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Tangent Spaces of Parametrized Sets - McInerney, Defn 3.3.5

  1. Feb 19, 2016 #1
    I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ...

    I am currently focussed on Chapter 3: Advanced Calculus ... and in particular I am studying Section 3.3 Geometric Sets and Subspaces of [itex]T_p ( \mathbb{R}^n )[/itex] ...

    I need help with a basic aspect of Definition 3.3.5 ...

    Definition 3.3.5 reads as follows:

    ?temp_hash=7aeccc5a693e5199bafcc0495a9943fc.png


    In the above definition we find the following:

    " ... ... Here, when we say that a parametrized curve [itex]c \ : \ I \longrightarrow S[/itex] is smooth, we mean that there is a smooth function [itex]\tilde{c} \ : \ I \longrightarrow U[/itex] such that [itex]c = \phi \circ \tilde{c} [/itex] ... ... "

    My question is as follows:

    Why do we need to bother defining [itex]\tilde{c}[/itex] ... the codomain of [itex]c[/itex] is defined as [itex]S[/itex] ... so we surely only need to stipulate that [itex]c[/itex] is continuously differentiable or [itex]C^1[/itex] ... that is the usual definition of 'smooth' so why isn't this enough ...

    ... ... so, my question is then, why do we bother defining [itex]\tilde{c}[/itex] and then go on to consider the composite function [itex]c = \phi \circ \tilde{c}[/itex] ... ?


    Hope someone can help ...

    Peter


    ===========================================================

    To give the context for McInerney's approach to this definition I am providing the introduction to Section 3.3 as follows:


    ?temp_hash=7aeccc5a693e5199bafcc0495a9943fc.png
    ?temp_hash=7aeccc5a693e5199bafcc0495a9943fc.png
     

    Attached Files:

  2. jcsd
  3. Feb 19, 2016 #2

    fresh_42

    Staff: Mentor

    You can imagine the following situation. Think of ##S## being the city in which you live, ##p## your home and ##U## a roadmap of it.
    The formula basically states that you can use a path ##I## on your map and project it to the actually city as well as finding it in reality.

    The point is that ##S## can be of any shape, e.g. curved (or with hills in the example above).
    Smoothness is a local property, i.e. it holds on small neighbourhoods around ##p##, even if for any ##p##. Instead to define what small neighbourhoods in ##S## are, we take a chart ##(U,\phi)## of ##S## in ##ℝ^n## where we already know what smooth means and define it with the help of ##\tilde{c}## and the requirements that ##\tilde{c}## is smooth and ##c=\phi \cdot \tilde{c}##.
     
    Last edited: Feb 19, 2016
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