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Partial derivative query - guidance needed

  1. Nov 11, 2012 #1
    I have a question but have not seen an example or find anything in my textbooks so would love some advice on how to understand the problem.

    Its a theory question on partial derivatives of the second order...

    T=T(x,y,z,t) with x=x(t), y=y(t), z=z(t).

    Find the second derivative of T wrt t

    So, first find the first derivative to give:

    [tex]\frac{dT}{dt} = \frac{\partial T}{\partial x}\frac{dx}{dt} +\frac{\partial T}{\partial y}\frac{dy}{dt}+\frac{\partial T}{\partial z}\frac{dz}{dt}+\frac{\partial T}{\partial t}[/tex]

    So I know the second derivative will be:
    [tex]\frac{d}{dt}\left(\frac{dT}{dt}\right)[/tex]

    So now my question. How do i take derivatives of products of mixed derivatives and write it in a correct manner, albeit a theoretical one.

    Your help means a lot.

    PT
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Nov 11, 2012
  2. jcsd
  3. Nov 11, 2012 #2

    Ray Vickson

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    Your dT/dt is a sum of four terms. Do you agree that you can take another d/dt of each term separately, then add the results? So, look at the first term
    [tex] \frac{\partial T}{\partial x} \frac{dx}{dt}.[/tex] This has the form G(x,y,z)*v(t), where
    [tex] G = \partial T/\partial x, \: v = dx/dt \text{ and } x = x(t), \:y = y(t), \:z = z(t).[/tex] Just apply your (d/dt) rule to G*v. Similarly for the other terms.

    RGV
     
  4. Nov 11, 2012 #3
    Thanks RGV. So would the first "set" be:

    [tex]\frac{\partial T}{\partial x}\frac{dx}{dt}+\frac{dx}{dt}\frac{dx}{dt}[/tex]
    or better still...
    [tex]\frac{\partial T}{\partial x}\frac{dx}{dt}+\frac{d^2x}{dt^2}[/tex]
     
  5. Nov 11, 2012 #4

    vela

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    Nope. Try again. I suggest you write the derivative first in terms of G(x,y,z,t) and v(t) and then use the fact that G=dT/dx and v=dx/dt.
     
  6. Nov 11, 2012 #5
    OK. I'm struggling but I think this might be it...

    [tex]\frac{\partial G}{\partial x}\frac{dx}{dt}+G\frac{dx}{dt}\frac{dx}{dt}[/tex]
     
  7. Nov 11, 2012 #6

    vela

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    Nope. It'll probably help to write out the variable dependencies explicitly. Applying the product rules gives
    $$\frac{d}{dt}[G(x,y,z,t)v(t)] = \left(\frac{d}{dt} G(x,y,z,t)\right) v(t) + G(x,y,z,t)\frac{dv}{dt}.$$ Now look at your expression for dT/dt in your original post. You'll get the same sort of thing for dG/dt. I'll let you take it from here.
     
  8. Nov 11, 2012 #7
    I get a different result everytime i try :(

    OK. Third time lucky!! For the first "element"



    [tex]\frac{\partial G}{\partial x}\frac{dx}{dt}v(t)+G\frac{dv}{dt} [/tex]

    Then substitute G and v for what is know....

    [tex]\frac{\partial G}{\partial x}\frac{dx}{dt}\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d}{dt}v[/tex]
    So
    [tex]\frac{\partial^2 T}{\partial x^2}2\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d^2x}{dt^2}+...[/tex]
     
  9. Nov 11, 2012 #8

    vela

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    You seem to keep missing the point: G is a function of y, z, and t as well as x.

    Also, are you really claiming that ##\frac{dx}{dt}\times\frac{dx}{dt} = 2\frac{dx}{dt}##?
     
  10. Nov 11, 2012 #9
    Im only writing for the very first term from the first derivative to see if I'm on the right track how does it look??....

    Sorry that was a typo for 2 times dx/dt
     
  11. Nov 11, 2012 #10
    Ahh, I think I see your point. For the first term in the first derivative, when taking the second derivative I must do it for x,y,z and t. The the same for the second term... etc, etc... Is this correct??
     
  12. Nov 11, 2012 #11

    vela

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    Yup.
     
  13. Nov 11, 2012 #12
    thats gonna be a big answer! OK. So here we go just for the first term of the first derivative...

    [tex]\frac{\partial^2T}{\partial x\partial x}\frac{dx}{dt}\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d^2x}{dt^2}
    +
    \frac{\partial^2T}{\partial x\partial y}\frac{dy}{dt}\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d^2x}{dt^2}

    +
    \frac{\partial^2T}{\partial x\partial z}\frac{dz}{dt}\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d^2x}{dt^2}
    +
    \frac{\partial^2T}{\partial x\partial t}\frac{dt}{dt}\frac{dx}{dt}+\frac{\partial T}{\partial x}\frac{d^2x}{dt^2}....




    [/tex]
     
  14. Nov 11, 2012 #13

    vela

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    Almost. It should be
    $$\left(
    \frac{\partial^2T}{\partial x^2}\frac{dx}{dt} +
    \frac{\partial^2T}{\partial y\partial x}\frac{dy}{dt} +
    \frac{\partial^2T}{\partial z\partial x}\frac{dz}{dt} +
    \frac{\partial^2T}{\partial t\partial x}\right)\frac{dx}{dt} +
    \frac{\partial T}{\partial x}\frac{d^2x}{dt^2}$$ The last term only appears once.
     
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