January 6th, 2015, 05:46 AM  #1 
Newbie Joined: Dec 2012 Posts: 17 Thanks: 0  Direction of gradient vector
Hello, Can the direction of the gradient vector be expressed by its length? In other words by: sqrt((PDz/PDx)²+(PDz/PDy)²) where PD = partial derivative I'm confused since our textbook mentions the following to calculate a slope: dz/ds = length of gradient vector * length of unit vector * cos(angle between those two) Since length of unit vector = 1 and cos = maximised when angle = 0: dz/ds = length of gradient vector = sqrt((PDz/PDx)²+(PDz/PDy)²). But does this mean that the length of the gradient vector equals its direction, since a gradient vector always points towards the direction of the maximum gradient? Thanks already! 
January 9th, 2015, 04:31 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
You should realize that "length of the gradient vectors equals its direction" makes no sense because "length" and "direction" cannot even be measured in the same units! It is true that the derivative in the direction in which the gradient vector is pointing is maximum value of length of the gradient. But that is not at all saying "the length is equal to the direction"! For example, the length of your shadow is shortest at noon but you cannot say "the length is equal to the time". You could, possibly, say "the time can be determined by the length of my shadow" but I am not sure that is what you mean by "expressed by". Last edited by Country Boy; January 9th, 2015 at 04:36 AM. 

Tags 
direction, gradient, vector 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
simple explanation for vector direction  ungeheuer  Linear Algebra  2  July 15th, 2014 03:51 PM 
Calculate departure vector(Direction)  LevyDee  Algebra  5  March 1st, 2013 10:22 PM 
Gradient Vector Check?  nabran  Linear Algebra  2  October 14th, 2011 05:23 AM 
Direction vector from two points  yaypotatoes  Algebra  2  September 26th, 2008 07:29 PM 
moving in the direction of eigen vector  mato  Linear Algebra  0  December 5th, 2007 03:05 PM 