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Proof involving homogeneous functions and chain rule

  1. Nov 20, 2009 #1

    hth

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    1. The problem statement, all variables and given/known data

    A function f is called homogeneous of degree s if it satisfies the equation

    f(x1, x2, x3,..... xn)=t^s*f(x1, x2, x3,..... xn) for all t

    Prove that the [tex]\sum[/tex] from i=1 to n of xi * df/dxi (x1, x2, x3,..... xn) = sf(x1, x2, x3,..... xn).

    2. Relevant equations



    3. The attempt at a solution

    Take A=f(t1,t2,..,tn)=t^kf(t1,t2,..,tn)

    Then find dA/dt...

    I keep lost in the differentiation. Am I even on the right track? Any help is appreciated.
     
    Last edited: Nov 20, 2009
  2. jcsd
  3. Nov 20, 2009 #2

    LCKurtz

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    Try something like this (3 variable example):

    Let g(t) = f(tx,ty,tz) = t3f(x,y,z)

    Now, using subscript notation for the partials of f with respect to its arguments, calculate g'(t) using the chain rule:

    g'(t) = xf1(tx,ty,tz)+yf2(tx,ty,tz)+zf3(tx,ty,tz)=3t2f(x,y,z)

    Now let x = u/t, y=v/t and z = w/t and see what happens. Should generalize to n variables.
     
  4. Nov 21, 2009 #3

    HallsofIvy

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    You mean "f(tx1, tx2, tx3,..... txn)=t^s*f(x1, x2, x3,..... xn) for all t"

     
  5. Nov 21, 2009 #4

    hth

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    You're right. I apologize for my mistake.
     
  6. Nov 21, 2009 #5

    hth

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    Alright, here's my attempt.

    f(tx1, tx2, tx3,..... txn)=t^s*f(x1, x2, x3,..... xn) for all t

    Prove that the [tex]\sum[/tex] from i=1 to n of xi * df/dxi (x1, x2, x3,..... xn) = sf(x1, x2, x3,..... xn).

    Proof.

    Let f = f (x1, x2, x3,...,xn).

    Then, by differentiating the function f(ty) = t^(s)f(y) by the chain rule,

    [tex]\partial[/tex]/[tex]\partial[/tex]x1f(ty)d/dt(ty1) + ..... + [tex]\partial[/tex]/[tex]\partial[/tex]xnf(ty)d/dt(tyn) = st^(s-1) f(y).

    So, y1[tex]\partial[/tex]/[tex]\partial[/tex]x1f(ty) + ..... + yn[tex]\partial[/tex]/[tex]\partial[/tex]xnf(ty) = st^(s-1)f(y).

    This is the best I could come up with. I'm pretty confused, honestly.
     
  7. Nov 21, 2009 #6

    LCKurtz

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    Are you responding to my suggestion? It isn't clear to me. Don't call all the xi the same thing y. You want to start with

    g(t) = f(tx1,tx2,...,txn) =tnf(x1, x2,..... xn)
    and differentiate with respect to t similar to what you have shown. Then do what I did.
     
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