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Taylor series with partial derivatives

  1. Feb 28, 2004 #1
    We were gievn a question in tutorial last week asking us to calculate the Taylor series of the function f(x,y) = e^(x^(2) + y^(2)) to second order in h and k about the point x=0, y=0

    I've got the forumla here with all the h's and k's in it and have it written down, but it's actually how to work it out that's confusing me.

    f(a,b) + 1/1! (hd/dx + kd/dy)f(a'b) etc....

    My confusion is do you multiply out the brackets so you'd have-

    f(a,b) + 1/1! (hdf(a,b)/dx + kdf(a,b)/dy)

    So you do the derivatives and then sub in the values of x and y

    Or, do you leave it as it is the first tiem I wrote it and end up with-

    1+ (hd/dx + kd/dx) +0.5(hd/dx + kd/dy)^2 etc

    I know there are more terms but I've so much trouble typing out mathematical terms on this computer! I know this is v.obvious etc but I just want to get this clear in my head cos I have a test this Wednesday at uni and I want to go in with a fighting chance! :smile: Thanks!
  2. jcsd
  3. Feb 28, 2004 #2

    matt grime

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    Your first statement is correct. I'm not sure I see what you've done in the second.

    (hd/dx +kd/dy) is an operator that acts on f, you then evaluate it at (0,0) in this case.

    the generic term of degree n in the expansion about (0,0) is

    [tex]\frac{1}{n!}\sum_{r=0}^n \binom{n}{r}h^rk^{n-r}\frac{\partial^nf}{\partial x^r \partial y^{n-r}}[/tex] with the function understood to be evaluated at (0,0)
    Last edited: Feb 28, 2004
  4. Feb 29, 2004 #3
    Thanks very much!
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