Double and triple integration are mathematical tools used to calculate the area and volume of regions in two and three-dimensional space, respectively. These integrals are extensions of single integration, which is used to calculate the area under a curve.

The calculation of double and triple integrals involves dividing the region of interest into smaller pieces and summing up the contributions from each piece. This process can become very complex, especially for irregularly shaped regions.

However, the use of integration in spacetime is a concept from physics, where time is treated as a fourth dimension in addition to the three spatial dimensions. In this case, integration in spacetime is used to calculate quantities such as the trajectory of a particle or the spacetime interval between two events.

In the curved spacetime of general relativity, integration is over the Riemannian spacetime. The volume element in this curved spacetime is 4-dimensional and, in principle, the general solution requires a quadruple integral (special cases integrate along a contour which is off-topic). The 4D volume element in the curved Riemannian spacetime necessarily includes the Jacobian determinant which transforms the flat 4D Galilean spacetime coordinates used in special relativity to the curved Riemannian spacetime coordinates. For the above transformation of coordinates, it is proven that the Jacobian is equal to sqrt(-g) where g is the determinant of the spacetime metric. Thus, all integrations include the 4D volume element dOmega multiplied by sqrt(-g) which corrects for curvature by the respective changes in the measurement units (for length and time).

The FDM technique (Finite Difference Method) is a numerical method used to solve differential equations. The basic idea of this method is to replace the differential equation with a difference equation, which is then solved using numerical techniques.

The FDM (Finite Difference Method) technique is a powerful tool for solving differential equations, especially for problems that do not have analytical solutions. However, the accuracy and stability of the solution depend on the step size of the grid used and the choice of finite difference approximations. Using the multiplier sqrt(-g) for the volume element changes the grid in correspondence with the curvature of space and time.

For more details, you can read the last section of my preprint Preprint Landau-Lifshitz stress-energy pseudotensor

We assume that there is a complex story deeply hidden behind double and triple integration.

This is the reason why the current definition of double and triple integration is incomplete.

Brief,

Time is part of an inseparable block of space-time and therefore,

geometric space is the other part of the inseparable block of space-time.

In other words, you can perform integration using the x-t space-time unit with wide applicability and excellent speed and accuracy.

On the other hand, the classical mathematical methods of integration using the FDM technique in the geometric Cartesian space alone x can still be applied but only in special cases and it is to be expected that their results are only a limit of success.

1-It is important to understand that mathematics is only a tool to quantitatively describe physical phenomena, it cannot replace physical understanding.

2-It is claimed that mathematics is the language of physics, but the reverse is also true, physics can be the language of mathematics as in the case of numerical integration and the derivation of the normal/Gaussian distribution law.

However, in a revolutionary technique, chains of B matrices are used to solve numerically single, double and triple integrals (Hypercube of volume L^4) as well as the general case of time-dependent partial differential equations with arbitrary Dirichlet BC and arbitrary initial conditions.

It is obvious that the new B matrix technique ignores 3D finite difference elements of small infinitesimal dx, dy and dz as if they never existed.

Dear Ismail Abbas

As an amateur in physics, I am deeply interested in space/time geometries and in their mathematical representations, and I have been reading this high level discussion with great interest.

One point drew a heightened level of interest in your statement number 2:

"2-It is claimed that mathematics is the language of physics, but the reverse is also true, physics can be the language of mathematics as in the case of numerical integration and the derivation of the normal/Gaussian distribution law."

The statement that mathematics is the language of physics always was a given for me, but the idea that the reverse could be true was quite unexpected, since from my analysis, all mathematical equations can only be generalized representations applicable to the whole set of idealized geometric shapes/volumes that emerge from our observations of physically existing "objects" and of the processes in which they are involved, that we then attempt to numerically map as precisely as possible.

From this perspective, the step of "numerical integration" does not come out as "physics becoming the language of mathematics", but becomes very specifically the ultimate level of application of the mathematical language, by means of what I name "first level equations", which are different from the typical idealized generalized equations applicable to the set of idealized geometric concepts in that all of their parameters can be numerically resolved with values extrapolated from physically carried out experiments.

The difference that I found between idealized equations and first level equations of the mathematical language is put in perspective in Section 2.27 of this article:

Article Advancement on the Mechanics of Conceptual Thinking (Republication PI)

If this issue is of interest to the debaters in this discussion, I would definitely appreciate any input about the perspective that even first level equations would still be part of the mathematical language.

Best Regards, André

Ismail Abbas André Michaud Lyudmil Antonov

Let us weigh in pros and cons:

Pros:

1. Complex story: Assuming that there is a complex story behind double and triple integration can inspire curiosity and encourage further exploration of the underlying concepts and principles. It can motivate researchers to delve deeper into the mathematical foundations and develop a more comprehensive understanding of integration.

2. Potential for new insights: By questioning the completeness of current definitions, researchers may be prompted to search for alternative approaches or perspectives that could lead to new insights and advancements in the field. This can potentially open up new avenues of research and stimulate innovation.

Cons:

1. Lack of consensus: Challenging the current definitions of double and triple integration without providing specific alternative definitions or evidence may lead to confusion or disagreement among mathematicians and educators. It is important to have a clear and widely accepted mathematical framework to ensure consistency and effective communication within the field.

2. Potential for misconceptions: Presenting integration as having a hidden complex story without adequate explanation or clarification may lead to misconceptions or misunderstandings among students and learners. It is crucial to provide accurate and accessible explanations of integration concepts to facilitate learning and comprehension.

3. Lack of empirical evidence: Without concrete evidence or specific examples demonstrating the incompleteness of current definitions, the assertion that there is a hidden complex story behind double and triple integration remains speculative. It is important to support claims with empirical evidence or rigorous mathematical analysis to ensure credibility and reliability.

In summary, considering the possibility of a complex story behind double and triple integration can be thought-provoking and may inspire further research. However, it is important to provide clear explanations, evidence, and alternative approaches to foster consensus and understanding within the mathematical/physics community.

Hope it helps: partial credit AI