Results 1 to 3 of 3

Thread: Prove solution exists

  1. #1
    Member
    Joined
    Aug 2009
    Posts
    130

    Prove solution exists

    Let $\displaystyle f: \mathbb{R}^{k+n} \rightarrow \mathbb{R}^n $ be of class $\displaystyle C^1 $; suppose that f(a) = 0 and that Df(a) has rank n. Show that if c is a point of $\displaystyle \mathbb{R}^n $ sufficiently close to 0, then the equation f(x) = c has a solution.

    I'm pretty sure that I have to use the implicit function theorem here, but I'm not sure how to proceed. Any ideas?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    3
    Quote Originally Posted by JG89 View Post
    Let $\displaystyle f: \mathbb{R}^{k+n} \rightarrow \mathbb{R}^n $ be of class $\displaystyle C^1 $; suppose that f(a) = 0 and that Df(a) has rank n. Show that if c is a point of $\displaystyle \mathbb{R}^n $ sufficiently close to 0, then the equation f(x) = c has a solution.

    I'm pretty sure that I have to use the implicit function theorem here, but I'm not sure how to proceed. Any ideas?

    As you can see in "statement of the theorem" in http://en.wikipedia.org/wiki/Implicit_function_theorem" , or

    perhaps a little clearer in Implicit Function Theorem -- from Wolfram MathWorld ,

    this is almost the exact wording of the implicit function theorem, so I don't understand what is it

    that you cannot understand here?

    Perhaps it is just a matter of writing $\displaystyle a=(x_1,...,x_n,x_{n+1},...,x_{n+m})\in\mathbb{R}^{ n+m}$ so that it'll look clearer...?

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2009
    Posts
    130
    Let $\displaystyle a = (a_1, a_2) $ The theorem says that since f(a) = 0 and $\displaystyle \frac{\partial f}{\partial y} (a)$ has rank n (this follows from Df(a) having rank n), then there is a neighborhood B of $\displaystyle a_1 $ in $\displaystyle \mathbb{R}^k $ and a $\displaystyle C^r $ map $\displaystyle g: B \rightarrow \mathbb{R}^n $ such that $\displaystyle g(a_1) = a_2 $ and f(x, g(x)) = 0 for all x in B.

    How does this show that if c is a point very close to 0 ( but not quite 0), that there is a solution to f(x) = c?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: Nov 22nd 2011, 05:23 PM
  2. Prove that exists m among ai whose sum is at least m
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: Apr 11th 2011, 06:11 PM
  3. How to prove that a limit exists at 0.
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Mar 27th 2011, 02:22 PM
  4. [SOLVED] How to prove that a limit exists?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: Oct 1st 2010, 06:02 AM
  5. m equations, rank of m, a solution exists?
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Oct 15th 2009, 06:46 PM

Search Tags


/mathhelpforum @mathhelpforum