1.设z=z(x,y)是由方程式e的z次方=xyz所含的隐函数,求dz 2.计算出曲面z=2-x^-y^2与xoy坐标面所围成的体积 设函数z=z(x,y)是由方程z+e的z次方=xy所确定的隐...
\u8bbez=z(x,y)\u662f\u7531\u65b9\u7a0b\u5f0fe\u7684z\u6b21\u65b9=xyz\u6240\u542b\u7684\u9690\u51fd\u6570,\u6c42dze^z=xyz
\u4e24\u8fb9\u5bf9x\u6c42\u504f\u5bfc
e^z*z'(x)=y(z+x*z'(x))
z'(x)=yz/(e^z-xy)
∂z/∂x=yz/(e^z-xy)
\u539f\u5f0f\u5bf9y\u6c42\u504f\u5bfc
e^z*z'(y)=x(z+y*z'(y))
∂z/∂y=xz/(e^z-xy)
dz=∂z/∂x*dx+∂z/∂y*dy
=yz/(e^z-xy)*dx+xz/(e^z-xy)*dy
\u4ee4F(x\uff0cy\uff0cz\uff09= z+z^e-xy=0
\u2234Fx=y Fz=-1+e^z\uff0c\u6709\u9690\u51fd\u6570\u8ba2\u7acbZ\u5148\u5bf9x\u504f\u5bfc=y/1+e^z
\u2234Fy=x \u6709\u9690\u51fd\u6570\u8ba2\u7acbZ\u5148\u5bf9y\u504f\u5bfc=x/1+e^z
\u6240\u4ee5Z\u5148\u5bf9x\u518d\u5bf9y\u6c42\u504f\u5bfc(y/1+e^z)dx+(x/1+e^z)dy
\u610f\u4e49\uff1a
\u5fae\u79ef\u5206\u5b66\u7684\u521b\u7acb\uff0c\u6781\u5927\u5730\u63a8\u52a8\u4e86\u6570\u5b66\u7684\u53d1\u5c55\uff0c\u8fc7\u53bb\u5f88\u591a\u7528\u521d\u7b49\u6570\u5b66\u65e0\u6cd5\u89e3\u51b3\u7684\u95ee\u9898\uff0c\u8fd0\u7528\u5fae\u79ef\u5206\uff0c\u8fd9\u4e9b\u95ee\u9898\u5f80\u5f80\u8fce\u5203\u800c\u89e3\uff0c\u663e\u793a\u51fa\u5fae\u79ef\u5206\u5b66\u7684\u975e\u51e1\u5a01\u529b\u3002
\u524d\u9762\u5df2\u7ecf\u63d0\u5230\uff0c\u4e00\u95e8\u5b66\u79d1\u7684\u521b\u7acb\u5e76\u4e0d\u662f\u67d0\u4e00\u4e2a\u4eba\u7684\u4e1a\u7ee9\uff0c\u800c\u662f\u7ecf\u8fc7\u591a\u5c11\u4eba\u7684\u52aa\u529b\u540e\uff0c\u5728\u79ef\u7d2f\u4e86\u5927\u91cf\u6210\u679c\u7684\u57fa\u7840\u4e0a\uff0c\u6700\u540e\u7531\u67d0\u4e2a\u4eba\u6216\u51e0\u4e2a\u4eba\u603b\u7ed3\u5b8c\u6210\u7684\uff0c\u5fae\u79ef\u5206\u4e5f\u662f\u8fd9\u6837\u3002
(e^z -xy)dz=(yz)dx+(xz)dy;
dz=[yz/(e^z-xy)]dx+[xz/(e^z-xy)]dy=[z/(xz-x)]dx+[z/(yz-y)]dy;
(2)曲面 z=2-x²-y² 为一伞形曲面,当 z=2 时,x=y=0;当 z=0(xoy 平面) 时,曲面与 xoy 平面的交线为圆 x²+y²=2,z=0~2 之间曲面围成的封闭空间是一圆锥体;
V=πr²*h/3=π(√2)²*2/3=4π/3;
1 e^z=xyz
e^zz'x=yz+xyz'x z'x=yz/(xy-e^z)=yz/(xy-xyz)=z/(x-xz)
类似 z'y=z/(y-yz)
dz=[z/(x-xz)]dx+[z/(y-yz)]dy
2.立体在xoy坐标面的投影D:x^2+y^2《2
V=∫∫(2-x^2-y^2)dxdy,用极坐标
=∫(0,2π)dθ∫(0,√2)r(2-r^2)dr
=2π(2-1)
=2π
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