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Math Help - tangent vector fields as linear partial differential operators

  1. #1
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    tangent vector fields as linear partial differential operators

    The vector field X is defined as X(f) = (∂f/∂x_i)v_i where X(x_i) = v_i. Also the directional derivative of f in the direction of v is defined as df(v) = (∂f/∂x_i)v_i. I am using einstein's summation notation so both X(f) and df(v) are sums.

    From this we can see that X(f) = df(v). The operator X is defined as
    X = (∂/∂x_i)v_i. The source of my confusion comes from the fact that i see df(X) written instead of df(v) where they take ∂/∂x_i to be the basis vectors. I'm confused about why vectors are denoted by ∂/∂x_i which is just an operator? i don't quite understand how an operator X can also act as a vector.

    for these operators to be vectors it must be true that they satisfy the properties defining a vector space for all differentiable functions right?

    When on a manifold do we always work in this basis of operators?
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  2. #2
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    In differential geometry, tangent vectors are defined as derivations. These are linear maps {\cal C}^\infty({\cal M})\rightarrow \mathbb{R} satisfying Leibniz's rule.

    For \mathbb{R}^n, the connection between the usual notion of a tangent vector and derivations is provided by the directional derivative.

    The reason for viewing tangent vectors as derivations is because the usual notion of a tangent vector doesn't generalize to abstract manifolds.
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