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Thread: Directional Derivatives and Gradient Vectors

  1. #1
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    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!
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  2. #2
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    Surely someone here has experience in these problems.
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  3. #3
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    Quote Originally Posted by charmedquark View Post
    Surely someone here has experience in these problems.
    Don't bump your post unless you have relevant information to add or a partial solution.
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    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?
    Last edited by charmedquark; Mar 21st 2011 at 01:28 PM.
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  5. #5
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    Quote Originally Posted by charmedquark View Post
    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 <f_x, f_y>. Taking the positive y axis as north and the positive x-axis east, a unit vector NW would be <-\sqrt{2}/2+ \sqrt{2}/2>. The derivative in that direction is given by -f_x\sqrt{2}/2+ f_y\sqrt{2}/2= 1/2. A unit vector NE would be <\sqrt{2}/2, \sqrt{2}/2> so the derivative in the direction is f_x\sqrt{2}/2+ f_y\sqrt{2}/2= -1/4.

    Solve those two equations for f_x and 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!
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