Results 1 to 3 of 3

Thread: Prove that the following is "exact"...

  1. #1
    Junior Member
    Dec 2009

    Prove that the following is "exact"...

    Problem: Let f: \Re \to \Re and \omega = f(||x||) (\sum_{i=1}^n{x_{i}dx_{i}}) \in A^1(\Re^n). Assuming f is continuous, prove that \omega is exact.

    Notation notes:

    A^1(\Re^n) is a vector space denoting the set of 1-forms on \Re^n.

    ||x|| is the length of the vector x, so ||x|| = \sqrt{(x_1)^2+(x_2)^2 + ... + (x_n)^2}.

    So this is how I tried to approach the problem. I need to find a function, say G, such that dG = \omega.

    So \omega = f(\sqrt{(x_1)^2+(x_2)^2 + ... + (x_n)^2})  (x_1dx_1 + x_2dx_2 + ... + x_ndx_n)

    \omega = f(\sqrt{(x_1)^2+(x_2)^2 + ... + (x_n)^2})x_1dx_1 + f(\sqrt{(x_1)^2+(x_2)^2 + ... + (x_n)^2})x_2dx_2 + ... + f(\sqrt{(x_1)^2+(x_2)^2 + ... + (x_n)^2})x_ndx_n

    Then I want to find \frac{\partial G}{\partial x_1}, \frac{\partial G}{\partial x_2}, ..., \frac{\partial G}{\partial x_n}, but I don't know how to take the derivative of f since I don't know what f is - and what I'm taking the derivative with respect to each time is within f.

    You can probably tell I'm not familiar with proofs dealing with sum notation. Any tips?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member
    Dec 2009
    I've been thinking about this, and maybe it would be okay to just integrate all of the last line of \omega and leave it in integral form. Instead of actually trying evaluate the integral, perhaps it's enough to just state that the integral exists (since f is continuous)? Then I could just let G equal that integral and have that complete the proof.

    I don't see it as an ideal solution, but could it be sufficient?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Feb 2010
    Basically, yea. Let g(x) := f(\sqrt{x}) so f(||x||) = g(||x||^2), and let G be a primitive for g. Then {\partial\over\partial x_i} (\frac12 G(||x||^2)) = g(||x||^2)x_i=f(||x||)x_i...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: Jun 4th 2011, 12:11 PM
  2. Replies: 2
    Last Post: Apr 24th 2011, 07:01 AM
  3. Replies: 1
    Last Post: Oct 25th 2010, 04:45 AM
  4. Replies: 2
    Last Post: Aug 31st 2010, 12:32 AM
  5. [SOLVED] Help with "Exact Differential Equations"
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: Feb 1st 2010, 06:05 AM

Search Tags

/mathhelpforum @mathhelpforum