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Thread: Gradient represented through fundamental form

  1. #1
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    Gradient represented through fundamental form

    If $\displaystyle E, F, G$ are the coefficients of the first fundamental form in a parametrization $\displaystyle \mathbf{x}:U \subset \mathbb{R}^2 \mapsto S$, then $\displaystyle grad \ f$ on $\displaystyle \mathbf{x}(U)$ is given by:

    $\displaystyle grad \ f =\frac{f_uG-f_vF}{EG-F^2}\mathbf{x}_u +\frac{f_vE-f_uF}{EG-F^2}\mathbf{x}_v$

    Now what I have come to realize is that we can represent $\displaystyle grad \ f$ as $\displaystyle (\mathbf{x}_u, \mathbf{x}_v)
    \begin{pmatrix}
    E & F\\
    F & G
    \end{pmatrix}^{-1} \begin{pmatrix}
    f_u\\
    f_v
    \end{pmatrix}$

    Now we get $\displaystyle \begin{pmatrix}
    E & F\\
    F & G
    \end{pmatrix}^{-1} $ from the second fundamental form.

    where we have
    $\displaystyle N_u = a_{11}\mathbf{x}_u+a_{21}\mathbf{x}_v $

    $\displaystyle N_v = a_{12}\mathbf{x}_u+a_{22}\mathbf{x}_v $

    Now is there a way that I can incorporate these elements to get it to work out? Can I for instance carry on with the following $\displaystyle II_p(a) = \langle D (grad \ f(v)),v\rangle$ where $\displaystyle v$ is a tangent vector on a surface?
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  2. #2
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    What is your question? From my understanding you're trying to express grad(f) using E, F, G and you did it. The result is correct so what are you really asking?
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  3. #3
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    I forget to write down the first part of the question, which defines $\displaystyle df_p= \langle grad \ f(p),v \rangle_p$ where $\displaystyle grad \ f(p) \in T_p(S) \subset \mathbb{R}^3$
    Then from the given definition I'm trying to derive the formula $\displaystyle grad \ f =\frac{f_uG-f_vF}{EG-F^2}\mathbf{x}_u +\frac{f_vE-f_uF}{EG-F^2}\mathbf{x}_v$
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  4. #4
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    since grad(f) is a tangent vector it is expressible by the base vectors:
    grad(f) = a Xu + b Xv, then we need to compute a and b by computing the inner products with Xu and Xv:
    Xu . grad(f) = a Xu . Xu + b Xu . Xv
    df( Xu) = a E + b F
    fu = a E + b F.
    Similarly:
    fv = a F + b G
    So we get a system of linear equations. Solve it we get
    (a, b)' = ( E F; F G)^{-1} (fu, fv)'
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