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

Eberhard Zeidler
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Eberhard Zeidler: Max Planck Institute for Mathematics in the Sciences

Chapter 11 in Quantum Field Theory III: Gauge Theory, 2011, pp 659-664 from Springer

Abstract: 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 $\mathbb{E}^{3}$ .

Keywords: Velocity Vector; Parallel Transport; Velocity Vector Field; Riemann Curvature Tensor; Intuitive Picture (search for similar items in EconPapers)
Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-22421-8_12

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DOI: 10.1007/978-3-642-22421-8_12

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