Differential geometry MOC

Vector field

Let be a -manifold. A -vector field formalizes the idea of assigning a vector smoothly to every point in . The set of all vector fields on is denoted , diff which forms a module over and thus in particular a (usually infinite-dimensional) vector space over . One can also consider a more general Tensor field. See also Smooth field.

Intrinsic manifold

The following characterizations of vector fields and are both useful.

As a section of

A vector field may be characterized as a smooth section of the Tangent bundle , thus

As a derivation on

A -vector field is a derivation on the algebra of smooth (scalar-valued) functions , thus

The evaluation of a vector field at a point is then the derivation

which gives a map .

Equivalence of charactrerizations

proof

When we wish to emphasize the latter view, we write for the derivation corresponding to .

Euclidean space

A vector field is a function assigning a vector to every point in space

Importantly, the domain represents Euclidean space whereas the codomain represents vectors in the physical sense of directional quantities (tangent space). There may also be a time dependence, which is treated separately.

Two special kinds of field are

• Conservative vector field, which is the gradient of a scalar potential
• Incompressible vector field, which is the curl of a vector potential

However, any vector field can be decomposed into conservative and incompressible parts, so that for any field there exists and such that¹

This is due to the Helmholtz theorem, and is consequently called the Helmholtz decomposition.

---

tidy | en | sembr

Footnotes

1. 2013. Introduction to electrodynamics, p. 54 (eqn. 1.105) ↩