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Math Help - Where is this statement true?

  1. #1
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    Where is this statement true?

    So, my calculus teacher just said in class today that, fxy = ((δ^2)f)/(δxδy) is true.

    Where exactly is this true?

    Thanks.
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  2. #2
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    Re: Where is this statement true?

    Quote Originally Posted by bagels0 View Post
    So, my calculus teacher just said in class today that, fxy = ((δ^2)f)/(δxδy) is true.

    Where exactly is this true?

    Thanks.
    Not sure exactly what you're saying, but it brings this to mind.
    Symmetry of second derivatives - Wikipedia, the free encyclopedia
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  3. #3
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    Re: Where is this statement true?

    I think what was meant was that f_{xy}= \frac{\partial^2 f}{\partial x\partial y}. If that is what was meant, it is always true- those are different notations for the same thing.

    If those really are \delta then I would guess that they refer to "small" changes in x, y, and f and so that is an approximation to f_{xy} but is exact in the case that f is a linear function of x and y.

    A third possibility is that your teacher was saying that f_{xy}= \frac{\partial^2 f}{\partial x\partial y}= f_{yx} which is the same as saying that \frac{\partial^2 f}{\partial x\partial y}= \frac{\partial^2 f}{\partial y\partial x}- that is that the "mixed" second derivative are equal- independent of the order of differentiation. That is true as long as the second partial derivatives are continuous.
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  4. #4
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    Re: Where is this statement true?

    Hello, bagels0!

    \text{So my calculus teacher said: }\:f_{xy} \:=\: \frac{\partial^2\!f}{\partial x\,\partial y}

    \text{Where exactly is this true?}

    What an UGLY way to write that identity!

    Note that: . f_{xy} \:=\:\frac{\partial f}{\partial y}\left(\frac{\partial f}{\partial x}\right) \:=\:\frac{\partial^2\!f}{\partial y\,\partial x}


    And so the claim is that: . \begin{Bmatrix} \dfrac{\partial^2\!f}{\partial y\,\partial x} \:=\: \dfrac{\partial^2\!f}{\partial x\,\partial y} \\ \text{or} \\ f_{xy} \:=\:f_{yx}  \end{Bmatrix}


    This is true for all functions, f(x,y).

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