Dot product between grad f and an arbitrary vector

[guiness]

Given a function f: R^n -> R, a point x in R^n, and an arbitrary vector v in R^n - is the dot product between grad f and v (evaluated at x) the same as df/dv?

If yes, it would be great if someone were to demonstrate a proof.

If no, what should be the correct interpretation of the dot product?

 

[Defennder]

You're referring to the directional derivative are you not? Also note you can't differentiate with respect to a vector. See here for a clearer explanation:
http://en.wikipedia.org/wiki/Directional_derivative

 

[HallsofIvy]

The first thing you will have to do is define df/dv! I know, for example, D_{vf as the directional derivative Defennder refers to- the rate of change of f in the direction of v which is independent of the length of v. The dot product of grad f with an arbitrary unit vector is the derivative in that direction. The dot product of grad f with an arbitrary vector is the derivative in that direction multiplied by the length of the vector.}

 

[guiness]

Thanks for the links! It makes sense now.