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Homework Help: Simple Gradient Question (funtion of two variables)

  1. Sep 20, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the gradient of the function at the given point. Then sketch the gradient together with the level curve that passes through the point.

    g(x,y) = x2/2 - y2/2; (√2, 1)

    2. Relevant equations
    ∇f = (∂f/∂x)i + (∂f/∂y)j

    3. The attempt at a solution
    ∇g = <x, -y>
    ∇g(√2, 1) = <√2, -1>

    Am I done with this solution? Or is there more I need to put for the gradient at the point?

    I'm not really sure if this needs to be reduced to a single number or not, I'm guessing that has to do with my lack of understand of the gradient itself. I thought I was supposed to have a direction vector, is it implied the direction vector is just u = i + j if it is not specified?

    I also have no idea how to draw what it's asking. I can't visualise any of this.
  2. jcsd
  3. Sep 20, 2012 #2


    User Avatar
    Science Advisor

    No, that is the gradient.

    You are confusing this with the derivative "in the direction of unit vector v". That is equal to [itex]v\cdot \nabla f[/itex].
  4. Sep 20, 2012 #3
    Yes, the gradient you calculated is correct.

    In order to visualize the gradient in your problem,
    depict a two dimensional graph and pick out arbitrary (x,y) points i.e. (2, 3) (0,1) (-1,0) (-1,-1) … and plug those into ∇(g). Use these corresponding vectors to depict the vector field. For example, at (1,0) the resulting vector would be <1,0>. Likewise…

    (2,3) <2, -3>
    (0,1) <0,-1>
    (-1,0) <-1,0>
    (-1,-1) ….
    …. ….

    To draw each vector, begin with the tail at the chosen (x,y) coordinates and move accordingly. For the first vector, begin at (2,3) then move in the positive x direction 2 units, then move down the negative y direction 3 units – there’s your endpoint. As you draw more vectors out, you should start to get an idea of the field’s behavior.
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