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Math Help - partial differentiation

  1. #1
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    partial differentiation

    partial differentiation-picture-44.png
    I have no idea what's going on here, perhaps tell me what topic heading this would fall under so I can look it over?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: partial differentiation

    We are given:

    PV=mRT

    hence:

    P=\frac{mRT}{V}\,\therefore\,\frac{\delta P}{\delta V}=-\frac{mRT}{V^2}

    V=\frac{mRT}{P}\,\therefore\,\frac{\delta V}{\delta T}=\frac{mR}{P}

    T=\frac{PV}{mR}\,\therefore\,\frac{\delta T}{\delta P}=\frac{V}{mR}

    and so:

    \frac{\delta P}{\delta V}\frac{\delta V}{\delta T}\frac{\delta T}{\delta P}=\left(-\frac{mRT}{V^2} \right)\left(\frac{mR}{P} \right)\left(\frac{V}{mR} \right)=-\frac{mRT}{PV}=-\frac{PV}{PV}=-1
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  3. #3
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    Re: partial differentiation

    thanks! how did you know which partials to take? can we just go right to left so partial derivatives: P/V then V/T then T/P ?
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