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Finding a direction to not change the level

  1. Mar 23, 2014 #1
    1. The problem statement, all variables and given/known data
    You are walking on the graph of ##f(x,y) = y cos(\pi x) - x cos(\pi y) + 10##, standing at the point ##(2,1,13)##. Find an x, y-direction you should walk in to stay at the same level.


    2. Relevant equations
    ##D_u f = \nabla \cdot \textbf{u}##


    3. The attempt at a solution
    The directional derivative is the rate of change in the direction of ##\textbf{u}##, so we want the rate of change in the direction of ##\textbf{u}## to not change, i.e. ##\nabla \cdot \textbf{u} = \textbf{0}##. So calculating the gradient gives ##<1,1>##. Then ##<1,1> \cdot \textbf{u} = \textbf{0}##. So that means ##\textbf{u} = <0,0>##, but this is incorrect. Why?
     
  2. jcsd
  3. Mar 23, 2014 #2

    Ray Vickson

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    Because going along ##\vec{u} = <0,0>## gets you nowhere: any multiple of 0 is still 0! What is the general criterion for the condition ##<u_x,u_y> \perp <1,1>##, expressed as an equation or equations involving ##u_x, \, u_y##?
     
    Last edited: Mar 23, 2014
  4. Mar 23, 2014 #3

    haruspex

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    By definition, you need a nontrivial solution of <1,1>.u = 0. u = <0,0> is not the only solution.
     
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