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.
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