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Real Analysis - Differentiation in R^n - Example of a specific function

  1. May 8, 2012 #1
    1. The problem statement, all variables and given/known data

    Give an example of a continuous function f:R^2→R having partial derivatives at (0,0) with
    f_1 (0,0)≠0,f_2 (0,0)≠0
    But the vector (f_1 (0,0),f_2 (0,0)) does not point in the direction of maximal change, even though there is such a direction.

    (If this is too difficult to read, please see the PDF for a nicer version)

    Note that this is a problem from TBB's Elementary Real Analysis

    2. Relevant equations

    none

    3. The attempt at a solution

    I have no idea how to attempt the construction of such a function. Any tips, suggestions, or a walkthrough of how to find such a function would be greatly appreciated. If you feel like giving me an answer, please explain it because understanding this is the most important part of this.
     

    Attached Files:

  2. jcsd
  3. May 11, 2012 #2
    I think you've recognized that this problem has you finding a "counterexample" to a well-known theorem in multivariate calc (it's not really a counterexample to the the theorem, because the theorem is true). If I were you, I would look carefully at the statement and proof of that theorem to see what conditions a function needs to satisfy in order for the theorem to apply. Then try to construct a function that doesn't have the necessary qualities.

    Also, the fact that the problem doesn't use the word "gradient" may be a hint.
     
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