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Math Help - Differentiation Simples

  1. #1
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    Differentiation Simples

    Why is this true?

    for \rho(\eta,t)

    <br />
\dfrac{\partial}{\partial \eta}(\ln(\rho))=\dfrac{1}{\rho}\cdot\dfrac{\parti  al \rho}{\partial\eta}<br />

    Is this needed to derive

    <br />
\dfrac{D\ln(\phi)}{Dt}=\nabla\cdot\bf U.<br />

    from the continuity equation

    <br />
\dfrac{\partial\rho}{\partial t}+\nabla\cdot(\rho \bf U)=0.<br />

    where \phi(\textbf{x},t)=1/\rho(\textbf{x},t) is the specific volume
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  2. #2
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    Quote Originally Posted by davefulton View Post
    Why is this true?

    for \rho(\eta,t)

    <br />
\dfrac{\partial}{\partial \eta}(\ln(\rho))=\dfrac{1}{\rho}\cdot\dfrac{\parti  al \rho}{\partial\eta}<br />

    Is this needed to derive

    <br />
\dfrac{D\ln(\phi)}{Dt}=\nabla\cdot\bf U.<br />

    from the continuity equation

    <br />
\dfrac{\partial\rho}{\partial t}+\nabla\cdot(\rho \bf U)=0.<br />

    where \phi(\textbf{x},t)=1/\rho(\textbf{x},t) is the specific volume
    <br />
\dfrac{\partial}{\partial \eta}(\ln(\rho))=\dfrac{d \ln (\rho)}{d \rho} \cdot \dfrac{\partial \rho}{\partial\eta}<br />
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  3. #3
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    Thank you. You have answered many of my questions in the past. You really do live up to your name. Any idea on the second part?
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