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    Directional Derivative and Tangent Vector

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    Give a simple proof or counterexample to disprove:

    If tangent vector p E Rn is such that the directional derivative of f by k vanished for every function f then k =0.

    If function f on Rn is such that the directional derivative of f by every tangent vector at every point vanishes, then f is constant.

    The directional derivative of the product of two functions product of their directional derivatives.

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    a) It is true.
    Proof. Without loss of generality, we assume that n=2. To the contrary, suppose that . Then for a function , its directional derivative with respect to at is given ...

    Solution Summary

    Directional derivatives and tangent vectors are investigated. The proof or counterexample to disprove is determined.