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Taylor expansion, of gradient of a function, in multiple dimensions

  1. Sep 11, 2011 #1
    Hello all,

    I understand that the taylor expansion for a multidimensional function can be written as

    [itex]f(\overline{X}[/itex] + [itex]\overline{P}[/itex]) = [itex]f(\overline{X}) + \nabla f(\overline{X}+t\overline{P})(\overline{P})[/itex]

    where t is on (0,1).

    Although I havent seen that form before, it makes sense.

    But I don't understand the integral in the following the Taylor expansion,

    [itex]\nabla f(\overline{X}[/itex] + [itex]\overline{P}[/itex]) = [itex]\nabla f(\overline{X}) + \int^{1}_{0} \nabla^{2} f(\overline{X}+t\overline{P})(\overline{P})dt[/itex]

    Could someone help me understand the derivation?

    Thank you,

    Last edited: Sep 11, 2011
  2. jcsd
  3. Sep 17, 2011 #2

    Stephen Tashi

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    I don't understand it. What kind of multiplication is going on in the last term? It appears to be two vectors multiplied together. Is it a dot product?

    I don't either, but this is an interesting formula and I would like to know where you saw it. Is this from a subject like fluid dynamics? Can you give a link to a page?
  4. Sep 17, 2011 #3
  5. Sep 17, 2011 #4

    Stephen Tashi

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