A constrained differential probelm

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The discussion centers on the constrained differential problem involving the function w(x,y,z) defined as zxe^y + xe^z + ye^z, with the constraint equation x^2y + y^2x = 1. The differential dy is given as - (2xy + y^2) / (2xy + x^2). A point of confusion arises regarding the substitution of z=0 when calculating ∂w/∂x, despite dz=0 being understood. Participants seek clarification on the reasoning behind setting z to zero and request the initial steps of the problem for better understanding. The conversation emphasizes the need for clarity in applying constraints and derivatives in differential problems.
Leo Liu
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Homework Statement
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Relevant Equations
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Define that $$w(x,y,z)=zxe^y+xe^z+ye^z$$
1614266061848.png

So the constraint equation is ##x^2y+y^2x=1##. And its differential is ##dy=-\frac{2xy+y^2}{2xy+x^2}##.

However, the solution plugs in ##z=0## when computing ##\frac{\partial w}{\partial x}## as shown in the screenshot below. While I understand that ##dz=0##, I can't see why ##z=0##. Could anyone explain?
1614266672731.png
 
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Can you show us the first parts of the problem?
 
Office_Shredder said:
Can you show us the first parts of the problem?
Sure, Here it is.
1614301330031.png
 

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