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Directional Derivative

  1. Apr 12, 2008 #1
    Hi:

    Can someone point how to approach this problem- we had 5 problems on directional derivatives and I solved 4. I understand the concept but in this question I don't know where to begin

    Problem Statement
    Assume that f:R[tex]^{n}[/tex] -> R[tex]^{m}[/tex] is a linear map, with matrix A with respect to the canonical bases. Show that Df(xo) = f for every xo [tex]\in [/tex] R[tex]^{n}[/tex]


    Plz advise - I will probably post follow-up questions to any answers

    Thanks

    Asif
     
  2. jcsd
  3. Apr 12, 2008 #2

    Hurkyl

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    You should never be at a loss of how to begin a problem -- definitions are almost always a reasonable starting point.
     
  4. Apr 12, 2008 #3

    HallsofIvy

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    Specifically, what is the definition of "Df" and what happens when you apply that definition to a linear map?
     
  5. Apr 13, 2008 #4
    I will give what I have done so far...

    Definition of a directional derivative is its partial derivatives wrt to all the variables in the given function.

    So in this case the question is for f:R[tex]^{n}[/tex]->R[tex]^{m}[/tex] there is an mxn matrix which would look like the following

    Df(x0) = (assume this is equation 1)

    df1/dx1 df1/dx2... df1/dxn
    df2/dx1 df2/dx2... df2/dxn
    . ........ . ........ .
    . ........ . ........ .
    dfm/dx1 dfm/dx2...dfm/dxn

    Where in above matrix dfm/dx1 is the partial derivative of function wrt x1,x2... I couldn't find symbol for partial derivative

    Now, if I use the definition of a linear map then I know that
    D([tex]\alpha1[/tex]+[tex]\alpha2[/tex] ) f(x0) = [tex]\alpha1[/tex]Df(x0) + [tex]\alpha2[/tex] Df(x0)

    I can also prove by continuity and as t->0 and [tex]\varsigma[/tex]->0 that
    Df(x0 + [tex]\varsigma[/tex]p1 + t[tex]\alpha2[/tex]p2) -> Df(x0) ----- equation 2

    Now since this is a canonical map the above matrix of Df(x0) in equation 1 reduces to the following
    1 0... 0
    0 1... 0
    . ...
    . ...
    0 0...1

    So in equation2 since D is essentially the above matrix, I can say the following:
    Df(x0 + [tex]\varsigma[/tex]p1 + t[tex]\alpha2[/tex]p2) -> f(x0) which is what I think the question wants.

    Is this correct?

    Thanks

    Asif
     
  6. Apr 13, 2008 #5

    HallsofIvy

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    Yes.

    (in LaTex, [ tex ]\partial[ /tex ] gives [tex]\partial[/tex].)
     
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