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Is the particle travelling in the direction of greatest increase in temperature

  1. Oct 17, 2009 #1
    1. The problem statement, all variables and given/known data
    A particle travels across a heated plate according to the path f(t) = (t^2, sint) at the time t seconds. The temperature T(x,y) at position (x,y) on the plate is given by T(x,y)= 200E^[-(x^2 + y^2)] degrees Celsius. At time t=pi seconds, is the particle traveling in the direction of greatest increase in temperature on the plate from position (pi^2, 0)? Explain why or why not.


    2. Relevant equations



    3. The attempt at a solution
    I have that at time t=pi the particle's rate of change in temperature is -800(pi^3)e^[-(pi^2+1)] - 400e^[-(pi^2+1)] or -0.479498546
    Also, I found the the gradient of T is the direction of the maximal rate of increase
    so i found that the gradient of T(pi^2,0) = <-400pi*e^(-pi^2), 0>
    But now i dont know what to do.
     
  2. jcsd
  3. Oct 17, 2009 #2

    Dick

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    You've found that the gradient vector of T is horizontal at (pi^2,0). That only way the particle can be travelling in the direction of greatest temperature increase if if it's tangent vector at t=pi is parallel to the gradient vector. Is it?
     
  4. Oct 18, 2009 #3
    hmmm okay thankyou:), but im unsure of how to calculate the tangent vector?

    thanks for your help
     
  5. Oct 18, 2009 #4

    Dick

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    The tangent vector is the vector f'(t).
     
  6. Oct 18, 2009 #5
    Oh okay:)! So then it wouldn't be parallel to the gradient vector because f'(pi) = <2pi, -1> which is not a multiple of the gradient i calculated. So, the particle isn't travelling in the direction of greatest temperature increase.

    Thanks so much!
     
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