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2.3 Gradient and Curvature of the Energy Continnum

2.3.1 Definition of Gradient and Curvature

The gradient (slope) and curvature of a line or surface are easy to visualize. For example, consider a line described in two dimensions by the expression, \(y=a_1 + a_2 x + a_3 x^2\). The gradient and curvature of this line is given by the first and second derivatives of this expression, as follows. \begin{align} gradient,\ \frac{\partial y}{\partial x} =&\ a_2 + 2 a_3 x \nonumber \\ curvature,\ \frac{\partial ^2 y}{\partial x^2} =&\ 2 a_3 \nonumber \end{align}

Euclidean Curvature is simply the gradient of the gradient. For a surface, which has two dimensions, the gradient and curvature will both be two dimensional at each point. The two gradients and curvatures are those of two lines intersecting at right angles to each other. A simple example if two dimensional curvature is that of a dome. At its highest point, a perfect dome has equal curvature components in any two orthogonal directions. An imperfect dome has an asymmetric appearance because the curvatures, in two directions at the apex, are not equal.

In three dimensions, there are three components of gradient and curvature. This cannot be visualized because curved volume is impossible to visualize. However, mathematical extrapolation is clear, and they exist even if we cannot 'see' them. In terms of the 'nabla' operator \(\nabla\), we have that \begin{align} gradient,\ = \nabla f(x,y,z) =i\frac{\partial f}{\partial x} + j\frac{\partial f}{\partial y} + k\frac{\partial f}{\partial z} \nonumber \end{align} where \(i,j,k\) are the unit vectors of three dimensional space.

For three dimensional curvature we have \begin{align} curvature\ \nabla^2 f(x,y,z) = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2} \nonumber \end{align}

It is important to appreciate that the gradient is a vector which denotes the direction of the line, surface or volume at each point. As a vector, its value depends on some reference direction. Using an xyz coordinate system in the form of angles, the gradient angle of each point in the system is the same as the coordinate angles. That is, the reference point for gradient is the origin of the coordinate system.

On the other hand, curvature is a scalar quantity independent of the location and orientation of the local coordinate system. In the ideal continuum, curvature at each point is simply, \(3/R\), where \(R\) is the radius of the hyper-sphere.

Finally, in this theory, time is an independent variable, so that the time differential, \(\partial t\), is assumed to have the same value at each point in the continuum and at each instant in time - hence a concept of Relativistic time is not required.

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