1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    HallsofIvy

    User Avatar
    Staff Emeritus
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook