Velocity Vector Fields on the Euclidean Manifold $\mathbb{E}^{3}$

Abstract

We want to study vector fields

$$\mathbf{w}=\mathbf{w}(P), \quad P \in \mathbb{E}^3$$

on the 3-dimensional Euclidean manifold $\mathbb{E}^{3}$. For example, this concerns velocity vector fields or force fields like

• Newton’s gravitational field w=F _grav,

• Maxwell’s electric field w=E, or

• Maxwell’s magnetic field w=B.

We will frequently use the intuitive picture of the velocity vector field of a fluid. For such vector fields w on $\mathbb{E}^{3}$, one has to distinguish between

• the covariant directional derivative D _v w, and

• the Lie derivative $\mathcal{L}_{\mathbf{v}}\mathbf{w}=D_{\mathbf{v}}\mathbf{w}-D_{\mathbf{w}}\mathbf{v}$. Here, v is the velocity field of the flow of fluid particles on $\mathbb{E}^{3}$.