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Vector of two variables

  1. May 15, 2014 #1
    For example the surface (x,y,x²+y²), can for example surfaces be considered as one abstract 'vector' in some abstract 'vector'-space? The ' ' because surfaces might not be a vector space. For surfaces we can exceptionally define normal vectors at every point.
     
  2. jcsd
  3. May 15, 2014 #2

    pasmith

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    For any non-empty set [itex]X[/itex], the set of real-valued functions with domain [itex]X[/itex] is a real vector space under the operations of pointwise addition [tex](f + g)(x) = f(x) + g(x)[/tex] and scalar multiplication [tex](af)(x) = af(x).[/tex]
    Thus the set of functions [itex]\mathbb{R}^2 \to \mathbb{R}[/itex] is a vector space under those operations, which has as a subspace the set of continuously differentiable functions [itex]\mathbb{R}^2 \to \mathbb{R}[/itex].

    Is that what you were after?
     
  4. May 15, 2014 #3
    Essentially, I was wondering whether a vector field could be considered as a surface or something more unitary in general. For example (x,y,(x²+y²)^(0.5)) is a cone and a (partial) vector field in space.
     
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