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Math Help - well defined partial derivatives everywhere in R2

  1. #1
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    well defined partial derivatives everywhere in R2

    What does it mean if a function has well-de ned partial derivatives everywhere in R2?

    Can the partial derivatives have a place where they are not defined?

    eg.
    f(x) = xy / (x^2 + y^2)

    is df/dx and df/dy "well defined"?

    Thanks
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  2. #2
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    Quote Originally Posted by swraman View Post
    What does it mean if a function has well-de ned partial derivatives everywhere in R2?

    Can the partial derivatives have a place where they are not defined?

    eg.
    f(x) = xy / (x^2 + y^2)

    is df/dx and df/dy "well defined"?

    Thanks
    Reading this might help: http://www.econ.ubc.ca/sart/320lec9.pdf
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  3. #3
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    Both first order differentiations of your function lead to the denominator (x^2+y^2)^2, as the result of the quotient rule of differentiation.

    For what values of x and y does a function  \frac{g(x)}{(x^2+y^2)^2} become undefined?

    Well... for all positive and negative values of x and y, the denominator is finite and positive. However... what happens for x = y = 0 ?
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