Asked 26th Jul, 2023

Is there any mathematical tool or other efficient way to transform from component form to vector form of partial differential equation?

In the field of solid mechanics, Navier’s partial differential equation of linear elasticity for material in vector form is:
(λ+G)∇(∇⋅f) + G∇2f = 0, where f = (u, v, w)
The corresponding component form can be evaluated by expanding the ∇ operator and organizing it as follows:
For x-component (u):
(λ+2G)*∂2u/∂x2 + G*(∂2u/∂y2 + ∂2u/∂z2) + (λ+G)*(∂2v/(∂x∂y) + ∂2v/(∂x∂z)) = 0
However, I find it difficult to convert from the component form back to its compact vector form using the combination of divergence, gradient, and Laplacian operators, especially when there are coefficients involved.
Does anyone have any experience with this? Any advice would be appreciated.

All Answers (2)

Doan Cong Dinh
Hanoi University of Science and Technology
Yi-De Liou
University of Texas at Arlington
Dear Doan Cong Dinh,
Thank you for your professional input, and I appreciate this valuable article.
I will try to grasp the concepts of quaternion analysis.

Similar questions and discussions

Does someone have any idea for proving or rejecting the Riemann Hypothesis?
3 answers
  • Shitephen WangShitephen Wang
Does someone have any idea for proving or rejecting the Riemann Hypothesis?
Mathematical proof of Euler product 
ζ(s) = Σ 1/n^s = 1/1^s + 1/2^s + 1/3^s + ... ... + 1/n^s -----[1]
s>=1, ζ(S) divergent
1/2^s ζ(S) = 1/2^s + 1/4^s + 1/6^s + ... ... + 1/2n^s -----[2]
=> (1-1/2^s) ζ(S) = 1 + 1/3^s + 1/5^s + 1/7^s + ... ... -----[3]
1/3^s (1-1/2^s) ζ(S) = 1/3^s + 1/9^s + 1/15^s + 1/21^s + ... ... -----[4]
=> (1-1/3^s) (1-1/s^s) ζ(S) = 1 + 1/5^s + 1/7^s + 1/11^s + 1/13^s + ... ...
... ...
(1-1/5^s)(1-1/3^s)(1-1/2^s) ζ(S) = 1 + 1/7^s + 1/11^s + 1/13^s + ... ...
... ...
∏(1-1/p^s) ζ(S) = 1
p(prime numbers)
=> ζ(S) = ∏(1-1/p^s)^(-1) = 1/1^s + 1/2^s + 1/3^s + ... ... + 1/n^s
s=1, ζ(S) divergent
So prime numbers are infinitas
Euler product is only meaningful when s>=1; the output will diverge if s<1
Riemann used analytic continuation to make the ζ(s) function meaningful on the complex plane 
when s<1.
ζ(s) = Σ 1/n^s = 1/1^s + 1/2^s + 1/3^s + ... ... + 1/n^s
analytic continuation -∞ <= s <= ∞
=> ζ(s) = Γ(1-s)/2𝝅i * ∫{-∞}^[∞] [(-Z)^s / (e^z - 1)] dZ/Z = Reimann ζ function (s)
Reimann ζ(s) = Σ 1/n^s, s∈C, n∈N
where s is any complex number, while n is any natural number.
Γ(s)= (s-1)!
ζ(s) = 2Γ(1-s)/(2𝝅)^(s-1) * sin (𝝅s/2) ζ(1-s)
when s = -2, -4, -6 ... ...
ζ(s) = 0 (trivial zeros)
Reimann hypothesis (1859)
all nontrivial zeros of ζ(s) function, their output of complex number with real part 1/2.
35 replies
  • Raphael NeelamkavilRaphael Neelamkavil
Beyond the Two Millennia
Raphael Neelamkavil, Ph. D., Dr. phil.
§§-- Without beings in existence (To Be), there is no science “on them”. Existence is not vacuous. Non-vacuous existents are in Extension (Existence-Category 1), i.e., they have parts, they are composed.
§§-- Both parts and wholes can interact. They cause impacts on a finite number of other existents and on themselves: Change (Existence-Category 2). Change involves motion, but is not motion. Parallel to these two physical-ontological Categories, no other characteristic is thinkable. Hence, Extension and Change ꞊ Exhaustive implications of To Be of Reality-in-total the highest natural kind.
§§-- If Extended+Changing (with parts and with impact formation) entities exist, this is causal existence. Every existent is such. Hence: Universal Causality...! Extension, Change, and Causality are pre-scientific Laws. Now, no Quantum Physicist can tell us that some (observable) processes are causal and the others (partial observables [unobservables] and non-observables) are merely statistically causal or non-causal…!
§§-- Smaller natural kinds (ordered and/or organized parts of Reality-in-total) also have characteristics. These are ontological universals (modes of being of processes). They are primarily in the natural kinds, and only thus in the token enities in the natural kind.
§§-- Space ꞊ measure of extension. Time ꞊ measure of change. These are epistemic concepts. Epistemic space-time cannot curve as physicists make us believe. Extension-Change-wise existent matter-energy conglomerations curve.
Centuries of violent and extremist discussions have taken place as to a Yes or No or Yes-and-No to causality in existent beings, namely, Reality-in-total. In the fray have been mainly philosophy, and only then physics. This state has changed after the genesis of quantum physics. In the above, I have “proved” in a very simple manner that Universal Causality is a pre-scientific Law.
The purely epistemic version of causality can only be a sort of concept and not be that of what happens in the world. It gets formulated due to the sense-related, conceptual, and logical conclusion towards a correlation of some sort between two or more events, but without recourse to the events’ existence.
Any further justification of the epistemological conclusion of causality without involving the purely physical-ontological aspect of existence of the event at question in total and its antecedent and consequent part-events may even be taken as an explanation of the experience of correlation. Historical examples abound, and Hume’s is the most famous example.
But this is not the case if the purely physical-ontological aspect of existence of the event at question in total and its antecedent and consequent part-events may be accepted as the conditio sine qua non of the sensation, conceptualization, and logical argument. Hence the fully physical-ontological status of causality.
Traditionally, causality is the relation between the antecedent and the consequent part-events of the one event at issue. And causation is the act of a cause-event in effecting an effect-event. This is the age-old manner of conceiving the ontology of causality. The former, the epistemic and the explanatory, have been the trend during most of the 20thcentury history of philosophical and physical-philosophical inquiry on causality.
But what I have proposed in the various parts of my five published books is a whole new manner of theorizing Universal Causality. I hope to finally suggest that this is also a game-changer in the history of the concept of causality.
27 replies
  • Raphael NeelamkavilRaphael Neelamkavil
Raphael Neelamkavil, Ph.D., Dr. phil.
The Background: The ultimate physical and cosmological significance of the Categories of Extension (“being extended / having parts” while in existence) and Change (“extended existents causing impacts on others and also on themselves”) must be seen in the context of warding off quantum-physical, cosmological, statistical, and other sorts of inexplicable and bizarre existence-related aberrations resulting from theories like those of (1) parallel universes, (2) extra dimensions, (3) vacuous universes, (4) total mutual disconnection of universes, (5) infinite number of positive-content physical universes taking origin like extra-fitted balloons from “technically / mathematically zero-valued” quantum vacua or quantum-vacuum universes without any iota of causal agency (because quantum vacua are merely of near-zero zero statistical expectations), (6) the presumed existence of space, time, and spacetime like physical things in mathematical fields, (7) the theoretical writing-off of time alone as unreal and unnecessary, etc.
This sort of aberrations renders some theories and their related concepts into theories about absolutely non-existent objects (in some analytic-logical philosophies, called also as “counterfactual possible worlds”) and into substitute theoretical entities that serve only to explain procedures and not to explain existent processes. These serve for physicists and cosmologists to temporarily save their face by use of irrational adherence to methods of maintenance of mere uncertainties in mathematical physics.
The Anomany in Theoretical Physics: I mentioned these above in order to speak of the anomaly in advanced mathematical physics. This curse is the confusion between (1) physical existents, (2) non-existent theoretical constructs, (3) theories representing small or large theoretical processes required only for theory, and (4) the lack of criteria of creating theories for describing existent processes with recourse to vacuous, non-existent, virtual objects and processes, but without turning these objects and processes into existent objects lacking the criteria of existence.
Positing ad hoc explanatory theories to clarify certain theoretical inaptitudes of notions or deviations in arguments is assuredly necessary for the progress of science. But these are sooner to be overwhelmed (not to be substituted) by more adequate and existentially non-aberrational unobservables and/or theoretical terms. As of now, physics, astrophysics, and cosmology are full of theoretical entities that cannot ever be proved to be existent unobservables. This is the curse of physics today – a graver problem today than previously.
New Grounded Physical-Ontological Categories behind Physics
26 replies
  • Raphael NeelamkavilRaphael Neelamkavil
1. Grounded Physical-Ontological Categories behind Physics
Grounding can be of various levels and grades. I speak of grounding all sorts of concepts, procedure principles, procedure methods, and theories in any system of thought and science. It is unnecessary in this context to discuss the grounding of highly derivative concepts that occur much later in theories than those that appear while founding them with best-grounded foundations. I go directly to the case of what should be called the most Categorial concepts behind physics, on which physics is grounded.
These Categorial concepts cannot be merely from within physics but should be directly related to and facilitating physics in as many of its aspects as possible. The success of foundational Categories consists in that they serve to ground as many aspects as possible of the particular science or system. Concepts strictly and exclusively physical or generally scientific cannot be as useful as notions from beyond in order to serve as Categories. Evidently, this is because no scientific discipline or system can be grounded on itself and hence on its own concepts. This is clearly also part of the epistemological and ontological implications of the work of Godel.
Grounded ontological Categories are such that they are inevitably and exhaustively grounded in the To Be of Reality-in-total as the only exhaustive implications of To Be. All other Categories, as far as possible, must be derivative of the most primary Categories. The more the number of Categories within the Categorial system that do not derive from the primary Categories the worse for the self-evidence of the science or system within it.
Grounding is exhaustive in the sense that the Categories that ground all physics need nothing else to be a concept than the To Be of Reality-in-total. To Be is the source of the Categories. It happens to be that there are two such Categories that are inevitably and exhaustively grounded. I call them Extension and Change. Clarifications of their meaning, ontological significance, and epistemological and physical implications and follow.
As I said, preferably grounding must be on the surest notion of all, which is existence. I prefer to term it To Be. As far as thought, feeling, and sensation are concerned, To Be is a notion in al of them. But principally To Be must belong to the whole of Reality, and not to a few things. If anything and/or all processes of Reality are existent, then what exist are the parts of existent Reality. The first minimum guarantee hereof should be that existence should be non-vacuous. Non-vacuous signifies that each possesses or contains whatever is possible within its existence in the given measurementally spatio-temporal context (which, as shall soon be clear, belong ontologically to the Extension-Change-wise existence of things).
3. Definitions of Universals, Extension-Change, Causality, and Unit Process
Even the minimum realism in thought, feeling, and sensation has for its principal conditions (1) the ontological primacy of universal qualities / natures that belong to groups of entities (processes), where the groups are also called natural kinds in the analytic tradition, and then (2) the ultimate simplicity and indivisibility of the universal characteristics that pertain to all existents. Contrary to the infinite divisibility of existent matter-energy, universals as the characteristics of existent matter-energy conglomerations (of togethernesses of unit Processes) are ontologically ideal universals, and hence indivisible. These universals are ideal not because of our idealisation of the characteristics, but instead because they are the general characteristics of the natural kinds to which each existent belongs. Thus, it is important to keep in mind that ontological universals are not our idealizations.
The properties of things are built out of these simple ontological universals in the natural kinds. The vague reflections of simple ontological universals within our minds are conceptually connotative universals, which are conceptual ideals. And their linguistic reflections in minds and all kinds of symbolic instruments are denotative universals.
Connotative and denotative universals are epistemological universals, formed epistemically from the little contact that minds have with the phenomena (“showings-themselves”) from some layers of processual objects from out there. The properties of existent processual things (matter-energy particulars) are vaguely reflected in minds and languages through the connotative and denotative instrumentalization of concepts in order to reflect the things via phenomena in terms of the data created by minds out of them. Any theory that permits ontological primacy to epistemological universals is one of a range of theories yielding primacy to the perceiving mind over the perceived objects. This is anathema in any scientific or philosophical science, because things are not vacua.
Non-vacuous existence implies that existents are extended. This is one of the most important characteristics of existents. Extension implies having parts, compositionality. Any extended existent’s parts impart impact to some others. This is Change. Only extended existents can exert impacts on any other. As a result, the object that exerts impact gets in itself some impact within, which is nothing but the proof that an impact by one extended part implies movements and impact formation by its parts too, as a result of the overall impact formation in question which contains the inner parts’ impact formation within. The latter need not always have its effects merely within the parts but instead also outwards.
Extension and Change are the highest, deepest, and most general characteristics of all existents. Interestingly, existence in Extension-Change is itself the process that we have so far named causation. Hence, anything non-vacuously existent has Extension and Change not separately but together. This is the meaning of Universal Causality. Physics cannot dispense with this pre-scientific universal Law. No more shall quantum physicists or scientists from other disciplines tell us that quantum physics has some sort of non-causality within! Any causal unit of existents in which the causal part and the effect part may be termed a process. Processuality is yet another important characteristic of existents, but we formulate it as Process, which represents the matter-energy units that there can be.
By this have clearly been set up three physical-ontological Categories of physics: Extension, Change, Causality, and Process. Space and time are merely epistemic categories. They cannot characterize existent processes. Ontological universals, as the characteristics of existent matter-energy conglomerations, are of togethernesses of unit Processes. Ontological universals are therefore ontologically ideal universals belonging (pertaining) to some natural kinds. The Categories as ontological universals belong to Reality-in-total, and not merely some natural kinds.
How Does Physics Know? The Epistemology Presupposed by Physics and Other Sciences
69 replies
  • Raphael NeelamkavilRaphael Neelamkavil
Raphael Neelamkavil, Ph.D., Dr. phil.
1. The Logic of Physics (See the previous discussion's lead-text)
2. The Epistemology behind Physics
The whole of logic, epistemology, ontology, etc. are not the exclusive property of physics, or of any other particular science, or of all the sciences together. Each of them may apply the various general logical, epistemological, and ontological principles in ways suitable to their disciplines, but cannot claim that theirs is the genuine or the possibly best logic, epistemology, ontology, etc.
There is yet another manner, beyond the sciences, wherein (1) the object range and viewpoint range become the broadest possible in epistemology, and (2) the epistemological manner in which the two are connected becomes satisfactory enough to explain both the aspects and the procedures involved between them. This is a philosophical version of epistemology. Even this manner is not complete without including the various logics, epistemologies, and ontologies of the particular sciences.
Before pointing out the special manner in which physics could use the more general aspects of epistemology in itself, let me mention a general trend in science, especially physics. I have seen many students of physics and mathematics mistaking the logical ways in which they do experiments and theories as the same as the conceptual foundations of physics and mathematics.
They do not even think of the epistemology of physics. The clear reason for this is that their epistemology is a crude correspondence theory of truth, and this is outdated. Take any of the best physicists, and we can see in their works the underlying undefined epistemology being closer to the correspondence theory of truth than anything else. I would like to suggest in the following a clear spine of epistemological rudiments for physics.
The pragmatism and scientism at the foundations of practical physics does not accept anything other than the correspondence theory as prescriptive of all the truths of science. Of course, the amount of finality achieved in truths will be the measure of tenability of their truth-probability. But this is to be reserved to the most general truths derivable from any science or philosophy. Low-level truths are much beyond the purview of correspondence between the objectual and the theoretical. Unaware of these facts, most physicists take the difference lightly.
It is a pity that the students of the sciences and also philosophy students with scientistic orientations even think of their ways of permitting truth correspondence to all their truths as the sole possession of scientists, which they suppose are being usurped from philosophy in the course of the past centuries in such a way that philosophy will have ever less reason to exist, or no more reason to exist. Imaginably, in this pride they are encouraged by their presumption of possession of the scientific temper in an exceptional manner.
More evidently, there were and there are physicists holding that their use of logic, epistemology, ontology, etc. is final and that all other details being done by other sciences, especially by philosophy, are a mere waste of time. If you want me to give an example, I suggest that you watch some of the YouTube interviews with Stephen Hawking, where he declares philosophy as a waste of time, or as an unscientific affair. The same sort of claim is to be seen being made by many mathematicians: that logic is a by-product of mathematics, and that philosophers are falsely proud of having logic as their methodology.
The reason why the whole of logic does not belong to the sciences is that the viewpoint from which sensation, thought, and feeling may be exercised in the broadest possible manner is not exhausted even by totaling all the object ranges of all the sciences. Each of them does logic in a manner limited by its object range. How then can their logic be the best possible? There is one and only one general science of which the viewpoint is the broadest. It is that science in which the viewpoint is that of the direct implications of the To Be of Reality-in-total.
Against this backdrop, although the following definition might seem queer for many physicists, mathematicians, and other scientists, there are reasons why I define here epistemology for use in physics. The following definition itself will clarify the reasons:
The epistemology behind physics is (1) the science of justifications (2) for the systemic fact, the systemic manner of achieving, the enhancement of the systemic manner of achieving, and the foundations of systems (3) of rationally derivable and explicable theoretical consequences of human efforts (4) to grasp the connection between physically existent reality and their pertinent realities of all sorts (5) in an asymptotic approach of truth-correspondence from the procedures of knowing (in terms of the pertinent realities of existent realities) onto the physically existent processes of reality, (6) in a spirally broadening and deepening manner of truth probability, (7) which serves to achieve ever better approximations of the epistemological ideal of knowing, namely, Reality-in-general, (8) starting from reality-in-particular, and (9) by use of the highest theoretical generalities pertaining to Reality-in-total and its parts, namely, reality-in-particular.
The epistemology of physics does not take the viewpoint of the To Be of Reality-in-total. But it must obey the primary implications of To Be and the viewpoint of the To Be of Reality-in-total. What these implications are, will be treated below, under “3. The Ontology of Physics”. Epistemology in philosophy may be slightly more general than the epistemology of physics, in the sense that philosophy takes the viewpoint of all physical processes that exist and attempt to view every reality from that viewpoint alone. If not, philosophy has no justification for existence.
Naturally, the epistemology of the sciences will not be so general as that of philosophy. But obedience to it is better for the epistemology of physics; and the advantages of such obedience will be seen in the results of such physics and such sciences.
The epistemology of physics, therefore, will attempt to theorize, know, and predict all that exist, but from the viewpoint exclusively of experimentally / empirically verifiable methods based on what is directly or indirectly before us, namely, the physical processes at our reach. The epistemology of systematically and systemically (i.e., systematically of systems of systems … ad libitum) moving in the use of logic from the given existent physical processes to the details of the not immediately given but ever more minute or ever more distant physical existents is the epistemology of physics. The above definition would, in my opinion, be sufficient to cover as broad and minute procedures as possible in physics. Time has come to appropriate it in physics, lest much advantage be lost for too long.
Not that philosophy does not trust this approach of physics. But philosophy looks for the Categorial presuppositions of existence behind all that is verifiable or verified empirically and empirical-theoretically. These presuppositions are the starting points and guiding principles of philosophy. There is a stark difference between a methodology of this kind and the methodology of basing everything on the truths derived from empirical and empirical-theoretical research. Now from this viewpoint you may judge the following suggestions and determine whether the epistemology of doing physical science is as broad as that of philosophizing.
Every moment, our body-brain nexus is continuously but finitely in contact with itself and with a finite extent of the environment, more or less simultaneously, but in differing intensities, no matter however elementary. The primary mode of this is through sensation, using all available and necessary aspects of it as the case may be. Thought and feeling are possible only in continuity with sensation, and never without it.
But one special characteristic of the human brain differentiating it from others is that sensation, feeling, and thought can very consciously induct into, and consequently deduce from the presuppositions of, all that exist – no matter whether they are a finite environment or infinite – and all these solely from the finite experience from the finite environment at hand. This seems to be absent in less human living beings.
Moreover, the second, but more forgotten, characteristic of the human brain differentiating it from others is that sensation, thought, and feeling are affective, tending to itself and to others, in the broadest sense of the term ‘affective’. It is the manner in which every human being tends in his/her sensation, feeling, and thought. Hence, all processes of knowing will be coloured by affection.
The manner and then the so-constructed broader background in which sensation, feeling, and thought take place is affection, which we term also love in a very general sense. Sensation, feeling, and thought are the three interconnected modes of tending of the body-brain to itself and to the environment, tend always to connect itself with the environment.
But here too the important differentiating characteristic in human body-brains is their capacity to tend to the environment beyond the immediate environments, and further beyond them, etc. ad libitum. There is nothing wrong in theoretically considering that there is the tendency in humans to converting this sort of ad libitum to ad infinitum, irrespective of whether these environments can really go ever broader at infinity in the content of matter-energy within Reality-in-total. Infinity is another term here for generalizing.
Reality consists of existent reality and realities that pertain to existent realities in their groups. Existent realities are clear enough to understand. Realities pertinent to existent realities are never to be taken as belonging to just one existent reality. They are always those generalities that belong to many existent realities in their respective natural kind. These generalities are what I call ontological universals.
All generalizations tend beyond onto the infinite perfection of the essential aspects of the concepts pertaining to the object-range. Not that the object-range must be infinite. Instead, the tending presumes an infinitization due to the idealization involved in generalizations. This is a kind of infinitization that does not need an infinite Reality-in-total in existence. All the concepts that a human being can use are based in the infinitization of the essential aspects of the concepts in their ideality. But behind these mental ideals there are the ideals, namely, the ontological universals pertaining to the groups (natural kinds) of processual entities in the environment. These are the ideals in the things and are not in us. These too are idealizations at the realm of the natural kinds that form part of Reality-in-total.
Without loving in the sense of tending to, as human do, to the inner and outer environments in their generalities there is no sensation, feeling, and thought. The tending to need not be due to the love of the objects but due to the love of something that pertains to them or to the ontologically universal ideals pertaining to the objects. From this it is clear that the relation between the processual objects and the sensing-feeling-knowing mind is set by the ontological universals in the natural kinds of existent physical processes.
At the part of the mind there should be idealized universals of conceptual quality, because the ontological universals in natural kinds cannot directly enter and form concepts. This shows that the conceptual universals (called connotative universals) are the mental reflections of ontological universals that are in the natural kinds. In short, behind the epistemology of sensation, feeling, and thought there are the ontology and epistemology of loving in the sense of tending to, due to the otherness implied between oneself and the environment.
There may be philosophers and scientists who do not like the idea of love. I say, this is due to the many psychology-related prejudices prevalent in their minds. We need to ask ourselves what the major mode of exercitation of any activity in human beings, and none can doubt the role of love in epistemology. The physical foundations of love too are commonly to be shared with the foundations of other aspects of physical existence.
Such tending by the person is mediated within the person by the connotative universals. Their expression is always in terms of symbols in various languages. These are called denotative universals. Connotative universals get concatenated in the mind in relation to their respective brain elements and form thoughts and feelings. Their expression in language is by the concatenation of denotative universals and get formulated in languages as theories and their parts.
To put in gist the latter part of “2. The Epistemology of Physics”, I suggest that the ontological, connotative, and denotative universals and the love of human agents to these and the very existent processual entities are what facilitate knowledge. The psychological question as to what happens when one has no love does not have any consequence here, because psychology differentiates between love and non-love in terms of certain presumed expressions of love and non-love.
In the case of the natural course of life of humans, the choice is not between love and non-love, but instead, between increasing or decreasing love. We do not speak here of loving other human beings as a matter of ethical action. Instead, the point is that of the natural love that humans have for everything including for sensing, feeling, knowing, etc.
One might wonder here why I did not discuss mathematics as an epistemologically valid tool of physics and other sciences. I have already dealt with this aspect in many other discussion texts in ResearchGate, and hence do not expatiate on it here.
3. The Ontology behind Physics (soon to be given as a separate RG discussion session)

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