We assume that the Nabla^2 expression in 3D geometry is quite old and its lifespan is almost expired.
B-matrix chains suggest adding a fourth dimension (mainly time t) woven into the 3D geometric space to form a 4D unit space for two fundamental reasons:
i- The classic expression in 1D,
Nabla^2 Y(x)={Y(x+ h)-2 Y(x)+Y(x-h)}/2 h^2
and similar for 2D and 3D,
is a rough approximation because it only uses 3 geometric points and requires a small interval h.
On the other hand, the same expression suggested by the statistical matrix-B chains is much more precise and uses as many geometric points “free nodes” as necessary with small or large intervals h.
ii- What is quite surprising is that the physical expression of Nabla^2 also turns out to be a differential and integral operator.
Single, double and triple finite integrals can be realized via a modern 4D expression[1].
1-Effective unconventional approach to statistical differentiation and statistical integration, Researchgate, IJISRT journal, Nov 2022.
The laplacian, square of the gradiant operator.you wtite an expression used in
numerical calculations. Ie a second vative in 1d. IIts approximate.
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