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Dependence between derivatives

  1. Jan 14, 2014 #1
    Hellow everybody!!

    A form how various ODE are intercorrelated can be sinterized like this:

    ##t = t##
    ##y = y(t)##
    ##y' = y'(t,\;y)##
    ##y'' = y''(t,\;y,\;y')##
    ##y''' = y'''(t,\;y,\;y',\;y'')##

    Until here, no problems!

    But, how is such relationship wrt the PDE?

    Would be this:

    ##x = x##
    ##y = y##
    ##u = u(x,\;y)##
    ##u_x = u_x(x,\;y,\;u)##
    ##u_y = u_y(x,\;y,\;u)##
    ##u_{xx} = u_{xx}(x,\;y,\;u,\;u_x,\;u_y)##
    ##u_{yy} = u_{yy}(x,\;y,\;u,\;u_x,\;u_y)##
    ##u_{xy} = u_{xy}(x,\;y,\;u,\;u_x,\;u_y)##
    ##u_{yx} = u_{yx}(x,\;y,\;u,\;u_x,\;u_y)##

    Ie, the derivative of order n wrt x no depends of the derivative of order n wrt y (and vice versa), or depends? If depends, so the relationship would be this:

    ##x = x##
    ##y = y##
    ##u = u(x,\;y)##
    ##u_x = u_x(x,\;y,\;u,\;u_y)##
    ##u_y = u_y(x,\;y,\;u,\;u_x)##
    ##u_{xx} = u_{xx}(x,\;y,\;u,\;u_x,\;u_y,\;u_{yy},\;u_{xy},\;u_{yx})##
    ##u_{yy} = u_{yy}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{xy},\;u_{yx})##
    ##u_{xy} = u_{xy}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{yy},\;u_{yx})##
    ##u_{yx} = u_{yx}(x,\;y,\;u,\;u_x,\;u_y,\;u_{xx},\;u_{yy},\;u_{xy})##

    Wich 2 last relation is correct?
     
  2. jcsd
  3. Jan 15, 2014 #2
    BTW, my question make sense???
     
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