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

[JanEnClaesen]

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.

 

[pasmith]

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?

 

[JanEnClaesen]

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.