Differential geometry MOC

Vector field

Let (𝑀,𝒜) be a 𝐶𝛼-manifold. A 𝐶𝛼-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), diff which forms a module over 𝐶𝛼(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 TM

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

𝔛(𝑀):=Γ𝛼(𝑀,𝑇𝑀).

As a derivation on C^\alpha(M)

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

𝔛(𝑀):=𝔡𝔢𝔯(𝐶𝛼(𝑀)).

The evaluation 𝑣𝑝 of a vector field 𝑣 ∈Vect⁡(𝑀) at a point 𝑝 ∈𝑀 is then the derivation

𝑣𝑝:𝐶𝛼(𝑀)→ℝ𝑓↦𝑣(𝑓)(𝑝)

which gives a map 𝔛(𝑀) →Γ(𝑇𝑀).

Equivalence of charactrerizations

proof

When we wish to emphasize the latter view, we write ∇𝑣 for the derivation corresponding to 𝑣 ∈𝔛(𝑀).

Euclidean space

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

⃗𝐅:𝔼𝑛→ℝ𝑛

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=−⃗∇𝑉+⃗∇×⃗𝚿

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

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Footnotes

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