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Homework Help: Proving Characteristics of Partial Derivatives

  1. Nov 12, 2011 #1
    1. The problem statement, all variables and given/known data
    Let F(x,y) be a twice differentiable function such that

    4 * Fx2 + Fy2 = 0.

    Set x = u2 - v2 and y = u*v. Show that

    Fu2 + Fv2 = 0

    3. The attempt at a solution
    Fu = Fx*2u + Fy*v
    Fv = Fx*-2v + Fy*u

    Fu2 = 4v2Fx2-4FxFy*u*v+Fy2u2

    Fv2 = 4u2Fx2+4FxFy*u*v+Fy2v2

    Adding these two together gives

    4Fx2*(u2v2) + 4Fy2*(u2v2),

    Which from our first equation, can clearly equal zero if we factor out the (u2v2) and divide. Then, we see that, indeed, Fu2 + Fv2 = 0.

    Sound about right?
  2. jcsd
  3. Nov 12, 2011 #2


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    Staff Emeritus
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    You have some typos in your post.
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