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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.