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Thread: Rearranging

  1. #1
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    Rearranging

    What does the rearranging lead to?

    $\displaystyle \frac{\partial a}{\partial x} \partial y = \frac{\partial b}{\partial y} \partial x$

    Dividing through by $\displaystyle \partial x$, which would it lead to?

    (1)
    $\displaystyle \frac{\partial a}{\partial x^2} \partial y = \frac{\partial b}{\partial y} $

    (2)
    $\displaystyle \frac{\partial ^2 a}{\partial x^2} \partial y = \frac{\partial b}{\partial y} $

    Equation (1) or (2)?
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  2. #2
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    Hello, Simplicity!

    What does the rearranging lead to?

    . . $\displaystyle \displaystyle \frac{\partial a}{\partial x} \partial y \:=\: \frac{\partial b}{\partial y} \partial x$

    Dividing through by $\displaystyle \partial x$, which would it lead to?

    $\displaystyle \displaystyle (1)\;\frac{\partial a}{\partial x^2} \partial y \:=\: \frac{\partial b}{\partial y} $
    . . . . . . . or
    $\displaystyle \displaystyle (2)\;\frac{\partial ^2 a}{\partial x^2} \partial y \:=\: \frac{\partial b}{\partial y} $

    We have: .$\displaystyle \displaystyle \frac{\partial a}{\partial x}\partial y \;=\;\frac{\partial b}{\partial y}\partial x $


    Dividing through by $\displaystyle \partial x$, we have: .$\displaystyle \displaystyle \frac{\partial a}{\partial x}\cdot \frac{\partial y}{\partial x} \;=\;\frac{\partial b}{\partial y}\cdot\frac{\partial x}{\partial x}$

    Then we have: .$\displaystyle \displaystyle\frac{\partial a}{\partial x}\cdot\frac{\partial y}{\partial x} \;=\;\frac{\partial b}{\partial y}$

    . . which can be written: .$\displaystyle \displaystyle \frac{\partial a\partial y}{\partial x^2} \;=\;\frac{\partial b}{\partial y}$ . . . answer (1)

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