Vector field

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

Intrinsic manifold

The following characterizations of vector fields and $𝔛 (𝑀)$ are both useful.

As a section of

𝑇 𝑀

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

𝔛 (𝑀) : = Γ 𝛼 (𝑀, 𝑇 𝑀) .

As a derivation on

𝐶 𝛼 (𝑀)

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

𝔛 (𝑀) : = 𝔡 𝔢 𝔯 (𝐶 𝛼 (𝑀)) .

The evaluation $𝑣 𝑝$ of a vector field $𝑣 \in V e c t (𝑀)$ at a point $𝑝 \in 𝑀$ is then the derivation

𝑣 𝑝 : 𝐶 𝛼 (𝑀) \to ℝ 𝑓 \mapsto 𝑣 (𝑓) (𝑝)

which gives a map $𝔛 (𝑀) \to Γ (𝑇 𝑀)$ .

Equivalence of charactrerizations

#missing/proof

When we wish to emphasize the latter view, we write $\nabla 𝑣$ for the derivation corresponding to $𝑣 \in 𝔛 (𝑀)$ .

Euclidean space

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

⃗ 𝐅 : 𝔼 𝑛 \to ℝ 𝑛

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 $𝑉$ and $Ψ$ such that¹

⃗ 𝐅 = - ⃗ \nabla 𝑉 + ⃗ \nabla \times ⃗ 𝚿

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

#state/tidy | #lang/en | #SemBr

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

Vector field

Intrinsic manifold

Euclidean space

Footnotes