Recent content by Alex86

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    Graduate Explicit and Implicit Matrix Notation Question

    Ok so this is a fairly stupid question I'm sure, but I'm not quite clear about the following: Given a Lorentz transformation we require the following to hold: g_{\sigma\rho}\Lambda^{\sigma}_{\mu}\Lambda^{\rho}_{\nu} = g_{\mu\nu} In other notation this is written: \Lambda^{T}g\Lambda...
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    Determining Smoothness Of A Function

    I'm not entirely sure what you mean by the first question... why would it only be defined in the domain of and relative to a single coordinate chart? Could there not be another chart in an open neighbourhood which also includes p? The chart in the question seems very general to me. As for the...
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    Determining Smoothness Of A Function

    I forgot to point out that the function "f(p)" is defined on a manifold where p is a point on the manifold.
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    Determining Smoothness Of A Function

    1. The problem statement: Given a chart \varphi, define a function f by f(p) = x^{k}(p), the k-th coordinate of p (where k is fixed). Is f smooth? 2. Homework Equations : f is smooth (C^{k}) iff F is smooth (C^{k}), where F: \Re^{n} \rightarrow \Re, F = f \circ \varphi^{-1} The...