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Total derivative -> partial derivative

  1. Apr 20, 2005 #1
    Under what conditions can you replace a total differential with a partial?

    dx/dy -> partial(dx/dy)

    in the context of 2 independant variables and multiple dependant variables.

  2. jcsd
  3. Apr 20, 2005 #2
    let x(u,v),and u,v are functions of y
  4. Apr 20, 2005 #3
    but under what conditions can i say

    dx/dy = ∂x/∂y ?
  5. Apr 20, 2005 #4


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    Science Advisor

    Just as dvs77 said: precisely when x depends only on y!

    However, that was that really your question? You originally asked "Under what conditions can you replace a total differential with a partial?" A "total differential" is not " a derivative". In other words, not dx/dy. In terms of two independent variables, x and y, the total differential of a function f(x,y) is [tex]df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy[/tex].

    Notice that, in the case you are describing, x and y are not reall "independent".
  6. Apr 20, 2005 #5

    And I was looking for the total derivative, sorry about the mistake in the op.
  7. Apr 20, 2005 #6


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    Homework Helper

    There's total derivative and there's total differential...Which one are u after...?Halls gave the simplest example of a (total) differential.

    The really interesting case is when dependece upon a variable is both explicite & implicite

    [tex] z=z\left(\frac{x}{t},yt^{2},u(t),t\right) [/tex]

    and u want to compute

    [tex]\frac{dz}{dt} [/tex]

  8. May 2, 2005 #7
    To be more precise with notations this should be

    [tex] \frac{\partial x}{\partial u}\frac{du}{dy}+\frac{\partial x}{\partial v}\frac{dv}{dy} [/tex]

    Note that if z=f(x,y) and y=g(x) then d/dx and \partial_x both exist but they are different.

    Another interesting case is : z=f(x,z)...(implicit functions)
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