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Minimizing a curve

  1. Mar 30, 2009 #1
    I've been given a continuously differentiable function C(u1, u2) in 2-D Euclidean space. Now i have that for u a function of s, that
    [tex] \frac{ dC}{ds} = \frac{\partial C}{\partial u_1}\frac{ du_1}{ds} + \frac{ \partial C}{\partial u_2} \frac{ du_2}{ds} [/tex]
    which is obviously just an extension of the chain rule. Now I'm told to minimize this subject to the constraint that
    [tex] \left( \frac{\partial u_1}{\partial s} \right)^2 + \left( \frac{\partial u_2}{\partial s} \right)^2 = 1 [/tex]
    and given that the answer is
    [tex] \frac{ du}{ds} = - \frac1{\left\| \frac{\partial C}{\partial u} \right\| } \frac{\partial C}{\partial u} [/tex]

    My problem is that I don't see how we arrived at the answer. What am I missing?
     
  2. jcsd
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