In vector calculus, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space.
Given a subset $S$ in ${\bf R}^n$, a vector field is represented by a vector-valued function $V: S \rightarrow {\bf R}^n$ in standard Cartesian coordinates $(x_1, \cdots , x_n)$. If each component of $V$ is continuous, then $V$ is a continuous vector field, and more generally $V$ is a $C^k$ vector field if each component of $V$ is $k$ times continuously differentiable.
Given two $C^k$-vector fields $V,\ W$ defined on $S$ and a real valued $C^k$-function $f$ defined on $S$, the two operations scalar multiplication and vector addition $$(fV)(p) := f(p)V(p) $$ $$(V+W)(p) := V(p) + W(p)$$ define the module of $C^k$-vector fields over the ring of $C^k$-functions.
