5411\u903c\u8fd1\\( P_0 \\)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n\n\n\n

\u53ef\u4ee5\u5230\u5982\u4e0b\u7ed3\u8bba\uff08\u8be6\u7ec6\u8bc1\u660e\u53c2\u8003\u540e\u7eed\u5c0f\u8282\u201c\u65b9\u5411\u5bfc\u6570\u7684\u8ba1\u7b97\u4e0e\u8bc1\u660e\u201d\uff09\uff0c\u5982\u679c\u65b9\u5411\\( l = (u_0,v_0) \\)\u4e0e \\( x \\)\u8f74\u7684\u5939\u89d2\u662f\\( \\theta \\)\uff0c\u90a3\u4e48\\( z = f(x,y) \\)\u5728\u70b9\\( (x_0,y_0) \\)\u5904\uff0c\u5728\u65b9\u5411 \\( l = (u_0,v_0) \\)\u7684\u5bfc\u6570\u53d6\u503c\u5982\u4e0b\uff1a<\/p>\n\n\n\n

$$ \\frac{\\partial z}{\\partial l} |_{(x_0,y_0)} = \\frac{\\partial z}{\\partial x} |_{(x_0,y_0)} cos(\\theta) + \\frac{\\partial z}{\\partial y} |_{(x_0,y_0)} sin(\\theta) \\tag{1} $$<\/p>\n\n\n\n

\u6839\u636e\u67ef\u897f\u4e0d\u7b49\u5f0f\uff0c\u6211\u4eec\u6709\u5982\u4e0b\u7ed3\u8bba\uff1a<\/p>\n\n\n

$$ \\frac{\\partial z}{\\partial l} |_{(x_0,y_0)} = \\frac{\\partial z}{\\partial x} |_{(x_0,y_0)} cos(\\theta) + \\frac{\\partial z}{\\partial y} |_{(x_0,y_0)} sin(\\theta)
\n\\\\
\n\\le \\sqrt{ ((\\frac{\\partial z}{\\partial x} |_{(x_0,y_0)})^2 + (\\frac{\\partial z}{\\partial y} |_{(x_0,y_0)})^2)(sin^2(\\theta)+cos^2(\\theta)) }
\n\\\\
\n= \\sqrt{ (\\frac{\\partial z}{\\partial x} |_{(x_0,y_0)})^2 + (\\frac{\\partial z}{\\partial y} |_{(x_0,y_0)})^2 }
\n$$<\/p>\n\n\n\n

<\/p>\n\n\n\n

\u4e0a\u9762\u8868\u793a\u7684\u6781\u503c \\( \\sqrt{ (\\frac{\\partial z}{\\partial x} |_{(x_0,y_0)})^2 + (\\frac{\\partial z}{\\partial y} |_{(x_0,y_0)})^2 } \\) \u6b63\u662f\u504f\u5bfc\u6570\u5411\u91cf\u7684\u201c\u8303\u6570\u201d\uff08\u957f\u5ea6\uff09\uff0c\u6839\u636e\u67ef\u897f\u4e0d\u7b49\u5f0f\u53d6\u6700\u5927\u503c\u7684\u6761\u4ef6\u4e5f\u6709\uff1a<\/p>\n\n\n

$$
\n\\frac{cos(\\theta)}{\\frac{\\partial z}{\\partial x}} = \\frac{sin(\\theta)}{\\frac{\\partial z}{\\partial y}}
\n\\\\
\ntan(\\theta) = \\frac{\\frac{\\partial z}{\\partial y} } { \\frac{\\partial z}{\\partial x} } = \\frac{\\Delta y}{\\Delta x}
\n$$<\/p>\n\n\n\n

\u6240\u4ee5\uff0c\u5373\uff0c\u5373\u5f53\u65b9\u5411\u6070\u597d\u4e3a\u504f\u5bfc\u6570\u5411\u91cf\u65f6\uff0c\u65b9\u5411\u5bfc\u6570\u53d6\u6700\u5927\u503c\u3002\u4e5f\u5c31\u662f\uff0c\u6211\u4eec\u7ecf\u5e38\u4f1a\u8bf4\u7684\uff0c\u4f1a\u770b\u5230\u7684\uff0c\u201c\u504f\u5bfc\u6570\u5411\u91cf\u662f\u6240\u6709\u65b9\u5411\u4e2d\u6700\u4e3a\u9661\u5ced\u7684\u65b9\u5411\u201d<\/span>\u6216\u8005\u8bf4\u201c\u68af\u5ea6\u662f\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u53d8\u5316\u7387\u6700\u5927\u7684\u65b9\u5411<\/span>\u201d\u3002<\/p>\n\n\n\n

<\/p>\n\n\n\n

3. \u65b9\u5411\u5bfc\u6570\u7684\u8ba1\u7b97\u4e0e\u8bc1\u660e<\/h3>\n\n\n\n

\u5728\u524d\u9762\uff0c\u6211\u4eec\u662f\u76f4\u63a5\u7ed9\u51fa\u4e86\u5982\u4e0b\u7684\u7ed3\u8bba\u7684\uff1a<\/p>\n\n\n

$$ \\frac{\\partial z}{\\partial l} |_{(x_0,y_0)} = \\frac{\\partial z}{\\partial x} |_{(x_0,y_0)} sin(\\theta) + \\frac{\\partial z}{\\partial y} |_{(x_0,y_0)} cos(\\theta)$$<\/p>\n\n\n\n

\u8fd9\u4e2a\u7ed3\u8bba\u7684\u83b7\u5f97\uff0c\u662f\u9700\u8981\u6709\u4e00\u4e9b\u6bd4\u8f83\u590d\u6742\u7684\u8ba1\u7b97\u6216\u8005\u8bf4\u8bc1\u660e\u7684\u3002\u8fd9\u91cc\uff0c\u5176\u4e3b\u8981\u8bc1\u660e\u6b65\u9aa4\/\u65b9\u6cd5\u4e4b\u4e00\uff0c\u5982\u4e0b\uff1a<\/p>\n\n\n

\\( \\frac{\\partial z}{\\partial l} |_{(x_0,y_0)} = \\lim\\limits_{P->P_0}\\frac{f(P)-f(P_0)}{|P-P_0|} = \\lim\\limits_{P->P_0}\\frac{f(x_0+\\Delta{x},y_0+\\Delta{y})-f(x_0,y_0)}{\\sqrt{\\Delta{x}^2+\\Delta{y}^2}}
\n\\)<\/p>\n\n\n\n

\u7531\u62c9\u683c\u6717\u65e5\u4e2d\u503c\u5b9a\u7406\uff1a\u5b58\u5728\\( \\alpha \\; \\beta \\)\uff0c\u4f7f\u5f97\u4e0b\u5f0f\u6210\u7acb\uff0c\u4e14 \\( 0 \\le \\alpha \\le 1 \\; and \\; 0 \\le \\beta \\le 1 \\)\uff1a<\/p>\n\n\n

\\(
\nf(x_0+\\Delta{x},y_0+\\Delta{y})-f(x_0,y_0)
\n\\\\
\n= [f(x_0+\\Delta{x},y_0+\\Delta{y}) – f(x_0,y_0+\\Delta{y})] + [f(x_0,y_0+\\Delta{y}) -f(x_0,y_0)]
\n\\\\
\n= f_x'(x_0 + \\alpha\\Delta{x} ,y_0+\\Delta{y})\\Delta{x} + f_y'(x_0, y_0 + \\beta\\Delta{y} )\\Delta{y}
\n\\)<\/p>\n\n\n\n

\u5bb9\u6613\u6709\uff0c\u8fd9\u51e0\u4e2a\u6761\u4ef6\u662f\u7b49\u4ef7\u7684\uff1a \\( P \\to P_0 \\)\u3001\\( \\Delta{x} \\to 0 \\, and \\, \\Delta{y} \\to 0 \\) \u3001\\( \\sqrt{\\Delta{x}^2+\\Delta{y}^2} \\to 0 \\)<\/p>\n\n\n\n

\u8003\u8651\\( \\frac{\\partial z}{\\partial x} \\)\u5728\\( (x_0,y_0)\\)\u5904\u8fde\u7eed\uff08\u8fd9\u662f\u4e00\u4e2a\u6761\u4ef6\uff09\uff0c\u5219\u6709\uff1a $$ \\lim\\limits_{\\Delta{x} \\to 0 \\\\ \\Delta {y} \\to 0 }f_x'(x_0 + \\alpha\\Delta{x} ,y_0+\\Delta{y}) = f_x'(x_0,y_0) $$<\/p>\n\n\n\n

\u6545\uff1a<\/p>\n\n\n

$$
\n\\begin{align}
\n\\frac{\\partial z}{\\partial l} |_{(x_0,y_0)} & = \\lim\\limits_{P->P_0}\\frac{f(P)-f(P_0)}{|P-P_0|}
\n\\\\
\n& = \\lim\\limits_{P->P_0}\\frac{f(x_0+\\Delta{x},y_0+\\Delta{y})-f(x_0,y_0)}{\\sqrt{\\Delta{x}^2+\\Delta{y}^2}}
\n\\\\
\n& =\\lim\\limits_{P->P_0}\\frac{f_x'(x_0+\\alpha\\Delta{x},y_0+\\Delta{y})\\Delta{x} + f_y'(x_0,y_0+\\Delta{y})\\Delta{y}}{\\sqrt{\\Delta{x}^2+\\Delta{y}^2}}
\n\\\\
\n& =\\lim\\limits_{P->P_0}\\frac{f_x'(x_0+\\alpha\\Delta{x},y_0+\\Delta{y})\\Delta{x}}{\\sqrt{\\Delta{x}^2+\\Delta{y}^2}} + \\frac{f_y'(x_0,y_0+\\Delta{y})\\Delta{y}}{\\sqrt{\\Delta{x}^2+\\Delta{y}^2}}
\n\\end{align}
\n$$<\/p>\n\n\n\n

\u6839\u636e\u4e0a\u9762\u7684\u56fe2\uff0c\u5bb9\u6613\u6709\uff1a<\/p>\n\n\n

$$
\n\\frac{\\Delta{x}}{\\sqrt{\\Delta{x}^2+\\Delta{y}^2}} = cos(\\theta) \\quad \\frac{\\Delta{y}}{\\sqrt{\\Delta{x}^2+\\Delta{y}^2}} = sin(\\theta)
\n$$<\/p>\n\n\n\n

\u6240\u4ee5\uff1a<\/p>\n\n\n

\\( =\\lim\\limits_{P->P_0}\\frac{f_x'(x_0+\\alpha\\Delta{x},y_0+\\Delta{y})\\Delta{x}}{\\sqrt{\\Delta{x}^2+\\Delta{y}^2}} + \\frac{f_y'(x_0,y_0+\\Delta{y})\\Delta{y}}{\\sqrt{\\Delta{x}^2+\\Delta{y}^2}}
\n\\\\
\n=f_x'(x_0,y_0)cos(\\theta) + f_y'(x_0,y_0)sin(\\theta)
\n\\\\
\n\\)<\/p>\n\n\n\n

\u597d\u4e86\uff0c\u8fd9\u5c31\u8bc1\u660e\u5b8c\u6210\u4e86\u3002<\/p>\n\n\n\n

4. \u5173\u4e8e\u4e0a\u8ff0\u8bc1\u660e<\/h3>\n\n\n\n

\u4e0a\u8ff0\u8bc1\u660e\uff0c\u5728\u4e00\u822c\u7684\u300a\u6570\u5b66\u5206\u6790\u300b\u6559\u7a0b\u7684\u201c\u591a\u5143\u51fd\u6570\u5fae\u5206\u201d\u76f8\u5173\u7ae0\u8282\u90fd\u4f1a\u6709\uff0c\u6216\u8005\u4f1a\u6709\u7c7b\u4f3c\u7684\u95ee\u9898\u8bc1\u660e\u3002\u8fc7\u7a0b\u8fd8\u662f\u6bd4\u8f83\u5de7\u5999\u7684\uff0c\u5148\u662f\u201c\u65e0\u4e2d\u751f\u6709\u201d\u65b0\u589e\u4e86\u4e00\u4e2a\u9879\uff08\\( f(x_0,y_0+\\Delta{y}) \\)\uff09\uff0c\u5206\u522b\u6784\u9020\u4e86\u5173\u4e8e \\( x \\)\u548c\\( y \\)\u7684\u504f\u5bfc\u6570\uff0c\u7136\u540e\u4f7f\u7528\u4e86\u201c\u4e2d\u503c\u5b9a\u7406\u201d\uff0c\u5c06\u5dee\u503c\u53d8\u6210\uff0c\u5bfc\u6570\u548c\u5fae\u5206\u53d8\u91cf\u7684\u79ef\uff08\u51c6\u786e\u7684\u8bf4\uff0c\u8fd8\u8981\u52a0\u4e0a\u4e00\u4e2a\u5173\u4e8e\\( \\rho \\)\u7684\u9ad8\u9636\u65e0\u7a77\u5c0f\uff09\u3002<\/p>\n\n\n\n

<\/p>\n\n\n\n

5. \u5411\u91cf\u5f62\u5f0f\u5316\u8868\u8fbe<\/h3>\n\n\n\n

\u4f7f\u7528\u5411\u91cf\u5f62\u5f0f\u5316\u8868\u8fbe\uff0c\u770b\u8d77\u6765\u4f1a\u7b80\u6d01\u5f88\u591a\u3002\u5bf9\u4e8e\u65b9\u5411\u5411\u91cf\uff08\u8fd9\u4e5f\u662f\u4e00\u4e2a\u5355\u4f4d\u5411\u91cf\uff09 \\( \\mathbf{l} = (u,v)\\)\uff0c\u51fd\u6570\\( f \\)\u7684\u504f\u5bfc\u6570\u5411\u91cf\u8bb0\u4e3a\\( \\nabla f = (\\frac{\\partial z}{\\partial x} , \\frac{\\partial z}{\\partial y} ) \\) \uff0c\u90a3\u4e48\u65b9\u5411\u5bfc\u6570\u4e3a \\( D_{\\mathbf{l}}f(P_0) = \\nabla f \\cdot \\mathbf{l} \\) \uff0c\u8fd9\u4e0e\u4e0a\u9762\u8868\u8fbe\u5f0f\u7684\u610f\u4e49\u662f\u76f8\u540c\u7684\u3002<\/p>\n\n\n\n

\u6839\u636e\u70b9\u51fb\u7684\u6027\u8d28\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n\n\n

\\( D_{\\mathbf{l}}f(P_0) = \\nabla f \\cdot \\mathbf{l} = ||\\nabla f|| ||\\mathbf{l} || cos\\theta = ||\\nabla f|| cos\\theta \\)<\/p>\n\n\n\n

<\/p>\n\n\n\n

\u4ece\u8fd9\u91cc\uff0c\u66f4\u5bb9\u6613\u770b\u51fa\uff0c\u65b9\u5411\u5411\u91cf\u4e0e\u68af\u5ea6\u5411\u91cf\u76f8\u540c\u65f6\uff0c\u65b9\u5411\u5bfc\u6570\u53d6\u6700\u5927\u503c\uff0c\u6700\u5927\u503c\u5373\u4e3a\u68af\u5ea6\u5411\u91cf\u7684\u6a21\u3002<\/p>\n\n\n\n

6. \u591a\u7ef4\u573a\u666f\u6269\u5c55<\/h3>\n\n\n\n

\u5728\u5f88\u591a\u7684\u6750\u6599\u4e2d\uff0c\u5728\u524d\u9762\u7684\u8868\u8fbe\u5f0f\u4e2d\uff0c\u7ecf\u5e38\u4f1a\u770b\u5230\u7684\u662f \\( cos(\\alpha) \\; cos(\\beta) \\)\uff0c\u800c\u4e0d\u662f\u672c\u6587\u4e2d\u7684 \\( sin(\\theta) \\; cos(\\theta) \\)\u3002\u8fd9\u91cc\u7684 \\( \\alpha \\)\u662f\u65b9\u5411\u5411\u91cf\u4e0ex\u8f74\u6b63\u65b9\u5411\u7684\u5939\u89d2\uff0c \\( \\beta \\)\u662f\u65b9\u5411\u5411\u91cf\u4e0ey\u8f74\u6b63\u65b9\u5411\u7684\u5939\u89d2\uff1b\u5728\u5b9a\u4e49\u57df \\( \\mathbb{R}^2 \\)\u4e0a\u6709\uff1a\\( \\alpha + \\beta = 90^{\\circ} \\)\uff0c\u5373\u6709 \\( cos^2\\alpha + cos^2\\beta = 1 \\)\u3002<\/p>\n\n\n\n

\u8fd9\u79cd\u5199\u6cd5\u6709\u7740\u66f4\u597d\u7684\u6269\u5c55\u6027\uff0c\u5f53\u5728\u66f4\u591a\u5143\u7684\u60c5\u51b5\u4e0b\uff0c\u4f8b\u5982\u4e09\u5143\u573a\u666f\u4e0b\uff0c\u5373 \\( z = f(x_1,x_2,x_3) \\)\uff0c\u65b9\u5411\u5411\u91cf\u4e0e x\uff0cy\uff0cz\u8f74\u7684\u5939\u89d2\u5206\u522b\u662f\uff1a\\( \\alpha \\; \\beta \\; \\gamma \\)\uff0c\u5219\u6709\uff1a \\( cos^2\\alpha + cos^2\\beta + cos^2 \\gamma = 1 \\)\u3002<\/p>\n\n\n\n

\u4efb\u610f\u7ef4\u5ea6\uff0c\u4e5f\u6709\u7c7b\u4f3c\u7684\u7ed3\u8bba\uff0c\u5e76\u4e14\u5e94\u7528\u67ef\u897f\u4e0d\u7b49\u5f0f\u65f6\uff0c\u4e0a\u8ff0\u7ed3\u8bba\u4e5f\u662f\u7c7b\u4f3c\u7684\u3002<\/p>\n\n\n\n

<\/p>\n\n\n\n

7. \u8bf4\u660e\uff1a\u76f4\u89c9<\/h3>\n\n\n\n

\u672c\u6587\u5185\u5bb9\u9700\u8981\u6216\u8005\u53ef\u4ee5\u5efa\u7acb\u5982\u4e0b\u7684\u201c\u76f4\u89c9\u201d\uff1a<\/p>\n\n\n\n

    \n
  • \u5728\u4e00\u7ef4\u7a7a\u95f4\uff08\u5373\\( \\mathbb{R}\\)\u4e0a\u7684\u51fd\u6570\uff0c\u5728\u67d0\u4e00\u70b9\u4e0a\u7684\u4e00\u9636\u5bfc\u6570\u7684\u7b26\u53f7\uff08\u6b63\/\u8d1f\uff09\uff0c\u53ef\u4ee5\u4ee3\u8868\u5728\u8be5\u65b9\u5411\u4e0a\uff0c\u51fd\u6570\u7684\u8d8b\u52bf\u662f\u589e\u957f\u8fd8\u662f\u4e0b\u964d\uff0c\u201c\u6b63\u53f7\u201d\uff0c\u5219\u662f\u589e\u957f\uff1b\u201c\u8d1f\u53f7\u201d\uff0c\u5219\u662f\u4e0b\u964d\u3002<\/li>\n\n\n\n
  • \u5728\u4e00\u7ef4\u7a7a\u95f4\uff08\u5373\\( \\mathbb{R}\\)\u4e0a\u7684\u51fd\u6570\uff0c\u5728\u67d0\u4e00\u70b9\u4e0a\u7684\u4e00\u9636\u5bfc\u6570\u7684\u7edd\u5bf9\u503c\u5927\u5c0f\uff0c\u5373\u4e3a\u5176\u201c\u9661\u5ced\u7a0b\u5ea6\u201d\uff08\u66f4\u591a\u7684\u65f6\u5019\u7406\u89e3\u4e3a\uff0c\u53d8\u5316\u7387\u5927\u5c0f\uff09<\/li>\n<\/ul>\n\n\n\n

    \u4e0a\u8ff0\u4e24\u4e2a\u7ed3\u8bba\uff0c\u57fa\u672c\u4e0a\u8ba4\u4e3a\u662f\u663e\u7136\u7684\u3002\u4e0b\u9762\u6269\u5c55\u5230\u591a\u7ef4\u573a\u666f\uff0c\u4e5f\u51e0\u4e4e\u662f\u663e\u7136\u7684\uff1a<\/p>\n\n\n\n

      \n
    • \u5728\u9ad8\u7ef4\u7a7a\u95f4\/\u591a\u7ef4\u53d8\u91cf\uff08\u5373\\( \\mathbb{R}^n\\)\u65f6\uff0c\u5728\u67d0\u4e00\u70b9\u7684\u4efb\u610f\u65b9\u5411\u4e0a\uff0c\u90fd\u6709\u5bfc\u6570\uff0c\u79f0\u4e3a\u65b9\u5411\u5bfc\u6570\uff0c\u8be5\u65b9\u5411\u5bfc\u6570\u7684\u7b26\u53f7\uff08\u6b63\/\u8d1f\uff09\uff0c\u53ef\u4ee5\u4ee3\u8868\u5728\u8be5\u65b9\u5411\u4e0a\uff0c\u51fd\u6570\u7684\u8d8b\u52bf\u662f\u589e\u957f\u8fd8\u662f\u4e0b\u964d\uff0c\u201c\u6b63\u53f7\u201d\uff0c\u5219\u662f\u589e\u957f\uff1b\u201c\u8d1f\u53f7\u201d\uff0c\u5219\u662f\u4e0b\u964d\u3002<\/li>\n\n\n\n
    • \u5728\u9ad8\u7ef4\u7a7a\u95f4\/\u591a\u7ef4\u53d8\u91cf\uff08\u5373\\( \\mathbb{R}^n\\)\u65f6\uff0c\u5728\u67d0\u4e00\u70b9\u7684\u4efb\u610f\u65b9\u5411\u4e0a\uff0c\u90fd\u6709\u5bfc\u6570\uff0c\u8be5\u5bfc\u6570\u7684\u7edd\u5bf9\u503c\u5927\u5c0f\uff0c\u5373\u4e3a\u5176\u201c\u9661\u5ced\u7a0b\u5ea6\u201d\uff08\u66f4\u591a\u7684\u65f6\u5019\u7406\u89e3\u4e3a\uff0c\u53d8\u5316\u7387\u5927\u5c0f\uff09<\/li>\n\n\n\n
    • \u66f4\u8fdb\u4e00\u6b65\u7684\uff0c\u4e5f\u5c31\u662f\u672c\u6587\u4e2d\u7684\u4e00\u4e2a\u7ed3\u8bba\uff1a\u9ad8\u7ef4\u7a7a\u95f4\/\u591a\u7ef4\u53d8\u91cf\uff08\u5373\\( \\mathbb{R}^n\\)\u65f6\uff0c\u51fd\u6570\u7684\u6240\u6709\u7684\u65b9\u5411\u5bfc\u6570\uff0c\u5728\u504f\u5bfc\u6570\u5411\u91cf\u65b9\u5411\u4e0a\uff0c\u53d6\u503c\u6700\u5927\uff0c\u5373\u662f\u6700\u4e3a\u201c\u9661\u5ced\u201d\u7684\u65b9\u5411\u3002<\/li>\n<\/ul>\n\n\n\n

      <\/p>\n\n\n\n

      8. \u6240\u4ee5\uff0c\u6700\u540e<\/h3>\n\n\n\n

      \u6240\u4ee5\uff0c\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\u4e2d\uff0c\u603b\u662f\u503e\u5411\u4e8e\u9009\u62e9\u504f\u5bfc\u6570\u5411\u91cf\u65b9\u5411\u8fdb\u884c\u4e0b\u4e00\u6b21\u8fed\u4ee3\u3002<\/p>\n\n\n\n

      <\/p>\n\n\n\n

      \u5728\u672c\u79d1\u6bd5\u4e1a\u540e\uff0c\u6700\u540e\u7559\u4e86\u51e0\u672c\u4e66\uff1a\u300a\u6570\u5b66\u5206\u6790\u300b\uff08\u4e0a\u4e0b\u518c\uff09\u3001\u6982\u7387\u8bba\uff0c\u4e00\u76f4\u5230\u7814\u7a76\u751f\u6bd5\u4e1a\u3001\u518d\u5230\u5de5\u4f5c\u90fd\u4e00\u76f4\u5e26\u7740\uff0c\u8fd8\u4ece\u5317\u4eac\u90ae\u5bc4\u5230\u4e86\u676d\u5dde\u3002\u672c\u60f3\u53ea\u662f\u505a\u4e2a\u7eaa\u5ff5\u7684\uff0c\u6ca1\u60f3\u5230\u7adf\u7136\u8fd8\u80fd\u7528\u4e0a…<\/p>\n","protected":false},"excerpt":{"rendered":"

      \u68af\u5ea6\u4e0b\u964d\u6cd5\uff08\u6216\u8005\u5176\u6539\u8fdb\u7b97\u6cd5\uff09\u662f\u673a\u5668\u5b66\u4e60\u7684\u57fa\u7840\u7b97\u6cd5\u4e4b\u4e00\u3002\u5728\u4e86\u89e3\u68af\u5ea6\u4e0b\u964d\u7b97\u6cd5\u7684\u8fc7\u7a0b\u4e2d\uff0c\u4f1a\u7ecf\u5e38\u770b\u5230\u4e00\u53e5\u8bdd\uff1a\u201c\u68af\u5ea6\u662f\u51fd\u6570\u5728\u67d0\u4e00\u70b9\u53d8\u5316\u7387\u6700\u5927\u7684\u65b9\u5411\u201d\u3002\u672c\u6587\u4ece\u8f83\u4e3a\u4e25\u683c\u6570\u5b66\u8bc1\u660e…<\/p>\n","protected":false},"author":1,"featured_media":22461,"comment_status":"open","ping_status":"closed","sticky":false,"template":"wp-custom-template-a-1440-px-width-template","format":"standard","meta":{"_eb_attr":"","inline_featured_image":false,"_tocer_settings":[],"footnotes":""},"categories":[5,137],"tags":[],"class_list":["post-14512","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-code-staff","category-learning-more"],"_links":{"self":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/14512","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/comments?post=14512"}],"version-history":[{"count":172,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/14512\/revisions"}],"predecessor-version":[{"id":22466,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/posts\/14512\/revisions\/22466"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/media\/22461"}],"wp:attachment":[{"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/media?parent=14512"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/categories?post=14512"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.orczhou.com\/index.php\/wp-json\/wp\/v2\/tags?post=14512"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}