高数 曲面方程 切平面方程 高等数学中,如何求一个曲面的切平面方程,能有例题最好了,谢谢...
\u9ad8\u6570\uff0c\u6c42\u66f2\u9762\u5728\u67d0\u70b9\u5904\u7684\u5207\u5e73\u9762\u65b9\u7a0b\u8bbe\u66f2\u9762\u65b9\u7a0b\u4e3a F\uff08X\uff0cY\uff0cZ\uff09\u3002
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\u5c06\u70b9\uff08a\uff0cb\uff0cc\uff09\u4ee3\u5165\u5f97 n\uff1d\uff3bFx\uff0cFy\uff0cFz\uff3d \uff08\u5207\u5e73\u9762\u6cd5\u5411\u91cf\uff09\u3002
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\u6269\u5c55\u8d44\u6599\uff1a
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\u5b83\u4eec\u76f8\u4e92\u5782\u76f4\u5b83\u4eec\u5c31\u7b49\u4e8eA1A2 \uff0b B1B2 \uff0b C1C2 \uff1d 0\u3002
\u5e73\u884c\u6216\u91cd\u53e0\u7684\u5e73\u9762\u4e5f\u5c31\u662fA1\uff0fA2\uff1dB1\uff0fB2\uff1dC1\uff0fC2\u3002
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\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u5e73\u9762\u65b9\u7a0b
\u8bbe\u4efb\u4e00\u70b9\uff0c\u7528\u4e00\u4e2a\u672a\u77e5\u6570\uff0c\u7136\u540e\u6c42\u5176\u5bfc\u6570\uff0c\u5373\u4e3a\u4ed6\u7684\u5207\u7ebf\u65b9\u7a0b
F(x,y,z)=z-e^z+2xy-3,aF/ax=2y,aF/ay=2x,aF/az=1-e^z,
于是法向量为(4,2,0),
故切平面方程是
4(x-1)+2(y-2)=0,化简为
2x+y-4=0。
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