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Hi Maschen:

This actually doesn't have anything to do with an article, but I was hoping you might have an idea. I'm trying to make a visualization of some transformations of the complex plane. The plane is carved up into six regions, and I'd like to see how each of them changes under these transformations. Actually, I'd like to be able to indicate any region of the plane that I like, and see the resulting image after the transformation.

Coloring a region is useful, but the challenge is that some transformations (e.g. 1/z) also flip the region, so something else has to be added. The best I've come up with so far is to strategically add stripes (to expose how the region has bent) and a gradient color (to detect flipping.) Do you have any idea what tools might work for this project? Thanks in advance: Rschwieb (talk) 21:18, 30 December 2015 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2015-12-30T21:18:00.000Z","author":"Rschwieb","type":"comment","level":1,"id":"c-Rschwieb-2015-12-30T21:18:00.000Z-wondering_if_you_can_help_me_with_graphics","replies":["c-Maschen-2015-12-30T21:54:00.000Z-Rschwieb-2015-12-30T21:18:00.000Z"]}}-->

Hi, Mathematica is a good program if you have access (I used to at uni because they had it installed there, but not my own copy). See for example [1] and [2]. Yes, colour gradients are used according to the phase and modulus of the complex number, so that each complex number has a unique colour, and a complex function returns a distorted pattern, see Domain coloring or [3]. There is also SageMath [4] which I have never encountered before, but it looks pretty good.

If the domain of the function is a closed curve shape in the plane (even with "holes"), you could also indicate the orientation of the domain; on the boundaries draw an arrow (like bivectors in geometric algebra). Then applying a complex function to it, the shape will distort according to the mapping, and if the domain reflects the orientation reverses. If the shape is open (say the entire upper-half plane z = x + iy for any −∞ < x < ∞ and all y ≥ 0), I'm not sure, maybe draw a closed curve inside the region with the orientation. It should definitely work for conformal transformations because angles are preserved, areas are distorted.

Hope this helps and happy new year! M∧Ŝc²ħεИ_τlk 21:54, 30 December 2015 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2015-12-30T21:54:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2015-12-30T21:54:00.000Z-Rschwieb-2015-12-30T21:18:00.000Z","replies":[],"displayName":"M"}}-->

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Hi, I am hopelessly inept at this, including the copyright reassurance parts, and you are an expert... I posted a "plea" for one or two figs in Talk:Wake#Plea for graphing help regarding the Kelvin wave pattern. Perhaps somebody you know could do this amended 12.2 , and if easy, 12.3 more easily than me, or else they exist someplace in Wikimedia and i could not find them... 12.3 looks like a classic, so... Thanks. No rush whatever! Cuzkatzimhut (talk) 19:58, 30 January 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-01-30T19:58:00.000Z","author":"Cuzkatzimhut","type":"comment","level":1,"id":"c-Cuzkatzimhut-2016-01-30T19:58:00.000Z-Since_you_are_an_expert_on_this....","replies":["c-Maschen-2016-01-31T09:00:00.000Z-Cuzkatzimhut-2016-01-30T19:58:00.000Z"]}}-->

Thanks for posting, will reply there. M∧Ŝc²ħεИ_τlk 09:00, 31 January 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-01-31T09:00:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-01-31T09:00:00.000Z-Cuzkatzimhut-2016-01-30T19:58:00.000Z","replies":[],"displayName":"M"}}-->

Here: Byron & Fuller, Mathematics of classical and quantum physics

It has just the right level of rigor. You probably know half or more of the contents. It is still a "must have". Vector spaces, Hilbert spaces of physics, Complex analysis taking it further than intro courses, Green's functions, two fat chapters on infinite-dimensional spaces + intro to group theory. YohanN7 (talk) 16:37, 1 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-01T16:37:00.000Z","author":"YohanN7","type":"comment","level":1,"id":"c-YohanN7-2016-02-01T16:37:00.000Z-Bargain","replies":["c-Maschen-2016-02-01T22:13:00.000Z-YohanN7-2016-02-01T16:37:00.000Z"]}}-->

Thanks for pointer, looked through the contents in the link and it also has integral equations, which are interesting and rarely covered. Will invest later. M∧Ŝc²ħεИ_τlk 22:13, 1 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-01T22:13:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-02-01T22:13:00.000Z-YohanN7-2016-02-01T16:37:00.000Z","replies":["c-YohanN7-2016-02-04T09:37:00.000Z-Maschen-2016-02-01T22:13:00.000Z","c-YohanN7-2016-02-04T10:14:00.000Z-Maschen-2016-02-01T22:13:00.000Z"],"displayName":"M"}}-->

Yes, the second half of the book is really a treat. For instance, the Hilbert transform is put to use in dispersion relations (ch. 6). Green's functions (ch. 7) are used to quite literally turn differential equations inside out to become integral equations. For example, the Scrödinger equation becomes the Lippmann–Schwinger equation. The propagators of quantum mechanics and QFT are Green's functions. Then (ch. 8–9) these integral equations are investigated and solved in a quite general setting (Banach space) and finally in Hilbert space. The "introductory" chapter on group theory is more comprehensive than one might think, as it includes applications to perturbation theory, and includes representation theory for finite groups, which is pretty much "the same" as for compact Lie groups. YohanN7 (talk) 09:37, 4 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-04T09:37:00.000Z","author":"YohanN7","type":"comment","level":3,"id":"c-YohanN7-2016-02-04T09:37:00.000Z-Maschen-2016-02-01T22:13:00.000Z","replies":[]}}-->

And oh, (ch 5), the theory of second order polynomial Sturm–Liouville problems problems is covered comprehensively in the setting of Hilbert space. What is usually a complete total mess of "special functions" (Legendre, Hermite, Chebyshef, ...) in the differential equation setting becomes logical and uniform. YohanN7 (talk) 10:14, 4 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-04T10:14:00.000Z","author":"YohanN7","type":"comment","level":3,"id":"c-YohanN7-2016-02-04T10:14:00.000Z-Maschen-2016-02-01T22:13:00.000Z","replies":[]}}-->

While at it: Simmons, Differential Equations with Applications and Historical Notes. Same thing here. You know it, but having e.g. second order linear equations nailed is a good thing. Price/Quality and Quantity ratio not as whopping as the previous one, but still very very good. The historical notes are marvelous. (Newton's foreseeing of special relativity is spooky.) YohanN7 (talk) 14:42, 4 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-04T14:42:00.000Z","author":"YohanN7","type":"comment","level":1,"id":"c-YohanN7-2016-02-04T14:42:00.000Z-Bargain","replies":["c-Maschen-2016-02-05T11:59:00.000Z-YohanN7-2016-02-04T14:42:00.000Z"]}}-->

Thanks, doesn't seem previewable on amazon but on google books there are no previews of the modern editions but snippets of the 1970s edition, I'll see into investing later. Would you recommend any particular edition (there seems to be loads)? M∧Ŝc²ħεИ_τlk 11:59, 5 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-05T11:59:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-02-05T11:59:00.000Z-YohanN7-2016-02-04T14:42:00.000Z","replies":["c-YohanN7-2016-02-05T12:24:00.000Z-Maschen-2016-02-05T11:59:00.000Z","c-YohanN7-2016-02-06T11:15:00.000Z-Maschen-2016-02-05T11:59:00.000Z"],"displayName":"M"}}-->

Strange indeed. Very strange. I swear that the Amazon page has changed since I posted. The price was $15.50 or so, with used ones for $12 (now you have the option to pay $195 for a new one - gah). I'll see which edition I have when I get home. YohanN7 (talk) 12:24, 5 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-05T12:24:00.000Z","author":"YohanN7","type":"comment","level":3,"id":"c-YohanN7-2016-02-05T12:24:00.000Z-Maschen-2016-02-05T11:59:00.000Z","replies":[]}}-->

I have a 1974 edition (McGraw-Hill), probably the original 1972 text. The virtue of the book is that it is easy. Even when the author warns that the material is "rather formidable", it is easy, even the problems are easy (they have answers, but no solutions). Maybe it would be formidable to the undergraduate for whom the book is intended? Don't know. (The Byron & Fuller book is decidedly "graduate".) My edition doesn't contain anything (not much) on partial diff eqns. Whether this is a downside or not is a matter of taste. I think PDE's form a subject of their own. YohanN7 (talk) 11:15, 6 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-06T11:15:00.000Z","author":"YohanN7","type":"comment","level":3,"id":"c-YohanN7-2016-02-06T11:15:00.000Z-Maschen-2016-02-05T11:59:00.000Z","replies":["c-Maschen-2016-02-06T13:20:00.000Z-YohanN7-2016-02-06T11:15:00.000Z"]}}-->

Seems to be published in 1972, 1980, 1991, 2003, 2006, each possibly reprinted. Anyway its cool: found a 1972 edition on alibris fairly cheap (sort of, cost £21.80 = US $31.62 second hand "good" non-exlibrary condition). Should be here in a couple of weeks or so (has to ship from the US to here in the UK), and looking forwards to it. All the other editions are too expensive anyway. It would be perfectly logical if the scope was entirely ODEs, PDEs are a huge topic on their own. M∧Ŝc²ħεИ_τlk 13:20, 6 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-06T13:20:00.000Z","author":"Maschen","type":"comment","level":4,"id":"c-Maschen-2016-02-06T13:20:00.000Z-YohanN7-2016-02-06T11:15:00.000Z","replies":[],"displayName":"M"}}-->

Normally I'd post on your page, but since you're watching here, another good-looking ordinary differential equations book appears to be

• Martin Braun Differential equations and their applications 4th edition (1993) Springer amazon.com

and while at it, for multivariable calculus including differential forms two also good-looking books at an advanced graduate level are

• Wendell Fleming Functions of Several Variables 2nd edition (1977) Springer amazon.com
• Edwards Advanced Calculus of Several Variables (1994) Dover, originally (1973) Academic Press amazon.com

More to look forwards to later this year. M∧Ŝc²ħεИ_τlk 00:33, 7 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-07T00:33:00.000Z","author":"Maschen","type":"comment","level":1,"id":"c-Maschen-2016-02-07T00:33:00.000Z-Bargain","replies":[],"displayName":"M"}}-->

Hi Maschen. Thanks for your re-work of angular momentum, particularly the lede. Your first sentence in the lede contains "... is a physical quantity is a measure of ..." There appears to be a word missing, or misspelt, or perhaps some of the words should be deleted. Dolphin (t) 05:45, 4 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-04T05:45:00.000Z","author":"Dolphin51","type":"comment","level":1,"id":"c-Dolphin51-2016-02-04T05:45:00.000Z-Angular_momentum","replies":["c-Maschen-2016-02-04T08:39:00.000Z-Dolphin51-2016-02-04T05:45:00.000Z"],"displayName":"Dolphin"}}-->

Thanks for feedback. It is too long but I'm not entirely sure what to remove. I'll try again now, and maybe later. M∧Ŝc²ħεИ_τlk 08:39, 4 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-04T08:39:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-02-04T08:39:00.000Z-Dolphin51-2016-02-04T05:45:00.000Z","replies":[],"displayName":"M"}}-->

Talk:Observer_effect_(physics)#Revert back with a vengeance. Sigh... Cuzkatzimhut (talk) 11:51, 20 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-20T11:51:00.000Z","author":"Cuzkatzimhut","type":"comment","level":1,"id":"c-Cuzkatzimhut-2016-02-20T11:51:00.000Z-Metaphysics_again","replies":["c-Maschen-2016-02-20T22:09:00.000Z-Cuzkatzimhut-2016-02-20T11:51:00.000Z"]}}-->

Commented there and asked for input on Wikipedia talk:WikiProject Physics‎. M∧Ŝc²ħεИ_τlk 22:09, 20 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-20T22:09:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-02-20T22:09:00.000Z-Cuzkatzimhut-2016-02-20T11:51:00.000Z","replies":[],"displayName":"M"}}-->

Thanks, I should have done that myself. As I mumbled, I bought RPF's point in college, and cannot bring myself to undo that now... Cuzkatzimhut (talk) 22:18, 20 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-20T22:18:00.000Z","author":"Cuzkatzimhut","type":"comment","level":1,"id":"c-Cuzkatzimhut-2016-02-20T22:18:00.000Z-Metaphysics_again","replies":[]}}-->

I was reading your bio and you seem very interesting. The tropes of your profile not only were hysterical to me, but also had a sort of resonance. I have questions regarding the World Lines theory and how it ties into Einsteins theory of relativity. In simple terms can you summarize world lines for someone who doesn't possess key background information. As I have not started my undergrad, the article on Wikipedia regarding world lines is hard to follow.

Sincerly,

a cog  — Preceding unsigned comment added by 166.173.61.89 (talk) 17:44, 23 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-23T17:44:00.000Z","author":"166.173.61.89","type":"comment","level":1,"id":"c-166.173.61.89-2016-02-23T17:44:00.000Z-Inquiry_for_the_Machine","replies":["c-Maschen-2016-02-23T22:28:00.000Z-166.173.61.89-2016-02-23T17:44:00.000Z"]}}--> 

A little confused about why you find me interesting. Anyway about world lines, think instead of a point particle moving in 1d only. The motion of the particle can be tracked by plotting its position at every instant of time on a distance-time graph (time on horizontal axis, distance on vertical axis), the result will be a curve in general, possibly with segments of straight lines. The velocity at any instant is the tangent to the curve at the instant. Now flip the time-distance axes, so time increases "upwards (vertically)", the position of the particle is somewhere along the distance axis "left/right (horizontally)", and the particle's velocity is the reciprocal of the gradient in the new axes. The curve showing the particle's position at every instant is the world line, and can be thought of as the "trajectory" of the particle in spacetime.

Normally the distance-time graph is used in the context of Newtonian mechanics, just 1 spatial dimension in this case (3 spatial dimensions more usually), with an absolute time. However, the background of special relativity is spacetime, in this case just 1 space dimension + 1 time dimension, see Minkowski space. In SR, time is not absolute. To track the motion of a particle, a spacetime diagram is used. This is essentially looks like a reflected distance-time graph on a screen/paper, but the physical spacetime is different to the Euclidean geometry to draw the distance-time axes. The space and time coordinates change under Lorentz transformations.

Someone just standing still in a lab watching the particle move around will measure the coordinate time (time on vertical axis of spacetime diagram). However, a clock attached to the particle will record a different time called the proper time of the particle. The "arc length" of the world line is proportional to the proper time of the particle, and can be used to parametrize the world line.

For objects in 2 space + 1 time dimensions, every point will trace out a world line in spacetime, the result will be a "world tube". Likewise for objects in 3 space + 1 time dimensions, the result is a higher dimensional analogue, a "world hypertube".

Hope this helps. M∧Ŝc²ħεИ_τlk 22:28, 23 February 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-02-23T22:28:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-02-23T22:28:00.000Z-166.173.61.89-2016-02-23T17:44:00.000Z","replies":[],"displayName":"M"}}-->

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Hi, just a note to steer your attention to an ANI discussion on rogue but coddled IP users. The last item at present on my talk page steers you there. Cuzkatzimhut (talk) 00:47, 5 June 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-06-05T00:47:00.000Z","author":"Cuzkatzimhut","type":"comment","level":1,"id":"c-Cuzkatzimhut-2016-06-05T00:47:00.000Z-Discussion_on_IP_users","replies":["c-Maschen-2016-06-07T16:20:00.000Z-Cuzkatzimhut-2016-06-05T00:47:00.000Z"]}}-->

Hi, sorry, I did notice this, but my own laptop has died so editing is much restricted for a while. Hope this stupidiotic and pointless fuss over nothing is settled soon. M∧Ŝc²ħεИ_τlk 16:20, 7 June 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-06-07T16:20:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-06-07T16:20:00.000Z-Cuzkatzimhut-2016-06-05T00:47:00.000Z","replies":[],"displayName":"M"}}-->

Hope so too. An illustration of the point is today's rants at Talk:Observer effect (physics). Cuzkatzimhut (talk) 16:29, 7 June 2016 (UTC) Should I or you invite Tommy to put his case at the Physics Tea room or whatever that page is called? Cuzkatzimhut (talk) 21:27, 7 June 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-06-07T16:29:00.000Z","author":"Cuzkatzimhut","type":"comment","level":1,"id":"c-Cuzkatzimhut-2016-06-07T16:29:00.000Z-Discussion_on_IP_users","replies":["c-Maschen-2016-06-10T07:21:00.000Z-Cuzkatzimhut-2016-06-07T16:29:00.000Z","c-Cuzkatzimhut-2016-06-08T14:36:00.000Z-Cuzkatzimhut-2016-06-07T16:29:00.000Z"]}}-->

I thought the "tea room" was for new users, and ANI for escalating disputes. If Tommy has stopped then there is no need to do anything, otherwise Wikipedia talk:WikiProject Physics or ANI I guess. M∧Ŝc²ħεИ_τlk 07:21, 10 June 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-06-10T07:21:00.000Z","author":"Maschen","type":"comment","level":3,"id":"c-Maschen-2016-06-10T07:21:00.000Z-Cuzkatzimhut-2016-06-07T16:29:00.000Z","replies":[],"displayName":"M"}}-->

Thanks for your good judgement; even to myself, some of my impatient statements often come across as "unusual". I will never again use the obvious metaphor "quasi-vandalism" to mean the obviously intended "reckless damage"... I had not appreciated how literalists glom on metaphorical adjectives. Archive....Cuzkatzimhut (talk) 14:36, 8 June 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-06-08T14:36:00.000Z","author":"Cuzkatzimhut","type":"comment","level":2,"id":"c-Cuzkatzimhut-2016-06-08T14:36:00.000Z-Cuzkatzimhut-2016-06-07T16:29:00.000Z","replies":[]}}-->

hi maschen,

i am a huge fan of your figures. i think i have said that before, but nonetheless i am wondering if you could make a figure (or two) because i think it would be helpful.

• specifically i am hoping you could make figures that would assist readers in understanding Constantin Carathéodory's axiomatic approach to thermodynamics that implicitly uses measure theory.^[1][2]

if you check out page 13 in the translated link (page 369 in the original), he gives an explanation for constructing the complex plane in space, and introduces a cylinder to facilitate geometric reasoning.

given your amazing figures are all over wikipedia math pages, i was hoping if you were up for a challenge that you'd CONSIDER trying to make one? if you're too busy (believable), i get it. thought i'd ask though. thanks 174.3.155.181 (talk) 23:57, 13 July 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-07-13T23:57:00.000Z","author":"174.3.155.181","type":"comment","level":1,"id":"c-174.3.155.181-2016-07-13T23:57:00.000Z-asking_for_a_favour_next_time_you_plan_to_make_wiki_contributions?","replies":["c-Maschen-2016-07-15T11:50:00.000Z-174.3.155.181-2016-07-13T23:57:00.000Z","c-Maschen-2016-07-15T14:02:00.000Z-174.3.155.181-2016-07-13T23:57:00.000Z","c-Maschen-2016-07-16T06:46:00.000Z-174.3.155.181-2016-07-13T23:57:00.000Z"]}}-->

Hi, sorry for a late reply, I'll have a look and try this weekend. Thanks for posting, M∧Ŝc²ħεИ_τlk 11:50, 15 July 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-07-15T11:50:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-07-15T11:50:00.000Z-174.3.155.181-2016-07-13T23:57:00.000Z","replies":[],"displayName":"M"}}-->

Having read p. 13 of this I can see why you want a picture (or two), the rambling description is horrible. To summarize so a simpleton like me can understand easier...

• We have the Pfaffian equation (no article? I think Pfaffian system is the closest)

${\displaystyle dx_{0}+\sum _{i=1}^{n}X_{i}dx_{i}=0}$

• P = (a₀, a₁, ..., a_n) is a point, the neighbourhood of which is being considered,
• P_i are accumulation points of P, such that a solution of the equation does not pass through P and P_i,
• C_i are solutions of the equation which pass through P_i
• for the line G, I assume their "x = t" (with t presumably the parameter of the line) should be "x₀ = t", along with x_k = a_k, and the P_i are not on this line,
• the C_i (hence corresponding P_i) and G are all in a plane,
• Q_i are intersection points passing through C_i and G,
• G₁ is any line parallel to G,
• G₁ and G are lines defining any cylinder (two lines would define a family of them, a third parallel line is needed for a unique cylinder),
• M is an intersection point of: a solution to the equation (curve) passing through P and lies on the cylinder, and G₁
• F(x₀, x₁, ..., x_n) = C is the family of surfaces obtained by varying P (for fixed M?) and C a parameter.

Let me know if this is correct. Thanks, M∧Ŝc²ħεИ_τlk 14:02, 15 July 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-07-15T14:02:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-07-15T14:02:00.000Z-174.3.155.181-2016-07-13T23:57:00.000Z","replies":[],"displayName":"M"}}-->

I noticed yesterday after writing the above list (but didn't post until now) you are discussing this on Talk:Integrability conditions for differential systems. If that is the article you want the picture for, shall we continue there? M∧Ŝc²ħεИ_τlk 06:46, 16 July 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-07-16T06:46:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-07-16T06:46:00.000Z-174.3.155.181-2016-07-13T23:57:00.000Z","replies":[],"displayName":"M"}}-->

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Hi, the page you merged on October 2015 is being grown again by an IP user. Can you have a look. scope_creep (talk) 10:33, 20 December 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-12-20T10:33:00.000Z","author":"Scope creep","type":"comment","level":1,"id":"c-Scope_creep-2016-12-20T10:33:00.000Z-Field_equation","replies":["c-Maschen-2016-12-20T11:16:00.000Z-Scope_creep-2016-12-20T10:33:00.000Z"],"displayName":"scope_creep"}}-->

Thanks for the heads up, will investigate. It was field equation merged into classical field theory. user:RockMagnetist contributed also. M∧Ŝc²ħεИ_τlk 11:16, 20 December 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-12-20T11:16:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-12-20T11:16:00.000Z-Scope_creep-2016-12-20T10:33:00.000Z","replies":[],"displayName":"M"}}-->

I did some mostly routine copy-editing on Joos–Weinberg equation including fixing the punctuation error in one of the headings and standard use of \ldots and \cdots and positions of commas in things like ${\displaystyle a,\ldots ,z}$ (as opposed to ${\displaystyle a,\ldots z}$, etc.) and spacing in non-TeX notation (e.g. a = b instead of a=b, etc.). However I wonder about something of a less routine nature. You wrote:

${\displaystyle \Pi _{i=1}^{i=j}\otimes B_{[\alpha _{i},\beta _{i}]}}$

I did a completely routine correction, changing this to:

${\displaystyle \prod _{i=1}^{j=1}\otimes B_{[\alpha _{i},\beta _{i}]}}$

But I'm wondering if the following could also serve:

${\displaystyle \bigotimes _{i=1}^{i=j}B_{[\alpha _{i},\beta _{i}]}}$

When the binary operation is ${\displaystyle {\text{“}}{\otimes }{\text{''}}}$ then often ${\displaystyle {\text{“}}{\bigotimes }{\text{''}}}$ is used in that way. Michael Hardy (talk) 17:59, 29 December 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-12-29T17:59:00.000Z","author":"Michael Hardy","type":"comment","level":1,"id":"c-Michael_Hardy-2016-12-29T17:59:00.000Z-Joos\u2013Weinberg_equation","replies":["c-Maschen-2016-12-29T19:42:00.000Z-Michael_Hardy-2016-12-29T17:59:00.000Z","c-Maschen-2016-12-29T19:51:00.000Z-Michael_Hardy-2016-12-29T17:59:00.000Z"]}}-->

Thanks for clean up as always. I did start the article including the lead and first section but didn't write the dense section Lorentz covariant tensor description of Weinberg–Joos states (that was user:Physics-math explorer [5]). To answer your question, yes

${\displaystyle \prod _{i=1}^{j=1}\otimes \quad {\text{and}}\quad \bigotimes _{i=1}^{i=j}}$

are the same in this context (tensor product). The first does not seem standard at all. The second is (large circle-times symbol, not usually the small one). M∧Ŝc²ħεИ_τlk 19:42, 29 December 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-12-29T19:42:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-12-29T19:42:00.000Z-Michael_Hardy-2016-12-29T17:59:00.000Z","replies":[],"displayName":"M"}}-->

It is fixed now. M∧Ŝc²ħεИ_τlk 19:51, 29 December 2016 (UTC)__DTREPLYBUTTONSCONTENT__-->__DTELLIPSISBUTTON__{"threadItem":{"timestamp":"2016-12-29T19:51:00.000Z","author":"Maschen","type":"comment","level":2,"id":"c-Maschen-2016-12-29T19:51:00.000Z-Michael_Hardy-2016-12-29T17:59:00.000Z","replies":[],"displayName":"M"}}-->

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