Can a vector field be represented as a surface in a vector space?

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The discussion centers on the representation of vector fields as surfaces within vector spaces. It establishes that while surfaces may not inherently qualify as vector spaces, they can be associated with normal vectors at each point. The set of real-valued functions defined on a non-empty set X forms a vector space under pointwise addition and scalar multiplication. Specifically, the functions mapping from \mathbb{R}^2 to \mathbb{R} constitute a vector space, with continuously differentiable functions forming a subspace.

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  • Understanding of vector spaces and their properties
  • Knowledge of real-valued functions and their operations
  • Familiarity with normal vectors in differential geometry
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JanEnClaesen
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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.
 
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For any non-empty set X, the set of real-valued functions with domain X is a real vector space under the operations of pointwise addition (f + g)(x) = f(x) + g(x) and scalar multiplication (af)(x) = af(x).
Thus the set of functions \mathbb{R}^2 \to \mathbb{R} is a vector space under those operations, which has as a subspace the set of continuously differentiable functions \mathbb{R}^2 \to \mathbb{R}.

Is that what you were after?
 
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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