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.© BrainMass Inc. brainmass.com February 24, 2021, 2:36 pm ad1c9bdddf
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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 ...
Directional derivatives and tangent vectors are investigated. The proof or counterexample to disprove is determined.