# Thread: Directional Derivatives and Gradient Vectors

1. ## Directional Derivatives and Gradient Vectors

I would appreciate any help on the following problem:

Suppose I am descending a mountain, and for every 3 meters I travel NW, I climb 1/2 a meter, and for every 2 meters I travel NE, I descend 1/4 of a meter.
a)What direction should I start for fastest descent?
b)If I travel in the direction of fastest descent at 2 meters/sec., what will be my rate of descent?
c)derive an expression for this rate of descent as a function of the direction traveled and the speed in that direction.
d)in what directions should I start in order NOT to go up or down?

I know how to find directional derivatives and gradient vectors but I am used to questions in a format where I am given the functions. I know the gradient vector gives the direction of fastest increase of such a function (travelling perpendicularly down level curves). However, I am not very good at applying my knowledge!

2. Surely someone here has experience in these problems.

3. Originally Posted by charmedquark
Surely someone here has experience in these problems.
Don't bump your post unless you have relevant information to add or a partial solution.

4. I can start it, perhaps, but I wouldn't be surprised if i was heading in the wrong direction already (bad pun): well i know the units for my directional derivative are in meters in altitude per meters travelled, one directional derivative might look like a general gradient vector times 1/sqrt(2) times the unit vector <-1,1>=(1/2)/3, and perhaps the other directional derivative looks like another general gradient vector with 1/sqrt(2) times the unit vector <1,1>=(-1/4)/2, assuming i have laid out my N,E,S,W directions according to the + y axis, + x axis, - y axis, and -x axis, respectively. Did I do all this wrong and/or could I have help from here?

5. Originally Posted by charmedquark
I would appreciate any help on the following problem:

Suppose I am descending a mountain, and for every 3 meters I travel NW, I climb 1/2 a meter, and for every 2 meters I travel NE, I descend 1/4 of a meter.
Call the function f so that its gradient is $\displaystyle <f_x, f_y>$. Taking the positive y axis as north and the positive x-axis east, a unit vector NW would be $\displaystyle <-\sqrt{2}/2+ \sqrt{2}/2>$. The derivative in that direction is given by $\displaystyle -f_x\sqrt{2}/2+ f_y\sqrt{2}/2= 1/2$. A unit vector NE would be $\displaystyle <\sqrt{2}/2, \sqrt{2}/2>$ so the derivative in the direction is $\displaystyle f_x\sqrt{2}/2+ f_y\sqrt{2}/2= -1/4$.

Solve those two equations for $\displaystyle f_x$ and $\displaystyle f_y$.

a)What direction should I start for fastest descent?
b)If I travel in the direction of fastest descent at 2 meters/sec., what will be my rate of descent?
c)derive an expression for this rate of descent as a function of the direction traveled and the speed in that direction.
d)in what directions should I start in order NOT to go up or down?

I know how to find directional derivatives and gradient vectors but I am used to questions in a format where I am given the functions. I know the gradient vector gives the direction of fastest increase of such a function (travelling perpendicularly down level curves). However, I am not very good at applying my knowledge!