(Partial differentiation question)

Question/Solution: http://f.imgtmp.com/EJckp.jpg

I'm talking about the first part since the second part is the same just with different variables and stuff. I get what the solution is saying but:

1) What if I computed a Jacobian, with
F = x^2 + xy + y^2 - z = 0
G = 2r + s - x = 0
H = r - 2s - y = 0
would it work? Whether it works or not, why does it work or not?

2) Why am I getting it wrong when I substitute x and y with the r and s equivalents and then attempt taking ∂z/∂r? Am I just making a mistake when I compute it or is this attempt just

Question

(Partial differentiation question)

Question/Solution: http://f.imgtmp.com/EJckp.jpg

I'm talking about the first part since the second part is the same just with different variables and stuff. I get what the solution is saying but:

1) What if I computed a Jacobian, with
F = x^2 + xy + y^2 - z = 0
G = 2r + s - x = 0
H = r - 2s - y = 0
would it work? Whether it works or not, why does it work or not?

2) Why am I getting it wrong when I substitute x and y with the r and s equivalents and then attempt taking ∂z/∂r? Am I just making a mistake when I compute it or is this attempt just

anonymous · Answer

What did your Jacobian turn out as?

s3a · Answer

The file I did it in got corrupted but if I remember correctly 1/(10y + 5) unless i'm confusing it for another question.

s3a · Answer

What matters more to me is if the two methods I gave are correct and making sure I set them up correctly. The tedious algebra, I only care for the solution I attached (which I fully get).

s3a · Answer

Basically, those are three correct alternatives to solving this problem, right?

anonymous · Answer

Let me check :) Hang on

s3a · Answer

Alright :)

s3a · Answer

Part of the reason why I am doing this is because I am trying to grasp when Jacobians are better or worse to use such that the problem is as simple as possible to solve. As for the substituting the r and s values into x and y, that seems to be the most instinctive way I responded to this particular problem and I wanted to make sure it's correct as well.

anonymous · Answer

I can't get your method 2. to work

I'll give 1. a try now

s3a · Answer

Alright, but can you try to help me prove or disprove (not in the formal sense) whether they will work by using theory because just by computing we could both make silly mistakes and not know it?

anonymous · Answer

For 1. the Jacobian gives me the partial derivatives I need but I still had to use the chain rule to get the end answer

anonymous · Answer

I'm not really sure why they should or shouldn't work, sorry.

s3a · Answer

Ok, well at least you told me that the Jacobian attempt is not a dead end. Thank you. (But if you get an epiphany or if someone else knows detailed theoretical answers as to why both should/shouldn't work, I would greatly appreciate it.)

s3a · Answer

I'll check again tomorrow because I'm really tired and sick.

anonymous · Answer

No worries, I'll look into more :)