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Math Help - Calculating Derivative in Several Variables

  1. #1
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    Calculating Derivative in Several Variables

    Suppose  A \subset R^m . Then  f: A \rightarrow R^n is differentiable at a point  \vec{x} \in A if there is an n by m matrix  B such that  \lim_{h \rightarrow 0} \frac{ ||f(\vec{x} + \vec{h}) - f(\vec{x}) - B \vec{h}||}{||\vec{h}||} .

    My question is, does it matter HOW  \vec{h} goes to 0? For example, consider the function  f(x,y) = xy . Say I wanted to calculate its derivative.

    When taking the limit of the derivative difference quotient, is it permissible to let the entries of the matrix  \vec{h} \in R^2 approach 0 in any fashion I want?

    I am asking this because I am just learning multi-variable calculus, and when calculating the derivatives I've always let  \vec{h} = (h h) and was wondering if I am allowed to do this.
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  2. #2
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    The expression

    \lim_{h \rightarrow 0} \frac{ ||f(\vec{x} + \vec{h}) - f(\vec{x}) - B \vec{h}||}{||\vec{h}||}

    is confusing. What is the limit equal to?
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  3. #3
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    Ooops. I meant to say that f is differentiable if there is a n by m matrix B such that the expression you posted goes to 0.
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  4. #4
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    If you look here, you will see that all of the partial derivatives can exist, and the function still not be differentiable at the point. So, it must be that \vec{h} must be allowed to approach zero in any direction whatever. That is the condition of differentiability. Now, if you're asking how to compute the derivative, then I would think you could let it go to zero in any particular way want; you should be able to get the same answer regardless.
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  5. #5
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    Suppose that f is a function that isn't differentiable, but the derivative difference quotient goes to 0 if h goes to 0 along a straight line through the origin.

    If I am computing the derivative, and I let h go to 0 along a straight line through the origin, then the derivative difference quotient would go to 0, and I would think that the derivative of f is some matrix B, even though f isn't really differentiable, since if I let h go to 0 in another way, the limit wouldn't exist.

    How do I get past this problem?
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  6. #6
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    Well, I would think you'd need to convince yourself the derivative existed before you went looking for it. That, at least, is the typical thing a mathematician would do. According to the wiki, "It is known that if the partial derivatives of a function all exist and are continuous in a neighborhood of a point, then the function must be differentiable at that point,..."
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  7. #7
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    Thanks for the help!
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  8. #8
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    No problem. Have fun!
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