A vector field on a manifold \(M\) is a section on the fiber section bundle \((TM, \pi, M)\), where \(TM\) denotes the tangent bundle . To be more precise it assigns to each point \(p \in M\) a vector in the tangent space \(T_pM\).
Remark
- A vector field can be seen as a function which maps every point to point to a vector. This can be visualized by mapping a needle to each point.