Scalar and Vector Field definitions I want to ask somethings about scalar and vector fields.
 

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*Formally is the definition of a scalar field simply a function of the type $$f:V\rightarrow F$$ where V is an arbitrary vector space and F is an arbitrary algebraic field.
 

*Formally is the definition of vector field: a vector valued function of the type $$f:V\rightarrow W$$ where V and W are arbitrary vector spaces.
 

*If i am right about 1) and 2) then when drawing/visualising vector fields and scalar fields, is it simply a convention to use the domain of the functions as 'points' and the codomain of the function as 'vector arrows' for example: when drawing/plotting the assignment $$(2,3) \mapsto(5,6)$$ we think about this as 'to the point (2,3), we assign a vector arrow (5,6)' however a perfectly different way of thinking about this is to say 'the point(2,3) goes to the point(5,6)'(which would of course be a coordinate transformation) or 'the vector arrow (2,3) goes to the vector arrow(5,6)' these are different interpretations of what means "(2,3) maps to (5,6)" however from an abstract point of view these ways should mean the same thing i.e $$F(2,3)=(5,6)$$ where $$F:\mathbb{R}^2\rightarrow \mathbb{R}^2$$
 
 
Thank you for any clarifications

 A: It depends on the level of abstraction you're working at. At the level of abstraction commonly used in, say, physics,

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*a scalar field is a function $f : X \to K$ where $K = \mathbb{R}$ or $\mathbb{C}$ and $X$ in full generality may be an arbitrary set but in practice is a manifold. If $X$ is a smooth manifold then $f$ is often but not always required to be smooth.

*a vector field is an assignment, to each point $x \in X$ of a smooth manifold, of a tangent vector $v_x$ in the tangent space $T_x(X)$ at $x$. Formally, this is a section of the tangent bundle $T(X)$. $v_x$ is often but not always required to be smooth. If $X$ is an open subspace of $\mathbb{R}^n$ then the tangent space at any point can be canonically identified with $\mathbb{R}^n$ so we can just work with functions $X \to \mathbb{R}^n$ but in general no such identification is possible globally.

Technically one could imagine calling a function $f : X \to V$ where $V$ is a finite-dimensional vector space a "vector field" but this would be nonstandard. This corresponds to a section of the trivial bundle with fiber $V$ which will differ from the tangent bundle of $X$ in general.
As for your third question, yes, you're technically right, but the point of doing this is to think about a vector field as an object "living on" $X$ in some sense. This is a bit vague but it becomes a lot clearer when considering nontrivial tangent bundles.
