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Math Help - Limit Definition of Partial Derivative

  1. #1
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    Limit Definition of Partial Derivative

    So I have a function

    x3y - xy3
    __________
    x2 + y2

    for all (x,y) not equal to (0,0)
    and the function is equal to 0 for (x,y) = (0,0) (Basically to make the function continuous)

    I need to do a limit definition of the function above

    fx(0,0) =

    lim(h->0)

    f(h,0) - f(0,0)
    _____________
    h

    And the same for fy


    I need to show that fx(0,0) = fy(0,0) = 0

    I was never good at limit derivatives in Calculus 1 because i already knew the shortcuts, so I am completely lost on how to do this.

    Sorry for the format, I could not get it the align right
    Last edited by bustacap09; April 25th 2012 at 06:21 AM.
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  2. #2
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    Re: Limit Definition of Partial Derivative

    To evaluate \displaystyle \begin{align*} \lim_{(x, y) \to (0, 0)} \frac{x^3y - x\,y^3}{x^2 + y^2} \end{align*} is most likely easiest solved by converting to polars, because by converting to polars, it doesn't matter what path you take, you will always be making \displaystyle \begin{align*} r \to 0 \end{align*}, so it transforms the limit into a limit of one variable instead of two. Anyway, with \displaystyle \begin{align*} x = r\cos{\theta}, y = r\sin{\theta} \end{align*} and \displaystyle \begin{align*} x^2 + y^2 = r^2 \end{align*}, we find

    \displaystyle \begin{align*} \lim_{(x, y) \to (0, 0)}\frac{x^3y - x\,y^3}{x^2 + y^2} &= \lim_{r \to 0}\frac{\left(r\cos{\theta}\right)^3r\sin{\theta} - r\cos{\theta}\left(r\sin{\theta}\right)^3}{r^2} \\ &= \lim_{r \to 0}\frac{r^4\cos^3{\theta}\sin{\theta} - r^4\cos{\theta}\sin^3{\theta}}{r^2} \\ &= \lim_{r \to 0}\left(r^2\cos^3{\theta}\sin{\theta} - r^2\cos{\theta}\sin^3{\theta}\right) \\ &= 0 \end{align*}


    Edit: I hope this post has been helpful, though I now realise I should have read the question in full, because when skimming through I thought you were asking to show the continuity at the point (x, y) = (0, 0), instead of what I now realise you were actually asking, i.e. to evaluate the partial derivatives.
    Last edited by Prove It; April 25th 2012 at 06:30 AM.
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  3. #3
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    Re: Limit Definition of Partial Derivative

    Would I use polar to do the limit definition? I thought the question was asking for me to use that limit derivative formula if you used like

    x^2

    You would do (x+h)^2 - x^2 all divided by h

    or am i thinking of the wrong thing?
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  4. #4
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    Re: Limit Definition of Partial Derivative

    Limit Definition of Partial Derivative-image201204250001.jpg
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  5. #5
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    Re: Limit Definition of Partial Derivative

    Quote Originally Posted by bustacap09 View Post
    Would I use polar to do the limit definition? I thought the question was asking for me to use that limit derivative formula if you used like

    x^2

    You would do (x+h)^2 - x^2 all divided by h

    or am i thinking of the wrong thing?
    The difference is that this example has only the real variable x so we are on the real line. On a line, "h going to 0" means "from above" and "from below". With complex numbers, we are in the complex plane so there are many more ways to "approach 0". h can "go to 0" along any line through 0 or even through other curves such as a parabola or a spiral. It is simplest to use polar coordinates so the distance from the origin is given by the single variable r for any value of \theta.
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