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Express derivatives most intuitively

  1. Feb 28, 2013 #1
    I have given a function g(t)=∇(f(x(t))) , f: IR³->IR and x: IR-> IR³ and want to express the first 3 derivatives with respect to time most simply.
    I thought that g'(t)=Hessian(f(x(t)))dx/dt
    but how do I get the further derivatives. is there any chance to express those in terms of taking the gradient or the hessian matrix of a function several times?
     
  2. jcsd
  3. Feb 28, 2013 #2

    HallsofIvy

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    If f is a function from R3 to R3, then its derivative is, as you say, a linear transformation from R3 to R3 which you can represent as a 3 by 3 matrix, the Hessian, at each point in R3, which has 9 entries and so can be thought of as in R9.

    And that, in turn, means that the derivative function is from R3 to R9 so its derivative would be a linear tranformation from R3 to R9 which could be represented by a "3 by 3 by 3" matrix- a sort of three dimensional variation of a matrix. Higher derivatives, then, would involve higher dimensions.
     
  4. Feb 28, 2013 #3
    okay, thank you for your reply, but how do I write down these higher derivatives explicitely, so that I get the right answer?
     
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