Vector Field

It is a function $f$ mapping one Euclidean space to another Euclidean space. $$ f\colon \mathbf {R} ^{n}\to \mathbf {R} ^{m} $$

It is called a vector field because we can imagine it like that each point in the $\mathbf {R} ^{n}$ is assigned with an vector of $\mathbf {R} ^{m}$. It is a space of vectors, which is called a vector field.