vector field

Not to be confused: vector space vs. vector field

A vector space is an algebraic structure - a set with operations (addition and scalar multiplication) satisfying certain axioms.
A vector field, on the other hand, is a geometric object - a function that assigns vectors to points in space.
You could say a vector field uses vector spaces: at each point in space, the vector it assigns comes from some vector space (usually ${\mathbb{R}}^n$), it is about how vectors vary from point to point in space.

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Vector Field

A vector field (physics) on a region $U\subseteq {\mathbb{R}}^n$ is a function $F:U\to {\mathbb{R}}^n$ that assigns a vector $F(x)$ to each point $x\in U$. We write:

$$F(x)=\left(\begin{array}{c}{F}_1(x)\\ ⋮\\ F_n(x)\end{array}\right)$$

where each component $F_i:U\to \mathbb{R}$ is a scalar-valued function.

A classic example is the gravitational field, where each point in space is assigned a vector pointing toward the center of mass, with magnitude decreasing as $1/r^2$. In ${\mathbb{R}}^2$, this takes the form:

$$F(x,y)=-\frac{1}{(x^2+y^2)}\left(\begin{array}{c}x\\ y\end{array}\right)$$

The negative sign indicates the force is attractive, pointing toward the origin.

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