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Math Help - [SOLVED] Second Fundamental Form

  1. #1
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    [SOLVED] Second Fundamental Form

    Hi there. I'm trying to understand the Second Fundamental Form but I'm stuck on something. Let:

    \textbf{X}=(x(u^1,u^2),y(u^1,u^2),z(u^1,u^2)) is a parameterised form of a surface.

    \textbf{X}_i=\dfrac{\partial \textbf{X}}{\partial u^i}

    \textbf{X}_{i j}=\dfrac{\partial \textbf{X}^2}{\partial u^i \partial u^j}

    It's written \textbf{X}_{i j}=\Gamma_{ij}^{r}\textbf{X}_r +L_{ij}\textbf{U} where \textbf{U} is the normal vector to the surface.

    I don't understand what is L and \Gamma
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  2. #2
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    Quote Originally Posted by fobos3 View Post
    Hi there. I'm trying to understand the Second Fundamental Form but I'm stuck on something. Let:

    \textbf{X}=(x(u^1,u^2),y(u^1,u^2),z(u^1,u^2)) is a parameterised form of a surface.

    \textbf{X}_i=\dfrac{\partial \textbf{X}}{\partial u^i}

    \textbf{X}_{i j}=\dfrac{\partial \textbf{X}^2}{\partial u^i \partial u^j}

    It's written \textbf{X}_{i j}=\Gamma_{ij}^{r}\textbf{X}_r +L_{ij}\textbf{U} where \textbf{U} is the normal vector to the surface.

    I don't understand what is L and \Gamma
    This is just the decomposition of \textbf{X}_{ij} in a basis. Indeed, (\textbf{X}_1,\textbf{X}_2) is a basis of the tangent plane, while \textbf{U} is orthogonal to that plane, so that (\textbf{X}_1,\textbf{X}_2,\textbf{U}) is a basis of \mathbb{R}^3 (which depends on (u_1,u_2)). Then for instance \Gamma_{ij}^1 is the \textbf{X}_1-component of \textbf{X}_{i j}.
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  3. #3
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    Quote Originally Posted by Laurent View Post
    This is just the decomposition of \textbf{X}_{ij} in a basis. Indeed, (\textbf{X}_1,\textbf{X}_2) is a basis of the tangent plane, while \textbf{U} is orthogonal to that plane, so that (\textbf{X}_1,\textbf{X}_2,\textbf{U}) is a basis of \mathbb{R}^3 (which depends on (u_1,u_2)). Then for instance \Gamma_{ij}^1 is the \textbf{X}_1-component of \textbf{X}_{i j}.
    Never mind. It's explained 3 chapters ahead. It's the Christoffel symbol.
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  4. #4
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    Quote Originally Posted by fobos3 View Post
    Never mind.
    Thanks for your consideration...
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