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Homogeneous of Degree N

  1. Nov 2, 2011 #1
    1. The problem statement, all variables and given/known data
    Let n be a positive integer. A function F is called honogeneous of degree n if it satisfies the equation F(tx,ty) = tnF(x,y) for all real t. Suppose f(x,y) has continuous second-order partial derivatives.

    Show that if F is homogeneous of degree n, then

    x*F_x + y*F_y = n*F(x,y), where F_x,F_y are the partial derivatives
    with respect to x,y.

    2. Relevant equations



    3. The attempt at a solution
    Suppose I let u=tx and v=ty. Then,

    F_t = F_u * x + F_v * y,

    which should then be equal to

    n*t^(n-1) * F(x,y).

    This, I think, almost looks like what I want to prove. Dividing by
    t^(n-1) gives n*F(x,y) and (F_u * x + F_v * y)/(t^(n-1)), which I want
    to rewrite as

    x*F_x + y*F_y.

    But I have no idea if/why this should be true. Am I thinking about this correctly, or have I done it the wrong way?
     
  2. jcsd
  3. Nov 2, 2011 #2

    LCKurtz

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    Try setting t = 1 in your answer.:smile:
     
  4. Nov 2, 2011 #3
    If t=1, then n*t^(n-1)*F(x,y) becomes n * F(x,y), which still equals

    F_u * x + F_v * y,

    which must clearly be equal to

    F_x * x + F_y * y

    if our premise is true. But I don't find this solution particularly convincing. Why should t have to equal 1? As in the definition, t can be equal to any real number.
     
  5. Nov 2, 2011 #4

    LCKurtz

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    You are given F(tx,ty)=tnF(x,y). You differentiated both sides with respect to t and got:

    xFx(tx,ty)+yFy(tx,ty)=ntn-1F(x,y)

    This is an identity in t,x,and y. In particular it holds for t=1, which gives your result. What is left to explain?
     
  6. Nov 2, 2011 #5
    Tell me what an "identity" is, then maybe I'll understand better.
     
  7. Nov 2, 2011 #6

    LCKurtz

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    An identity is an equation that is true for all values of its variables. For example:

    (x-1)2= x2-2x+1

    is an identity because it is a true statement no matter what value x has. This is different than an equation like this:

    x2+x = 6

    which is true only for some values of x.
     
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