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Gradient Vector Problem

  1. Jan 13, 2013 #1
    1. The problem statement, all variables and given/known data
    For a hill the elevation in meters is given by z=10 + .5x +.25y + .5xy - .25x^2 -.5y^2, where x is the distance east and y is the distance north of the origin.

    a.) How steep is the hill at x=y=1 i.e. what is the angle between a vector perpendicular to the hill and the z axis?

    b.)In which compass direction is the slope at x=y=1 steepest? Indicate whether the angle you provide is the angle measured in the standard way in the counter-clockwise direction from the x axis (east) or whether it is the compass azimuth.



    2. Relevant equations



    3. The attempt at a solution
    a.) So I got the gradient to be ∇f(x,y,z)= .5i - .25j...simply by the definition of the gradient. So i figured to figure the angle between the gradient and the z axis i could use the formula cosθ= (A dotted into B)/(lAl*lBl) but wouldn't be the z axis be the unit vector in the k hat direction making the cosθ=0...which can't be right. Any help on what to do? Also, I was given an equation that says ∇f(x,y,z) perpendicular to a 3 dimensional surface f(x,y,z) = constant. But I don't know how to use that equation to generate an answer.

    b.)The gradient gives you the direction of the quickest altitude ascension (at least for this problem) so how do I use what I calculated for the gradient to answer this part?

    Thanks for any help!!
     
  2. jcsd
  3. Jan 13, 2013 #2

    haruspex

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    Think of the altitude as a potential in 2 dimensions, just x and y. So ∇ is ?
     
  4. Jan 13, 2013 #3

    rude man

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    grad z gives you the x-y direction of steepest ascent, but now you have to ask yourself: when going in that direction, how much does dz change when you go a distance ds in the aforederived direction.

    Think about how the x and y components of the gradient at your point (1,1) tell you what dz/ds would be.

    P.S. the angle the problem asks for happens to be (90 deg - the steepness angle). Also, once you did (a) you already have (b). In fact, (b) should come first ... at least the way I look at it. Probably you were not supposed to solve for grad z for part (a) ... I don't know.
     
  5. Jan 13, 2013 #4
    Dear Rude Man,

    Very nice job of doping it out. That's exactly the way I would have done it, including doing part (b) first.

    Chet
     
  6. Jan 13, 2013 #5

    rude man

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    Thanks Chet.
     
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