Question

设z=z(x,y)是由方程e^(-xy)+2z-e^z=2确定 求dz|(x=2...

设z=z(x,y)是由方程e^(-xy)+2z-e^z=2确定 求dz|(x=2,y=-1/2)

Answer 1

对方程e^(-xy)+2z-e^z=2两边微分,
有：e^(-xy)*d(-xy)
+
2*dz
-
e^z*dz
=
0
-e^(-xy)*(x*dy
+
y*dx)
+
2*dz
-
e^z*dz
=
0
移项,得：(e^z
-
2)*dz
=
-y*e^(-xy)*dx
-
x*e^(-xy)*dy
当x=2,y=-1/2时,代入e^(-xy)+2z-e^z=2,得：z
=
1
所以dz|(x=2,y=-1/2)
=
[-y*e^(-xy)*dx
-
x*e^(-xy)*dy]/(e^z
-
2)（其中x=2,y=-1/2,z=1）
所以dz|(x=2,y=-1/2)
=
e*(1/2
*
dx
-
2*dy)/(e
-
2)