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Problem in finding problem in proof

  1. Jun 3, 2007 #1
    Let [tex]G\subset\mathbb{R}^n[/tex] be an open set, and [tex]f:G\to\mathbb{R}[/tex] be a continuously differentiable function. The Schwarz's theorem says that if [tex]\partial_k\partial_i f[/tex] exists continuously in some environment of some [tex]a\in G[/tex], then also [tex]\partial_i\partial_k f[/tex] exists in the point and [tex]\partial_k\partial_if(a)=\partial_i\partial_kf(a)[/tex].

    I struggled with the proof of this for some time, and then came to a conclusion, that the proof that is in my lecture notes is unnecessarely complicated, and found a simpler one myself. My question now is, that is there anything wrong with the proof I have found. I couldn't find anything wrong myself, but the fact that it is shorter than the one in the lecture notes of course raises some suspicion.

    Firstly, I think there is unnecessary assumtions. The function does not need to be continuously differentiable, but it is enough that [tex]\partial_k f[/tex] is continuous in some environment of the [tex]a\in G[/tex].

    Define

    [tex]
    L(h,t):=\frac{f(a+he_i+te_k)-f(a+he_i)-f(a+te_k)+f(a)}{ht}
    [/tex]

    and now

    [tex]
    \partial_k\partial_i f := \lim_{t\to 0}\lim_{h\to 0} L(h,t)
    [/tex]

    and

    [tex]
    \partial_i\partial_k f := \lim_{h\to 0}\lim_{t\to 0} L(h,t)
    [/tex]

    Now we can think that for fixed h, the expression [tex]f(a+he_i+te_k)-f(a+te_k)[/tex] is a function of t, and according to the assumtion that [tex]\partial_k f[/tex] is continuous in an environment of a, and according to the intemediate value theorem, for sufficently small h, there exists [tex]\xi_t[/tex] so that

    [tex]
    \big(f(a+he_i+te_k)-f(a+te_k)\big)-\big(f(a+he_i)-f(a)\big) = \big(\partial_k f(a+he_i+\xi_t e_k) - \partial_k f(a+\xi_t e_k)\big)t
    [/tex]

    and so that [tex]0\leq \xi_t \leq t[/tex] or [tex]t\leq \xi_t\leq 0[/tex]. Then we have

    [tex]
    L(h,t)=\frac{\partial_k f(a+he_i+\xi_t e_k) - \partial_k f(a+\xi_t e_k)}{h}
    [/tex]

    and

    [tex]
    \lim_{h\to 0} L(h,t) = \partial_i \partial_k f(a+\xi_t e_k)
    [/tex]

    This limit exists, because it is a definition of [tex]\partial_i\partial_k f[/tex], and it exists continuously according to the assumtions. The final strike is then

    [tex]
    \partial_k\partial_i f(a) = \lim_{t\to 0}\lim_{h\to 0} L(h,t) = \lim_{t\to 0}\big(\partial_i\partial_k f(a+\xi_t e_k)\big) = \partial_i\partial_k f(a)
    [/tex]

    The proof that is in my lecture notes, uses the intermediate value theorem similarly, but is also uses it in the other direction, and then uses triangle inequality in the end.
     
    Last edited: Jun 3, 2007
  2. jcsd
  3. Jun 3, 2007 #2
    I think I believed in this proof for more than 3 months, and now when I typed it here, at the same instant I understood that [tex]\xi_t[/tex] would depend on h, and this ruins the proof. So typical for me... :cry:
     
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