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
 
Thread 'Use greedy vertex coloring algorithm to prove the upper bound of χ'

Thread 'Use greedy vertex coloring algorithm to prove the upper bound of χ'

Hi! I am struggling with the exercise I mentioned under "Homework statement". The exercise is about a specific "greedy vertex coloring algorithm". One definition (which matches what my book uses) can be found here: https://people.cs.uchicago.edu/~laci/HANDOUTS/greedycoloring.pdf Here is also a screenshot of the relevant parts of the linked PDF, i.e. the def. of the algorithm: Sadly I don't have much to show as far as a solution attempt goes, as I am stuck on how to proceed. I thought...
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