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synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A vector field is a section of a vector bundle.
More specifically, a vector field is a tangent vector field which is a section of a tangent bundle. Hence this is a function which picks a tangent vector at each point of a manifold, such that this assignment is suitbably suitablydifferentiable.
Equivalently this is a rank -tensor field on
For instance section 1.3 and 1.4 of