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Math Help - Basis of Tangent space

  1. #1
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    Basis of Tangent space

    Hello,

    I try to show this equation:
    Let M be a manifold. For a tangent space T_mM and a coordinate system (U, \phi) at m \in M, we have a basis for [LaTeX ERROR: Convert failed] whereas
    \frac{d}{d\phi_i}(f)=\frac{d(f\circ\phi^{-1})}{dx_i}(\phi(m))

    Now i want to prove the coordinate change, in the case, where we have two different coordinate systems in m. The Formula i want to proove is in the book of warner at page 15 Remark 1.20(c) see below. I hope you can see it in the link. Have you an Idea how to proof the statement? Some Hint perhaps...

    Thanks in advance!

    Foundations of differentiable ... - Google Bücher
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  2. #2
    Super Member Rebesques's Avatar
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    Consider a point p\in M belonging to the intersection of the coordinate charts (U,\phi),\ (V,\psi).
    Let f:M\rightarrow\mathbb{R}^{n} be smooth and denote \frac{\partial}{\partial x_i}(f)=\frac{\partial}{\partial x_i}(f\circ\phi^{-1})(\phi(p)), \frac{\partial}{\partial y_j}(f)=\frac{\partial}{\partial y_j}(f\circ\psi^{-1})(\psi(p)).
    Now, the map y=\psi\circ\phi^{-1}:\phi(U)\rightarrow\psi(V) is a diffeomorphism, and let J(p) be its Jacobian at p. Now, by using the chain rule on \mathbb{R}^n,
    \frac{\partial}{\partial x_i}(f)=\frac{\partial}{\partial x_i}(f\circ\psi^{-1}\circ\psi\circ\phi^{-1})(\phi(p))=\sum_j\frac{\partial}{\partial y_j}(f)\frac{\partial y_j}{\partial x_i}, or \frac{\partial}{\partial x_i}=J(p)\frac{\partial}{\partial y_j}.
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