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Math Help - Directional Derivative

  1. #1
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    Directional Derivative

    Quick Question;

    In Munkres' "Analysis on Manifolds", he defines the directional derivative of a map f at the point x, with respect to the vector u to be:

     \lim_{t \rightarrow 0} \frac{f(x + tu) - f(x)}{t} , provided this limit exists.

    Consider the function f(x,y) = |x| + |y|.  \lim_{t \rightarrow 0^{+}} \frac{f(x+tu) - f(x)}{t} \neq \lim_{t \rightarrow 0^{-}} \frac{f(x + tu) - f(x)}{t} and so the directional derivative at (0,0) with respect to u doesn't exist, according to the definition above (where it requires that both one-sided limits be equal to eachother). However, I have someone who is more knowledgeable in maths than me that says that it is only required that the limit exist when t approaches 0 through positive values. Am I right, or is he right?

    EDIT: Even Wikipedia says that it is only required t go to 0 through positive values. But it just doesn't seem right that Munkres would make a mistake in his definition.

    EDIT 2: http://mathworld.wolfram.com/DirectionalDerivative.html agrees with Munkres' definition.

    I guess the definition depends on the author of the book?
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  2. #2
    Senior Member Tinyboss's Avatar
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    Oops, misread and spoke too soon.
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