设z=u^2v^2,而u=x-y,v=x+y,求dz/dx,dz/dy

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设z=u^2v^2,而u=x-y,v=x+y,求dz/dx,dz/dy

设z=u^2v^2,而u=x-y,v=x+y,求dz/dx,dz/dy
设z=u^2v^2,而u=x-y,v=x+y,求dz/dx,dz/dy

设z=u^2v^2,而u=x-y,v=x+y,求dz/dx,dz/dy
z=(x-y)^2(x+y)^2=(x^2-y^2)^2=x^4-2x^2y^2+y^4
dz=4x^3dx-4xy^2dx-4yx^2dy+4y^3dy=4x(x^2-y^2)dx+4y(y^2-x^2)dy
所以:
dz/dx=4x(x^2-y^2)
dz/dy=4y(y^2-x^2)

dz=2uv²du+2vu²dv=2uv²(dx-dy)+2vu²(dx+dy)=(2uv²+2vu²)dx+(2uv²-2vu²)dy
∴dz/dx=2uv²+2vu²
∴dz/dy=2uv²-2vu²