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2.3.3 Gradient and Curvature of the real Universe

The gradient of the ideal and real continuum is a three dimensional vector relative to the origin of a xyz coordinate system. It can be characterized at each point as a function of the vector radius, \( \nabla R(x,y,z) \), or in terms of angles, \(\nabla R(\theta_x, \theta_y, \theta_z)\), where \(\nabla\) is the differential operator given by \begin{equation}
i \frac{\partial}{\partial x} +j \frac{\partial}{ \partial y}+k \frac{\partial}{\partial z} \nonumber
\end{equation} or in terms of angles, \begin{equation}
i \frac{\partial}{\partial \theta_x} +j \frac{\partial}{ \partial \theta_y}+k \frac{\partial}{\partial \theta_z} \nonumber
\end{equation}

The gradient of \(R(x,y,z)\) is then \begin{equation} \nabla R(x,y,z) = i \frac{\partial R}{\partial x} +j \frac{\partial R}{ \partial y}+k \frac{\partial R}{\partial z} \nonumber
\end{equation}

\(\nabla R\) is a vector, its three components representing the slope as the deviation from flatness along each of the three axes of the coordinate system.

The second space differential of R is the gradient of the gradient, \(\nabla (\nabla R)\), usually written as \(\nabla ^2 R\). It may be a scalar known as the "scalar Laplacian" or a vector, known as the "vector Laplacian". In this present context, the second space derivative is used as a measure of Euclidean curvature, which we initially assume to be a scalar quantity, (single valued). It is independent of the origin and orientation of the coordinate system used to measure it, and hence it is a property of the real continuum at each point. The three components of \(\nabla ^2 R\) are scalar quantities each representing the curvature in each of the three coordinate directions. It is given as
\begin{equation}
\nabla ^2 R = \frac{\partial ^2 R}{\partial x^2} +\frac{\partial ^2 R}{ \partial y^2}+ \frac{\partial ^2 R}{\partial z^2} \nonumber
\end{equation}

Euclidean curvature at a point on the circumference of a circle is the reciprocal of the radius of the circle. Hence, for the ideal continuum, the Euclidean curvature of each of the three axes is \(1/R\), so that total curvature is \(3/R\).

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