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Change of Variables

  1. Aug 31, 2011 #1
    Hi there

    I am given xi=x/s(t) and T=h(t)F(xi,t) and I need to tranform deltaT/deltat. How do I do it? Do I use the chain rule? The answer to it is : s*(dh/dt)*F+s*h*(deltaF/deltat)-xi*(ds/dt)*h*(deltaF/deltaxi) but I don't know how to get this answer. Please help me. Thank you
     
  2. jcsd
  3. Aug 31, 2011 #2

    pwsnafu

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    What's delta T / delta t? Is this from discreet math?
     
  4. Aug 31, 2011 #3
    Sorry it's [itex]\deltaT[/itex] /[itex]\deltat[/itex]
     
  5. Aug 31, 2011 #4
    Hi It's the greek letter delta. I want to tranform (delta T)/(delta t)
     
  6. Aug 31, 2011 #5
    I have to solve a heat equation but first I must change the variables.
     
  7. Aug 31, 2011 #6

    Stephen Tashi

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    I think pwsnafu is asking whether [itex] \frac{ \delta T}{\delta t} [/itex] means the derivative of T with respect to t.
     
  8. Aug 31, 2011 #7

    pwsnafu

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    Yeah. I've never seen lower case delta used in that way before. Do you mean partial derivative?
     
  9. Aug 31, 2011 #8
    Yes it is
     
  10. Aug 31, 2011 #9
    How do you do the change of variable when xi=x-s(t)/1-s(t) and T=(1-s(t))*F(xi,t). I want to transform partial derivative of T with respect to t.
     
  11. Aug 31, 2011 #10
    That's really messy . . . Candy. Looks like you have a chained list of variables:

    [tex]\Xi(x,s)=\frac{x}{s}[/tex]

    [tex]s=s(t)[/tex]

    [tex]T(h,F)=h(t)F(\Xi,t)[/tex]

    and you want to compute:

    [tex]\frac{dT}{dt}[/tex]

    so by the general chain-rule:

    [tex]\frac{dT}{dt}=h(t)\frac{\partial}{\partial t} F(\Xi,t)+F\frac{dh}{dt}[/tex]

    and:

    [tex]\frac{\partial}{\partial t} F(\Xi,t)=\frac{\partial F}{\partial \Xi}\frac{\partial \Xi}{\partial t}+\frac{\partial F}{\partial t}[/tex]

    anyway, doing all that and simplifying, I still don't get exactly what you posted as the answer (close though) so maybe I'm missing something. Maybe though you can clean it up for me.
     
  12. Aug 31, 2011 #11
    The equation have to transform is d^2T/dx^2=dT/dt
     
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