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Finding a Directional Derivative Given Other Directional Derivatives

  1. Mar 27, 2016 #1
    1. The problem statement, all variables and given/known data
    Suppose ##D_if(P) = 2## and ##D_jf(P) = -1##. Also suppose that ##D_uf(P) = 2 \sqrt{3}## when ##u = 3^{-1/2} \hat i + 3^{-1/2} \hat j + 3^{-1/2} \hat k##. Find ##D_vf(P)## where ##v = 3^{-1/2}(\hat i + \hat j - \hat k)##.

    2. Relevant equations


    3. The attempt at a solution
    $$2\sqrt{3} = ∇f ⋅ 3^{-1/2}(\hat i + \hat j - \hat k)$$
    $$= 3^{-1/2}(∂f/∂x + ∂f/∂y + ∂f/∂z)$$
    $$6 = (∂f/∂x + ∂f/∂y + ∂f/∂z)$$

    This is where I'm stuck. I would appreciate hints.
     
  2. jcsd
  3. Mar 27, 2016 #2

    Ray Vickson

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    ##v## is a linear combination of ##i, j## and ##u##.
     
  4. Mar 28, 2016 #3
    ##v = 1/\sqrt{3} <1, 0, 0> + 1/\sqrt{3} <0, 1, 0> - 1/\sqrt{3} <1/\sqrt{3}, 1/\sqrt{3}, 1/\sqrt{3}>##
    ##f(x,y,z) = 1/\sqrt{3} x + g(y,z)##
    ##f(x,y,z) = 1/\sqrt{3} y + h(x,z)##

    I'm not sure what ∂f/∂z would be.
     
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