Total differential

1. Apr 17, 2014

chemphys1

1. The problem statement, all variables and given/known data

If z(x,y) = f(x/y)
show that

x(∂z/∂x)y + y(∂z/∂y)x = 0

2. Relevant equations

so I understand z(x,y)
means I can write
dz = (∂z/∂x)ydx + (∂z/∂y)x dy

I do not understand the = f(x/y) bit though?
does that mean this?
df= (∂f/∂x)y dx+ (∂f/∂y)x dy
and (∂f/∂x)y = -y/x2 (∂f/∂y)x = 1/x

although that seems wrong can't manipulate to get the answer

any help on the method or explaining f(x/y) equalling z(x,y) would be appreciated. maths is not my strongest so if possible go as basic as it comes

2. Apr 17, 2014

LCKurtz

That isn't true. Try it for z = f(x/y) = x/y. Generally speaking, if everything is continuous, you would have $z_{xy} = z_{yx}$ and you wouldn't expect to multiply one by x and the other by y and have them be equal with opposite signs. Did you copy the problem correctly, parentheses and all?

3. Apr 17, 2014

chemphys1

complete question is attached, but the information in the original post is correctly copied as far as I can see

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4. Apr 17, 2014

LCKurtz

Like I said, it is false without more context. Check my example yourself.

5. Apr 17, 2014

chemphys1

Sorry, I really do not follow
I find it hard to understand mathematical notation, so re: z = f(x/y) = x/y
I can't see how to check the example

6. Apr 17, 2014

CAF123

You are familiar with the notation f(x) as describing a function of x. Similarly, f(x/y) just means you have some function in the variable x/y. You can instead consider u=x/y and then you are back to the more familiar f(u).

But with LCKurtz's example of f(x/y)=x/y, I get the equation to be satisfied. I also get it to be satisfied in general. I think LCKurtz simply misread the notation, that's all.

7. Apr 17, 2014

LCKurtz

Are you saying that $\left(\frac {\partial z}{\partial x}\right)_y$ means something other than $\frac \partial {\partial y}\left (\frac{\partial z}{\partial x}\right )$ or, as I wrote $z_{xy}$?

8. Apr 18, 2014

CAF123

Yes, I took $\left(\frac{\partial z}{\partial x}\right)_y$ to mean, say, differentiate z wrt x, keeping y held fixed. Similarly for the other case. I actually thought that was a standard notation, although I have seen cases where they simply suppress the variable being held constant because in a sense it is obvious from the problem.

9. Apr 18, 2014

LCKurtz

Well, that's a new one on me. In over 40 years of teaching calculus using many different texts, I never encountered that notation.