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Total differential

  1. Apr 17, 2014 #1
    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. jcsd
  3. Apr 17, 2014 #2

    LCKurtz

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    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?
     
  4. Apr 17, 2014 #3
    complete question is attached, but the information in the original post is correctly copied as far as I can see
     

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  5. Apr 17, 2014 #4

    LCKurtz

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    Like I said, it is false without more context. Check my example yourself.
     
  6. Apr 17, 2014 #5
    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
     
  7. Apr 17, 2014 #6

    CAF123

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    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.
     
  8. Apr 17, 2014 #7

    LCKurtz

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    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}##?
     
  9. Apr 18, 2014 #8

    CAF123

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    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.
     
  10. Apr 18, 2014 #9

    LCKurtz

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