a vector field

Picture the wind across a weather map: at every point there is an arrow showing which way and how hard the air is moving. That assignment of one arrow to every point is a vector field. On a manifold the arrows are not free-floating; the arrow at each point must lie in that point's own tangent space — it is a velocity you could actually have while standing there.

Precisely, a smooth vector field on a manifold M is a smooth rule that assigns to each point p a tangent vector X(p) in T_p M. In a chart it is written X = a^1 d/dx^1 + ... + a^n d/dx^n, where the components a^i are smooth functions of position. A vector field can be read two ways, both standard: as a field of arrows, and as a first-order differential operator that takes a function f to its directional derivative X(f) along the arrows. Following the arrows generates a flow — solve the differential equation dx/dt = X(x) and you get integral curves, the trajectories a particle would trace if its velocity at every point were the field's arrow there.

Vector fields are the language of dynamics and of geometry's machinery: velocity fields of flows, the gradient of a function (once a metric is chosen), and the building blocks of the Lie bracket that measures how two flows fail to commute. A famous honest limit is the Hairy Ball Theorem: on an even-dimensional sphere every continuous vector field must vanish somewhere — you cannot comb a hairy ball flat without a cowlick. So the topology of the manifold constrains which vector fields can exist, a first hint that shape and analysis are deeply intertwined.