Vector field

A vector field on a smooth manifold is a section of the tangent bundle $TM$. More intuitively it is a map that gives a tangent vector for every point in the manifold.

Definition

For a smooth manifold $M$, a vector field is a continuous map $X:M⟶TM\text{s.t. }X(p)\in T_{p}M\forall p\in M$

Types

If $X$ is a smooth map, we call it a smooth vector field (though these are usually what we study, so often they are just written as vector fields) If $X$ is a vector field that isn’t continuous, we call it a rough vector field.

Notes

Given a vector field, we can split it up into the basis vectors so $X_p=X^i(p)\frac{\partial }{\partial x_i}{|}_p$ Thus, evaluation of a vector field on a function $f\in C^{\infty }$ gives $X(f)=df(X)$

• The space of all vector fields is a vector space

References

@lee2013 was extremely helpful in understanding vector fields.