Results 1 to 5 of 5

Math Help - Derivative Trouble

  1. #1
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347

    Derivative Trouble

    Let S\subseteq\mathbb{R}^3 be a regular surface and let p\in S. then (U,F,V) is a local parametrisation of S at p. Let u = F^{-1}(p)
    Define \gamma: (-\delta,\delta) \rightarrow U, with \gamma(0)=u. Then \{\partial_1F(\gamma(t)), \partial_2 F(\gamma(t))\} is a basis of the tangent plane  T_{F(\gamma(t))}S

    Here \partial_i F(\gamma(t)) = DF(u)e_i , i.e the jacobian then applied to e_i
    I am having a lot of trouble trying to show that

     <br />
\frac{d}{dt} \partial_i F(u+t e_j) = \partial_{ji} F(u)<br />
    where \partial_{ji} F(u) = \frac{\partial^2 F}{\partial x_j \partial x_i }.

    I was wondering if anyone could give me some advice on how to think about this, I am very weak in this type of calculus. Thanks very much
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Quote Originally Posted by slevvio View Post
     <br />
\frac{d}{dt} \partial_i F(u+t e_j) = \partial_{ji} F(u)<br />
    This part should read \frac{d}{dt} \partial_i F(u+t e_j)|_{t=0} = \partial_{ji} F(u) otherwise it makes no sense, since they don't have the same domain. To solve this just apply the chain rule to the left side (maybe writing out all the partials of the components of F could help you see this).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    Sorry for my mistake. YEs thanks for your reply but I do not know how to apply the chain rule

    Lets take F_1 to be the first component of F.

    then I guess I am trying to work out \frac{d}{dt} (\frac{\partial F_1}{\partial x_i} \circ \gamma )(t) But how do I do this? I cannot use the normal chain rule since there is an intermediate step in \mathbb{R}^2 , so how do I do this? I have never used a higher dimensional chain rule before. Thanks for any help
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Apr 2009
    From
    México
    Posts
    721
    Okay, so F:U \subset \mathbb{R}^2 \rightarrow \mathbb{R} ^3 so \partial _i F(v)= (\partial _iF_1(v),\partial _iF_2(v), \partial _iF_3(v)) (why?). This is a maping g_i : U\subset \mathbb{R}^2 \rightarrow \mathbb{R}^3. Use of the chain rule gives \frac{d}{dt} g_i(u+te_j)=Dg_i(u+te_j)e_j=(\partial _{ji} F_1(u+te_j),\partial _{ji}F_2(u+te_j),\partial _{ji}F_3(u+te_j))
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347
    Thank you very much, I looked on wikipedia as well for the definition of the chain rule and used the jacobian matrices to work it out rigourously, thanks alot !!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Trouble Finding Derivative
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 17th 2011, 08:29 AM
  2. Derivative trouble # 2
    Posted in the Calculus Forum
    Replies: 9
    Last Post: May 25th 2011, 05:18 PM
  3. Derivative trouble # 1
    Posted in the Calculus Forum
    Replies: 12
    Last Post: May 25th 2011, 08:45 AM
  4. Having trouble with derivative..please help.
    Posted in the Calculus Forum
    Replies: 3
    Last Post: February 16th 2010, 01:31 PM
  5. derivative trouble
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 15th 2007, 06:12 PM

Search Tags


/mathhelpforum @mathhelpforum