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

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).