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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.

##### Solution Summary

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

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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 ...

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• BSc , Wuhan Univ. China
• MA, Shandong Univ.
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• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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