I guess you mean (∂x/∂u)v . Consider first the differentials

du = 2x dx - 2y dy, and dv = y dx + x dy. When v is constant, i.e. dv = 0, one has dy = -(y/x) dx, and hence du = (2x + 2y2/x) dx. I.e.,

(∂x/∂u)v = x/[2(x2 + y2)]

Or, as a general procedure, you may compute the Jacobi matrix relating (∂x , ∂y)  to (∂u , ∂v), and invert that relation.

Jacobian J of the transformation is 

2x     -2y

y          x

Inverse of the Jacobian, which is also the Jacobian of the inverse transformation, can be found using J J-1=1, which yields,

J-1 = 1/Det(J) [Jc]T = 1/(2x2+2y2) [Jc]T

11 element of J-1 is x / [2(x2 + y2)] which is what Kare has got.

@Robert J. Low

this certainly is a homework. This is a major problem for lecturers today. I don't know this type of question in RG is correct or not. I find out some of my students do same thing. I think we should adopt with new technology and find new ways for evaluation of student without need to avoid them using these tools. 

"I rather thought it was the second meaning that RG was all about, not saving people the bother of doing their own homework (or, in all too many cases, typing a simple query into google)."

I understand completely problem, but either RG must restrict student access to site (which isn't entirely bad idea) or we accept they have access to this site and can use it to solve their homework. Actually in some cases I encourage student use search tools and learn how to search properly, but in most cases I think they must do some math and use their mind to do homework. It will be very useful if people with lecture experience discuss this issue and find a good way to solve it. 

I don't see anything wrong with student access to RG, given the general access rules which seem to be applied. What is wrong is to use this access to (try to) make other members explicitly solve ones homework problems. It this case I did provide a solution, because I thought it might demonstrate a different (physicists) approach, complementary to the general formulas of calculus - which the questioner should have learned in some class.

Alternatives are to respond with a warning to the questioner about the dangers of cheating, and/or to flag the question(er) to RG. Students must learn to use new technology, but they should not forget appropriate behaviour and ethical issues.

Very interesting discussion.

@Kåre Olaussen

What you suggest is too much ideological, we can discuss about it more. But I am interested in sociological aspect of this phenomena. what I see, is this: it seems that in last decades internet grows and we deals with a global brain. Actually maybe it will better if we accept this and use it in appropriate way. For this purpose, we must reform our educational method too, not just force student to do everything in old ways.

@ Azizi, yes, new innovations in educational methods are needed along with phenomenal growth and penetration of internet in almost all walks of life. We should not force a poor student to learn in our older ways nowadays.

inefficient, unreliable and unfair to those people who do homework for money.

Takes long time. One cannot be sure that somebody actually answers. Those who do can do it all wrong.

And one could get a general reply: differentiate equations for u and v, you'll get a set of linear equations with respect to du, dv, dx etc, Solve it.

No, students who study management should get C minus for such an amateurish approach.  

Y.Azizi> What you suggest is too much ideological,

Well, I stand up for the ideology that appropriate and ethical behaviour is valuable in the long run -- even if that does not seem to be the case for presidential runs...  ;=D

Wikipedia, Google et. al., and Internet in general are tools which todays students certainly must learn to use (together with many other members of RG, judging from many questions posted here). Educators should adapt to this new environment as soon as possible; I think most of them try the best they can.

But, even with all (dis-)information in the universe three clicks away, some old-fashioned knowledge is required to make sensible use of that data. How can managers make use of their beloved h-indices if nobody can distinguish between papers written by  robots and Robert? :-) 

I am entertaining the idea of learning Spanish. But why should I, if I can post some text or phrases on RG (since I, for mysterious reasons, don't bother to enter it into Google translate myself), and ask some kind member here to provide an accurate translation?

Dear all,

Thankyou for all your discussions. This was not my homework question. In deriving something, i got struck with this and was confused. So at that instant i put this in RG to see if there are different solutions for this. Thankyou again.