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Math Help - Partial Derivative Proof (thermodynamics notation)

  1. #1
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    Partial Derivative Proof (thermodynamics notation)

    Show that:  \left(\frac{\partial z}{\partial y}\right)_{u}  =  \left(\frac{\partial z}{\partial x}\right)_{y} \left[ \left(\frac{\partial x}{\partial y}\right)_{u} - \left(\frac{\partial x}{\partial y}\right)_{z} \right]

    I have Euler's chain rule and "the splitter." Also the property, called the "inverter" where you can reciprocate a partial derivative.

    If I write Euler's chain rule, I only know how to write it when there are 3 variables, I usually write it in the form:
    \left(\frac{\partial x}{\partial y}\right)_{z} \left(\frac{\partial y}{\partial z}\right)_{x} \left(\frac{\partial z}{\partial x}\right)_{y}  =  -1

    Where I can write x,y,z in any order as long as each variable is used in every spot. However, I do not know how to work this chain rule if I have an extra variable (u in this case).

    I also tried using the "splitter" to do something like writing:
     \left(\frac{\partial z}{\partial y} \right)_{u}  =  \left(\frac{\partial z}{\partial x} \right)_{u} \left(\frac{\partial x}{\partial y}\right)_{u}

    However, I do not know what to do with this because I have the term
     \left(\frac{\partial z}{\partial x} \right)_{u} , which doesn't appear in the original problem.

    Any help would be appreciated.

    Thanks in advance.

    (This is for a thermodynamics course, but we are still in the mathematics introduction.)
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  2. #2
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    i too am an engineer but i only took it to the undergrad lvl and am not familiar with this notation... can you clarify?
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  3. #3
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    Sure thing.

    If I say something like  \left(\frac{\partial z}{\partial x}\right)_{y} .

    This means... The partial derivative of z with respect to x, holding y constant.
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  4. #4
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    Quote Originally Posted by Jacobpm64 View Post
    Sure thing.

    If I say something like  \left(\frac{\partial z}{\partial x}\right)_{y} .

    This means... The partial derivative of z with respect to x, holding y constant.
    There is no need to write,
     \left(\frac{\partial z}{\partial x}\right)_{y}

    Because the notation,
    \frac{\partial z}{\partial x}
    Means exactly that i.e. "you hold all variables constant except x".
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  5. #5
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    Oh, i thought it was different in thermodynamics, Please excuse me.
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  6. #6
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    ^ im not sure i agree with your take on this, as this would simply cancel out the RHS of his eqn


    i got the gerbils runnin, ill let u know if i come up with anything
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  7. #7
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    There's a good argument here about this.

    Scroll down to neutrino's post.

    Partial derivative notation

    I think that should clear things up about the notation.
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