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Math Help - How to evaluate the second total derivative of a 2D function

  1. #1
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    How to evaluate the second total derivative of a 2D function

    suppose i have F(x,y), and i want to evaluate d2F. is the following correct?

    dF = \frac{{\partial F}}{{\partial x}}dx + \frac{{\partial F}}{{\partial y}}dy

    d^2F = \frac{{\partial}}{{\partial x}} dF + \frac{{\partial}}{{\partial y}}dF

    \frac{{\partial}}{{\partial x}} dF = \frac{{\partial^2 F}}{{\partial x^2}}dx^2 + \frac{{\partial^2F}}{{\partial y \partial x}}dydx

    \frac{{\partial}}{{\partial y}} dF = \frac{{\partial^2 F}}{{\partial y^2}}dy^2 + \frac{{\partial^2F}}{{ \partial x \partial y}}dxdy


    \frac{{\partial^2F}}{{\partial x \partial y}} dxdy = \frac{{\partial^2F}}{{\partial y \partial x}} dydx \Rightarrow

    d^2F = \frac{{\partial^2 F}}{{\partial x^2}} dx^2 + \frac{{\partial^2F}}{{\partial y^2}} dy^2 + 2 \frac{{\partial^2 F}}{{\partial y \partial x}}dydx

    i don't believe i ever studied higher-order total derivatives in college, and now that i have a need to make use of it, i am unsure if 2 \frac{{\partial^2 F}}{{\partial y \partial x}}dydx should be \frac{{\partial^2 F}}{{\partial y \partial x}}dydx

    thanks in advance!


    edit:
    thanks for the clue-in on LaTeX plato.
    Last edited by method; August 11th 2009 at 09:57 AM.
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  2. #2
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    Quote Originally Posted by method View Post
    if someone could tell me how to do the sub/superscript formatting without the use of html sub / sup tags, please let me know!
    You can learn to use LaTeX.
    Typing [tex]\frac{{\partial ^2 f}}{{\partial x^2 }}[/tex] gives \frac{{\partial ^2 f}}{{\partial x^2 }} .
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  3. #3
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    aah. excellent. i'll spruce the OP up for legibility.

    do you happen to know the answer to the original Q ?
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  4. #4
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    let  f(t)=f\big(x(t), y(t)\big)

    then  \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}

     \frac{d^{2}f}{dt^{2}} = \frac{d}{dt}\Big(\frac{\partial f}{\partial x} \frac{dx}{dt}\Big)  + \frac{d}{dt} \Big(\frac{\partial f}{\partial y} \frac{dy}{dt} \Big) (sum rule)

     = \frac{dx}{dt}\frac{d}{dt} \Big(\frac{\partial f}{\partial x}\Big) + \frac{\partial f}{\partial x} \frac{d^{2}x}{dt^{2}} + \frac{dy}{dt}\frac{d}{dt} \Big(\frac{\partial f}{\partial y}\Big) + \frac{\partial f}{\partial y} \frac{d^{2}y}{dt^{2}} (product rule)

     = \frac{dx}{dt}\Bigg(\frac{\partial}{\partial x} \Big(\frac{\partial f}{\partial x}\Big)\frac{dx}{dt} + \frac{\partial}{\partial y} \Big(\frac{\partial f}{\partial x}\Big) \frac{dy}{dt} \Bigg)  + \frac{\partial f}{\partial x} \frac{d^{2}x}{dt^{2}} + \frac{dy}{dt}\Bigg(\frac{\partial}{\partial x} \Big(\frac{\partial f}{\partial y}\Big)\frac{dx}{dt} + \frac{\partial}{\partial y} \Big(\frac{\partial f}{\partial y}\Big) \frac{dy}{dt} \Bigg)+ \frac{\partial f}{\partial y} \frac{d^{2}y}{dt^{2}} (first total derivative)

     = \frac{dx}{dt}\Big(\frac{\partial^{2} f}{\partial x^{2}}\frac{dx}{dt} + \frac{\partial^{2} f}{\partial y \partial x} \frac{dy}{dt} \Big)  + \frac{\partial f}{\partial x}\frac{d^{2}x}{dt^{2}} + \frac{dy}{dt}\Big(\frac{\partial^{2} f}{\partial x \partial y}\frac{dx}{dt} + \frac{\partial^{2}f}{\partial y^{2}} \frac{dy}{dt} \Big)+ \frac{\partial f}{\partial y} \frac{d^{2}y}{dt^{2}}

     = \frac{\partial^{2} f}{\partial x^{2}}\Big(\frac{dx}{dt}\Big)^{2} + \frac{\partial^{2} f}{\partial y \partial x} \frac{dx}{dt} \frac{dy}{dt} + \frac{\partial f}{\partial x}\frac{d^{2}x}{dt^{2}} + \frac{\partial^{2} f}{\partial x \partial y} \frac{dy}{dt} \frac{dy}{dt} + \frac{\partial^{2} f}{\partial y^{2}}\Big(\frac{dy}{dt}\Big)^{2} + \frac{\partial f}{\partial y}\frac{d^{2}y}{dt^{2}}

    and assuming the mixed derivatives are equal

     = \frac{\partial^{2} f}{\partial x^{2}}\Big(\frac{dx}{dt}\Big)^{2} + 2 \ \frac{\partial^{2} f}{\partial x \partial y} \frac{dx}{dt} \frac{dy}{dt} + \frac{\partial f}{\partial x}\frac{d^{2}x}{dt^{2}} + \frac{\partial^{2} f}{\partial y^{2}}\Big(\frac{dy}{dt}\Big)^{2} + \frac{\partial f}{\partial y}\frac{d^{2}y}{dt^{2}}

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  5. #5
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    that's a lotta LaTeX!

    thanks! that does answer the question.
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