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Calculating when a covector is zero.

  1. Oct 1, 2011 #1
    1. The problem statement, all variables and given/known data
    Hopefully an easy question. Let M be an n-dimensional smooth manifold and [itex] f: M \to \mathbb R [/itex] a smooth function. Assume the covector field df is given in some coordinate basis as
    [tex] df = \sum_{i=1}^n a_i dx^i [/tex].
    Find all points [itex] p \in M [/itex] such that the [itex] df_p = 0 [/itex]

    3. The attempt at a solution
    I'm pretty sure all that needs to be done is find all point [itex] p \in M [/itex] such that [itex] a_i(p) = 0 [/itex] for all i=1,..n. However, I just want to make sure nothing tricky is going on here that I'm missing. My reasoning is that if there is an [itex] a_i(p) \neq 0 [/itex] then
    [tex] a_i(p) dx^i \left(\left. \frac{\partial }{\partial x^i}\right|_p \right) = a^i(p) \neq 0 [/tex]

    Are there other tricky cases that I might not be seeing? I can imagine that for fixed vector fields there are points where the differential could yield zero, but I take the statement [itex] df_p = 0 [/itex] to mean that the 1-form is identically the zero map at that point.
     
  2. jcsd
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