The Euclidean Manifold $\mathbb{E}^{3}$

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

Chapter 4 in Quantum Field Theory III: Gauge Theory, 2011, pp 321-354 from Springer

Abstract: Abstract Let us use the notation introduced at the beginning of Sect. 1.2 on page 71. Consider the motion $$P=P(t), \qquad t \in \mathbb{R}$$ of a particle (Fig. 4.1). Equivalently, we write $$\mathbf{x}= \mathbf{x}(t), \qquad t\in \mathbb{R}.$$ Here, x(t) denotes the position vector starting at the origin O at time t with the terminal point P(t)=O+x(t). Let E 3(P) denote the space of all the position vectors starting at the point P. This is a real 3-dimensional Hilbert space equipped with the inner product 〈u|w〉 P :=uw and the norm $|\mathbf{u}|_{P}:= \sqrt {\langle \mathbf{u}|\mathbf{u}\rangle_{P}}$ for all u,w∈E 3(P).

Keywords: Tangent Space; Cartesian Coordinate System; Cotangent Bundle; Parallel Transport; Noncommutative Geometry (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/978-3-642-22421-8_5

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