A single space is different from the classical concepts of four-dimensional space-time system, global vectors, operations which sometimes do not even require differentiation or of integration. For the same classical formulas, often with enough knowledge of vector algebra and geometry (example). In this app we will show only some of them, gradually supplementing this section.

Let's start with the non-obvious properties of the global velocity vector (GVV), which can be represented as: \[\mathbf{V} = \frac{c}{\gamma} \left\{\pm 1,\, \pm \beta,\, \pm \beta^2,\, \ldots,\, \pm \beta^n \right\} \qquad (1.1)\] where: \(\gamma = 1 / \sqrt{1 - \beta^2}\) (The Lorentz factor), and \(\beta = v/c\). We write this vector in a more convenient form: \[\mathbf{V} = \frac{1}{\gamma} \mathbf{v}, \quad \mathbf{v} = c \left\{\pm 1,\, \pm \beta,\, \pm \beta^2,\, \ldots,\, \pm \beta^n \right\} \qquad (1.2)\] Recall that: \(\mathbf{v}\cdot \mathbf{v} = c^2 \gamma^2\). Then I'll be done with the following equations: \[\mathbf{V}\cdot \mathbf{v} = \int \mathbf{V} \,\Bbb{d}\mathbf{v} = c^2 \gamma \qquad (1.3)\] For the first member of this formula the proof is obvious, and for the integral it would be: \[\int \mathbf{V} \,\Bbb{d}\mathbf{v} = \int \frac{1}{\gamma} \mathbf{v} \,\Bbb{d}\mathbf{v} = \int \frac{1}{2\gamma} \,\Bbb{d}(\mathbf{v}\cdot \mathbf{v}) = \int \frac{1}{2\gamma} \,\Bbb{d} (c^2 \gamma^2) = c^2 \gamma \qquad (1.4)\] Another also non-obvious property GVV looks like this: \[\int \mathbf{v} \,\Bbb{d}\mathbf{V} = 0 \qquad (1.5)\] it is derived by the same method as the formula (1.4). By the way, the properties (1.3) and (1.5) are not only working in a single space decomposition for a Lorentz factor, but also in any other (single).