A New Formulation of Maxwell’s Equations
Round 1
Reviewer 1 Report
Manuscript number: symmetry-1188751 Symmetry MDPI (type: Viewpoint)
TITLE: A new formulation of Maxwell’s equations
AUTHORS: Simona Fialová, František Pochylý
The first review of the manuscript
Overall description of the manuscript
In the manuscript entitled “A new formulation of Maxwell’s equations” by Simona Fialová and František Pochylý, the authors present new forms of Maxwell's equations in vector and scalar variants, which are based on the use of Gauss’s law for magnetic induction and electrical induction. The authors derives the equations in both differential and integral form. The most important results are derived in Section 3 based on Gauss's law for magnetism and Maxwell–Faraday equation (Faraday's law of induction). Next, in Section 4 authors considered all four Maxwell’s equations for a non-conductive environment. Sections 5 and 6 is devoted to “scalar version” of Maxwell’s equation presented in Section 3 and 4, respectively. Section 7 contains application to non-conductive magnetic liquid. The manuscript contains some mathematical background and formal proofs. The topic is interesting for a limited audience, but the paper can attract some attention in the community, particularly for “engineer” and “technical sciences” communities . However, in my opinion, it should be rewritten in such manner to be more specific and understandable for non-specialist reader.
The most important results of the paper is that “unsteady states of the magnetic induction intensity are generated on the borders of the area and vice versa”. This could result in a tool that can be used in the numerical finite volume method and optimization. The obtained equations will also allow the qualitative analysis of the influence of boundary conditions.
The paper fits the journal scope. The English language in the manuscript is good, however some language errors should be corrected before the final publication. The paper has 9 pages and includes 12 references (equivalent to 1/2 pages), and 1 figures (about 1/2 of a page) – effectively about 8 pages of the main text. The diagram (figure) is clear and essential and its captions are informative. The title clearly and concisely conveys the topic of the article. The abstract quite well describes the content of the manuscript. The results look mathematically correct. The discussion and conclusions are supported by the results.
In my opinion the manuscript cannot be published in the present form, but it might be published after some revisions. I believe that suggested improvements can increase the readability and quality of the paper. My comments are also associated to Section 3 of the manuscript, which is the most important section of the paper, because all the results presented are based on the equations presented there.
Some specific comments to the authors:
1) page 2, line 29: the authors wrote: “Because we know that the equation for symmetry (2.4.) can be used …”. I do not understand why equation (2.4.) is called as equation for symmetry. It is rather general relation between variables. In the following text equation (2.1) is called as “symmetry principles”. Please clarify.
2) I assume that in equation (2.2b) the authors use Einstein’s summation convention. It should be clearly indicated. In other case the summation is missing (over j and k indexes in the left equation and over i index in the right equation). Please add some comment to the main text of the manuscript.
3) page 3, lines 52-54: the authors wrote: “When we want to add in the symmetry conditions also the continuity equation (2.2a), the Gauss divergence theorem (2.2 a,b) must be applied.” Its quite misleading. What is the Gauss divergence theorem? I would rather say that it is right equations in equations numbered as (2.2 a) and (2.2.b). What the author call as continuity equation? It not clear enough.
4) Is equation (2.3) is some theorem? I is not obvious for me why first equation is fulfilled. Please add some justification. The next part of (2.3.) is, as I understand, just integration by part. Please confirm.
5) The equation given in line 58 is a consequence of left equation in (2.2.a) or (2.2.b). It should be clearly indicated, because for a potential reader it could not be so obvious as for the authors.
6) page 3, lines 60-62: the authors wrote: “Expressions (2.6a) and (2.6b) are very important, because it points out the fact that unsteady states of the magnetic induction intensity are generated on the borders of the area and vice versa.” But according the Section 1 “Nomenclature” B denotes “magnetic flux density”, whereas H is “magnetic field intensity”. But in equations (2.6.a) and (2.6.b) only B appears. This inconsistency should be clarified.
7) page 3, line 67-68: the authors wrote: “the above-mentioned criteria we obtain new shape of Maxwell equations.” Formally it is a new shape only of two out of four Maxwell equations: Gauss's law for magnetism and Maxwell–Faraday equation (Faraday's law of induction) from (2.2).
8) page 4, line 81-82: the authors wrote: “For the nonconductive space are the 82 derivations described within the Chapter 3”. Probably, it should be change into “Chapter 4”.
9) In my opinion it would to be useful to add text, which describes what is included in the sections of the manuscript (e.g. at the end of the introduction). It is very useful for a reader, when it is consistently described how the paper is organized.
10) I think that introduction part (Section 2) is too short. Also the bibliography is not very extended. It quite strange that after more than 200 years of electrodynamics development, the authors can find only 12 references. It is not sufficient. There is plenty monographs on electrodynamics in general and it would be good for a reader if some comprehensive review of the newest literature was given.
11) Please also organize references in the order of appearing in the text. For example, the first cited reference (in line 24, page 2) has number [10], and next Ref. [5] appears (line 27, page 1).
12) In my opinion, Sections 8 and 9 (Discussion and Conclusion) can be merged. In Section 8, I do not see any discussion, rather review of the result and discussion presented in the previous section. Maybe “Conclusions and final remarks” would be appropriate title for that merged final section.
To sum up, after these issues will be resolved, I strongly believe this paper could become suitable for publication in “Symmetry” MDPI journal as an regular article provided that the author introduces required changes. They should precisely and convincingly provide all clarifications for all points mentioned above, particularly, they should extend introduction and modify the text to be more specific and mathematically strict. The topic of the paper, which is strongly associated with basic concepts of electrodynamics, can be interesting for some scientist. However, in the present form the paper cannot be published. It should be reformulated and modified according points 1)-12) mentioned above. It is very good that it is formally mathematically correct, but the conclusions and the main message should be given in easy and clear way.
Author Response
Thank you for your comments, all were used in the corrected article. It helped us a lot!
Author Response File: 
Author Response.docx
Reviewer 2 Report
Results found in this manuscript are, in my opinion, interesting. However, I think it needs (a) much more input on why the subject is important and (b) an extensive revision of writing to be sure that it is clear to the general readership that can be interested in something as fundamental as a new formulation of Maxwell equations.
I recommend the authors to write a completely new version in which the computations are well motivated, the results are explained and the comparison with the standard form of Maxwell equations is discussed to verify the advantages of the new formulation. In its present form, I can not recommend publication of this manuscript in Symmetry.
I suggest the following points to be taken into account in the new version.
1. The abstract is, in my opinion, incomplete. Please, add brief information on relevance of non-stationary equations as opposed to stationary ones for concrete problems.
2. The introduction section is too short. It should be rewritten to explain more carefully the relevance of the subject. The purpose of the paper is clear by the last two sentences of the introduction, but why is it important should be better explained with examples and physical situations in which traditional form of Maxwell equations lead to issues that the new formulation could avoid.
3. Page 2, Lines 28-32. Sentences “ Because we know that the equation for symmetry (2.4.) can be used, it is useful to adjust the equation to a more appropriate form for optimization as well as qualitative analysis and numerical methods. Considering that N-S equations are partial differential and differential equations, whose solution depends significantly on boundary conditions”. I do not understand what the authors mean. Please clarify.
4. Section 7. This section is important, since it presents a physical situation in which the new form of Maxwell equations discussed in previous sections can be relevant. However, there is no introduction on what the physics the authors are studying here is, or what do they want to achieve. In my opinion, these questions should be answered in the text. At the end of this section, the authors say that the advantage of the new formulation is evident. Well, it is not so evident to me. I would need more information on what advantage, related to numerics or boundary conditions, they are thinking about.
5. The Conclusions section is clear but I would like to see much more details on the advantages of the new formulation. I recommend to rewrite this section and put more effort in the explanation of concrete problems and concrete details in which these advantages can be noticeable.
6. In general, I would encourage the authors to deeply revise the writing of the manuscript in order to be more reachable to the broad community of readers of an open access, high impact, journal as Symmetry.
Author Response
Thank you for your advices, we have corrected the article a lot.
Author Response File: Author Response.docx
Round 2
Reviewer 2 Report
In my opinion, this version is much better than the first one. I recommend publication.
