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Calculus - Differentials and Partial Derivatives

  1. Jun 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Find a differential of second order of a function [itex]u=f(x,y)[/itex] with continuous partial derivatives up to third order at least.


    Hint: Take a look at [itex]du[/itex] as a function of the variables [itex]x[/itex], [itex]y[/itex], [itex]dx[/itex], [itex]dy[/itex]:
    [itex]du= F(x,y,dx,dy)=u_xdx +u_ydy[/itex].



    2. Relevant equations
    3. The attempt at a solution
    I'll be glad to receive some guidance regarding the following:
    1) Does the second order differential is assumed to be:
    [itex] d^2 f = ( \frac{ \partial}{ \partial x_1} \Delta x_1 +\frac{ \partial}{ \partial x_2} \Delta x_2 ) ^n f [/itex] ?
    2) If so, how should I solve this question?


    Thanks in advance !
     
  2. jcsd
  3. Jun 25, 2012 #2
    You know what du is, so what's [tex]d^2 u = d(du) = ?[/tex]
     
  4. Jun 25, 2012 #3

    HallsofIvy

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    df= fxdx+ fydy.

    As bigplanet401 said, you want the differential of that- but you also need to know that differentials are skew commutative. That is, that dxdy= -dydx.
     
  5. Jun 25, 2012 #4
    Thanks a lot !
     
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