F(x,y) and change of variables

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Homework Statement

f(x,y) a function of two variables.
x = 2u
y = u-3v

Using a change of variables, transform the equation (d²f/dx²) + (df/dy) = 0
into the coordinates system {u,v}.

Homework Equations

We have kind of a replacement teacher for the session and it is his first time giving the course. He almost doesn't give examples and well he's pretty bad :( The problem is that the exams and homeworks are the same for every group, so whenever we get an homework much of what is in it is mostly unknown to us (questions made by the other teachers of the other groups).
So anyways, I'll stop moaning. I quite don't understand what I have to do. If someone could just point me in some direction, I'd be glad.

The Attempt at a Solution

Should I do the partial derivatives? Where do I use the change of variables?

Oyoyoy.

 

[Dick]

You need to use the chain rule for partial derivatives. E.g. df/du=(df/dx)*(dx/du)+(df/dy)*(dy/du) (all partial derivatives). Try looking that up.

 

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Oooh yes, that definitely rings a bell.
I'll try that tonight, thanks :D

 

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Hey hmm, the partial derivative in x of "e^(xy)", is it zero?

 

[Dick]

No, why would you think that? Is this a different problem?

 

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Ok now, sorry I was fooling around with the other numbers of the homework.
So should I do something like, assuming f = f(x,y), :

df/du = (df/dx)(dx/du) + (df/dy)(dy/du)
and
df/dv = (df/dx)(dx/dv) + (df/dy)(dy/dv)

But then what does it means to transform the equation into the coordinates system {u,v}?

 

[HallsofIvy]

It means to get a corresponding equation in the variable u and v rather than x and y!
$$\frac{\partial f}{\partial y}= \frac{\partial f}{\partial u}\frac{\partial u}{\partial y}+ \frac{\partial f}{\partial v}\frac{\partial v}{\partial y}$$
In particular, with your x= 2u, y= u- 3v, you will need u and v in terms of x and y so solve the 2equation: u= x/2, y= (x/2)- 3v: 3v= (x/2)- y: v= x/6- y/3.
From that
$$\frac{\partial u}{\partial y}= 0$$
and
$$\frac{\partial v}{\partial y}= -1/3$$
so
$$\frac{\partial f}{\partial y}= -\frac{1}{3}\frac{\partial f}{\partial v}$$

 

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And there was light.
Wow thanks now that's clear :D
So now I just have to find d²f/dx² = d/dx [(df/du)(du/dx) + (df/dv)(dv/dx)] (or I'm not sure, got to check my notes about composed functions) and rewrite the equation with the new values found? Like "what I'm about to find" + -1/3 (df/dv) = 0 ?
That is wonderful.

 

[HallsofIvy]

Yes. Be careful about the second derivative. Write out the first derivative and then apply the same chain rule when differentiating that.

 

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Ok so now I have :

d²f/dx² = d/dx [ 1/2 (df/du) + 1/6 (df/dv) ]
Is this right?
Damn I feel so stupid because I feel that it's not that hard :(
I'm not even sure about the next steps.
I think I need a drink.

 

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And wouldn't dy/du = 1 because they are independent or something?
f(2u, u-3v)
Or I'm talking crap.
oyoyoy