structural:reformatting, structural:expansion
Generalizations

Replacing vectors by p -vectors (p th exterior power of vectors) yields p -vector fields; taking the dual space and exterior powers yields differential k -forms, and combining these yields general tensor fields.

Algebraically, vector fields can be characterized as derivations of the algebra of smooth functions on the manifold, which leads to defining a vector field on a commutative algebra as a derivation on the algebra, which is developed in the theory of differential calculus over commutative algebras.