数学分析考研题

=1/(dy/du)

=(1+uv)^2

dv/dy

=1/(dy/dv)

=-(1+uv)^2/u^2

dz/dx

=dz/du*du/dx+dz/dw*dw/du*du/dx

=[(1+uw)-u(w+udw/du)]/(1+uw)^2*1-u^2/(1+uw)^2*dw/du*1

=1/(1+uw)^2

dz/dy

= dz/du * du/dy+dz/dw *(dw/du * du/dy+dw/dv * dv/dy)

=[(1+uw)-u(w+udw/du)]/(1+uw)^2*(1+uv)^2

-u^2/(1+uw)^2*((1+uv)^2*dw/du-(1+uv)^2/u^2*dw/dv)

x^2*dz/dx+y^2*dz/dy

=x^2*[1/(1+uw)^2]+y^2*{[(1+uw)-u(w+udw/du)]/(1+uw)^2*(1+uv)^2-u^2/(1+uw)^2*((1+uv)^2*dw/du-(1+uv)^2/u^2*dw/dv)}

=u^2*[1/(1+uw)^2]+[u/(1+uv)^2]^2*{[(1+uw)-u(w+udw/du)]/(1+uw)^2*(1+uv)^2-u^2/(1+uw)^2*((1+uv)^2*dw/du-(1+uv)^2/u^2*dw/dv)}=...=u^2/(1+uw)^2

= & gt(1-u^2*dw/du)=u^2*dw/du-dw/dv

= & gt2u^2*dw/du-dw/dv=1