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
Aug17-08, 04:55 AM
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
Aug17-08, 07:12 AM
The first thing you will have to do is define df/dv! I know, for example, D[sub]v[sub]f 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.