Equal partial derivatives

  1. Hi, in general can we tell anything about the partial derivatives of a differentiable function if they are equal?

    for example I would like them to have to equal some constant. Would this be true?
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,504
    Staff Emeritus
    Science Advisor

    Do you mean [itex]\partial f/\partial x= C[/itex] and [itex]\partial f/\partial y= C[/itex]? The same constant or different constants? From [itex]\partial f/\partial x= C[/itex], we get [itex]f(x,y)= Cx+ g(y)[/itex] where g can be any function of y. Differentiating that with respect to y, [itex]\partial f/\partial y= g'(y)= C[/itex] which tells us that g(y)= Cy+ C' where C' is an arbitrary constant of integration. That is, f(x,y)= Cx+ Cy+ C'.
     
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