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What is Euler's Criterion in plain simplified english

  1. Jun 29, 2008 #1
    What is "Euler's Criterion" in plain simplified english

    This sentence in textbook reads: "If ∂M/∂y = ∂N/∂x the differential is exact (Euler’s Criterion)."

    What does the equation mean, and also im not familiar with the backwards number 6 symbol --what is that?
    thanks
     
  2. jcsd
  3. Jun 29, 2008 #2
  4. Jun 29, 2008 #3
    Re: What is "Euler's Criterion" in plain simplified english

    Given that f is a function of several variables, [itex]\frac{\partial f}{\partial x}[/itex] is the partial derivative of f with respect to the variable x. It is the same as the normal derivative of f if we take all other variables to be constants.
    Suppose we have a function f(x,y). The differential of f(x,y) is defined to be [itex]df(x,y) = \frac{\partial f}{\partial x}(x,y) dx + \frac{\partial f}{\partial y}(x,y) dy[/itex]. If you have an expression M dx + N dy, it is called an exact differential if it is the differential of some f(x,y). The Euler criterion is one way of telling whether this is the case; it is easy to see that it should be the case.
     
  5. Jun 29, 2008 #4
    Re: What is "Euler's Criterion" in plain simplified english

    "If you have an expression M dx + N dy, it is called an exact differential if it is the differential of some f(x,y). "

    Could you please provide an example of this?
     
  6. Jun 29, 2008 #5
    Re: What is "Euler's Criterion" in plain simplified english

    Suppose you have the vector field g(x,y) = (-x/r3/2) dx - (y/r3/2) dy, where r = [itex]\sqrt{x^2 + y^2}[/itex]. Then g(x,y) is an exact differential, because g(x,y) = df(x,y) where f(x,y) = r-1/2. Ie., f(x,y) is the potential of a central force g(x,y). You may also see [itex]f = -\nabla g[/itex].
     
  7. Jun 29, 2008 #6

    HallsofIvy

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    Re: What is "Euler's Criterion" in plain simplified english

    If you don't know about partial derivatives, why are you worrying about "Euler's Criterion" and "exact differentials"? The latter requires that you be well versed in partial derivatives.
     
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