# Difference between revisions of "User talk:MarkGall"

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:Thanks! I think calculus might not be in there at all yet, but it probably ought to be. Maybe there can be a section called "basic tools" or something at the top, dealing with calculus and linear algebra. The outline of the article is growing faster than I have time to fill it in! Please do add your new suggestions to the page, and I'll keep working whenever I have time. It's the middle of the semester now, and things are getting busy again. --[[User:MarkGall|MarkGall]] 23:14, 17 October 2009 (EDT) | :Thanks! I think calculus might not be in there at all yet, but it probably ought to be. Maybe there can be a section called "basic tools" or something at the top, dealing with calculus and linear algebra. The outline of the article is growing faster than I have time to fill it in! Please do add your new suggestions to the page, and I'll keep working whenever I have time. It's the middle of the semester now, and things are getting busy again. --[[User:MarkGall|MarkGall]] 23:14, 17 October 2009 (EDT) | ||

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+ | == Good Bezout's theorem addition == | ||

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+ | Great addition of Bezout's theorem! | ||

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+ | I wonder if we have enough on advanced calculus, though admittedly that is often handled in engineering rather than math departments. Perhaps it is folded into differential geometry in most math departments.--[[User:Aschlafly|Andy Schlafly]] 15:37, 24 October 2009 (EDT) |

## Revision as of 19:37, October 24, 2009

**Useful links**

**Welcome!**

Hello, MarkGall, and * welcome* to Conservapedia!

We're glad you are here to edit. **We ask that you read our Editor's Guide before you edit.**

At the right are some useful links for you. You can include these links on your user page by putting "{{Useful links}}" on the page. Any questions--ask!

Thanks for reading, MarkGall!

**ṬK**_{/Admin}^{/Talk}00:50, 10 June 2009 (EDT)

Hello there, Mark! I am truly glad to see that you are adding new pages for Obama's "Czars." So far, so good for a newcomer to Conservapedia! -RKLuffy88

## Contents

## Czars...

Great work, Mark, but can we please consolidate these into one Obama Czar article? --**ṬK**_{/Admin}^{/Talk} 22:10, 11 June 2009 (EDT)

- As before, I'm happy to merge them -- I only created seperate pages since it was requested on the main page by DeanS to fill in red links he added to Obama Administration. What should the new article be titled, and how would you recommend laying it out? --MarkGall 16:44, 7 July 2009 (EDT)

- One would have to find the articles, as they are not linked from the Obama Administration page. If you can do that, provide the links, I can move them onto the same page, entitled Obama Administration Czars. --
**ṬK**_{/Admin}^{/Talk}21:31, 7 July 2009 (EDT)

- One would have to find the articles, as they are not linked from the Obama Administration page. If you can do that, provide the links, I can move them onto the same page, entitled Obama Administration Czars. --

- Here they are:
- Alan Bersin, Steven Rattner, Gil Kerlikowske, Paul A. Volcker, Carol Browner, Jeffrey Zients, Cameron Davis, Nancy-Ann DeParle, John Brennan, Dennis Blair, Kenneth R. Feinberg, Cass R. Sunstein, Vivek Kundra, Adolfo Carrion, Jr., Gary Samore
- These are all linked at Obama Administration (the sixth section, "czars"), which ought to be updated accordingly. I can go through and clean up the redundant information once your merge is finished. Thanks! --MarkGall 21:46, 7 July 2009 (EDT)

## String Theory

Thanks for adding the attributions to Givental, Lian, Liu, and Yau. Don't ask me why I left out such an important credit.--Lemonpeel 12:36, 16 June 2009 (EDT)

- Lotrsw86? --
**ṬK**_{/Admin}^{/Talk}05:13, 23 June 2009 (EDT)

- Who/what is Lotrsw86? If you suspect me of having another username, I assure you that's not the case. I hope it's clear from my contributions by now that I'm not a vandal. --MarkGall 11:12, 23 June 2009 (EDT)

## Math articles

If you please, try to place an example of a problem within the math articles that are posted. For instance, in your Fundamental group article there could be a problem within each subtopic, broken down in a step-by-step manner to lead the reader in understanding and solving it. Think this is possible? Karajou 01:42, 4 July 2009 (EDT)

- I like your idea about problems in the math articles. The issue here is that it's difficult to come up with problems on the fundamental group that could really be answered at this level. An exposition at this level doesn't give the reader the knowledge to solve any real problems. I don't even prove what the fundamental group of the circle is -- a proof requires much more theory than is in the article, but the intuition is clear. I could ask for intuitive guesses about fundamental groups, but just about every space that a new reader knows about is given as an example already! I will try to think something up, I'm sure it can be done.
- In the article differential geometry which I did a bit of work on a couple weeks ago, I inserted "motivating questions" in a few places, to indicate why someone would care about it. I can certainly do this in the fundamental group article. Actually, I was thinking that on the front page mathematics, it would be neat to put motivating questions under all the headers you created, to give a flavor of these areas of mathematics. What do you think of that idea? --MarkGall 09:33, 4 July 2009 (EDT)

- I added two problems/solutions to fundamental group. What do you think? One is for conceptual understanding, while one tests ability to manipulate the definitions. I'll try to add more. What do you think? --MarkGall 21:51, 7 July 2009 (EDT)

## big bang

You undid my edit; why? The list of astronomers I posted contained many people who are far more knowledgeable than either Arp or Burbidge, as evidenced by their continued research in the field of cosmology. While Arp and Geoff might be fine astronomers, they certainly do not understand the preponderance of data that contradicts their pet theories; if their's are notable names to be attached to an article in opposition to the Big Band Theory, why should the names of proponents not be included?

- You added a useless list of mostly obscure scientists. It did nothing to improve the exposition of the article and created the appearance that the theory is more widely supported among scientists than it actually is. --MarkGall 16:53, 2 September 2009 (EDT)

## Edit summaries

Hi, Mark. Could you be more specific than *reinserted some correct info* when you edit difficult or advanced articles like Infinity? Our readers and contributors need to know WHAT is being inserted and WHY. --Ed Poor ^{Talk} 10:36, 18 September 2009 (EDT)

- My apologies. I was reinserting some correct information which I had inadvertently removed when taking out some apparent vandalism, and didn't think to state what it actually was. That material's been there for some time so I hope it won't be controversial, but I'll try to pay more attention in the future. --MarkGall 11:28, 18 September 2009 (EDT)

## Possibility

Mark, it just dawned on me that you might enjoy teaching a math course here to high school homeschoolers. Just a thought for you to consider ...--Andy Schlafly 10:42, 19 September 2009 (EDT)

- Mr. Schlafly, I would certainly love to do that at some point. Unfortunately, for now I don't have time to put in the effort it would require to do it well -- I'm busy enough teaching undergrads (who I suspect are much less motivated than the homeschoolers here!) and keeping up with my own work. I'd consider it at some point in the future, perhaps next year so I can develop materials over the summer. I saw that you once tried to start up a math course here -- if enough interest materializes again, I'd be happy to be an assistant teacher and help out as much as I'm able! --MarkGall 10:49, 19 September 2009 (EDT)

- Mark, that's fine and I appreciate your work ethic! Maybe I'll get a math course started sometime early next year, and with your help it can grow.

- The advantage of teaching with the wiki software is that a course can improve dramatically each time it is taught. I'm seeing that with the Economics course now. Thanks again for your insights and effort.--Andy Schlafly 12:35, 19 September 2009 (EDT)

## Templates etc.

(Sorry about ping-ponging this thread all over my page, Andy's page, and your page, but it will be finished very soon.)

Thanks for weighing in on this. I've made the "H" change, calculus; I don't think there's any issue about that one.

On the "A" change (topology/Riemann), I've been going back and forth in my mind about how to do this. I don't think that it being an open question or not is really important. These things are really just decorations. Though I think the best description should be along the lines of "You should know what the equation to the left is talking about before reading this article." (Sort of like the "You must be this tall to go on this ride" signs at amusement parks :-) Or maybe it should be stronger, as in "You should know what this is talking about *and be able to solve it*." Which of course means Riemann is not suitable. And I think the higher homotopy groups would cut me out. :-( My topology study ended at about the point of the higher homotopy groups. :-( In any case, by the "and be able to solve it" criterion, which works for the other 3 templates, pi-1 of S-1 is about right.

In any case, let me know if this is OK, or if you want pi-m of S-n, and we can let Andy re-protect them. Unprotecting the two lower ones ("E" and "M") was not necessary; it was never my intention to modify those, but I was in too big a hurry last night, at 12 minutes past midnight, to deal with such subtleties. PatrickD 09:51, 22 September 2009 (EDT)

- Sure, sounds good to me. \pi_1(S^1) is about the right level. Thanks for fixing this! --MarkGall 14:19, 22 September 2009 (EDT)

## Incompleteness Theorem

Mark:

I'm planning to work on the issues of logic, completeness, undecidability, and the Gödel incompleteness theorem next. Basically, everywhere I look, I see things to do, and I happened to look at the Undecidable page and the Axiom page. The latter is fairly straightforward -- the characterization of the axioms of geometry being "challenged" by Non-Euclidean geometry needs to be clarified.

But the "undecidable" stuff needs a lot more care. I'm pretty sure I understand the issues, but I'm not 100% sure, so I want to run it by you. It seems to me that essentially all of the material about "Famous undecidable statements" is wrong. These may be statements that are at the edge of what ZF and ZFC mean, but they don't relate to the Gödel incompleteness theorem. And "undecidablility" is about the Gödel incompleteness theorem, *not* about the axioms of ZF.

My understanding is that the Gödel incompleteness theorem says, roughly

- Any logical system rich enough to express integer arithmetic is "incomplete" in the sense that it is not true that, for every well-formed formula F, either F or not F is a theorem (has a proof in that logic).

The logical systems being analyzed by Gödel are simple logics about integers, with nowhere near the sophistication of ZF or ZFC. The non-provability of AC in plain ZF is not really what the Gödel incompleteness theorem is about. You simply add AC, get ZFC, note that the result is still consistent (that is, you now have a trivial proof of AC because it is an axiom, but you do not have a proof of not-AC) and move on.

Hence, it seems to me that all the things in the article related to AC (Banach-Tarski, Continuum hypothesis, large cardinals, non-measurable sets) belong in a different article, since they distract from Gödel's contribution. Only the halting problem (a result from the closely related field of theoretical computer science) is directly relevant to Gödel. And that's what "decidability" should be about.

Of course, fact that neither AC nor not-AC has a proof in ZF is an example of incompletness, but it's not what Gödel was talking about. That is, plain ZF is incomplete, but incomplete logics are a dime a dozen. Gödel showed that *no* logic rich enough for integers could be complete.

Does this make sense?

PatrickD 23:07, 23 September 2009 (EDT)

- Hi Patrick,

- Logic's not my forte (you may have noticed I haven't really touched those articles), but I believe everything you say is correct. I am certain that the only thing on the list of "undecidable problems" which is actually undecidable is the halting problem. "Undecidable" is certainly not the same thing as "not provable within ZF", and choice is independent of ZF, not undecidable. Your statement of incompleteness is also good, and I think understandable.

- If you haven't seen it, there's a nice little book called "Godel's Proof" by Nagel that sketches the proof (in 100+ pages) in a fairly nontechnical way. I couldn't give a formal statement of the theorem, but your summary is more or less the same as the one there. I'm looking forward to seeing the article!

## Dirichlet's theorem

Thanks for explaining the requirement!--Andy Schlafly 00:38, 1 October 2009 (EDT)

- No problem... thanks for catching the error in the statement! Can't believe I missed that one. --MarkGall 00:48, 1 October 2009 (EDT)

## Functors

Sorry to mess up your elegant article on mathematical functors with low-class computer stuff, but there are probably more people familiar with the latter usage. :-(

Was the material on contravariant functors OK? I remember learning about this back in college. The cohomology functor is contravariant, right?

PatrickD 15:31, 1 October 2009 (EDT)

- My article's not elegant at all! It's just a cleanup until the CP treatment of category theory gets redone (if ever; I don't think it's a high priority). Thanks for adding the CS definition -- I'd heard the term, but never knew what it was before! I think the ideal situation would be a single page about basic notions in category theory -- definition of category, examples, definition of functor, examples, definition of natural transformation, examples, .... It doesn't make sense to me for functor to have its own page. I'd put things like "pushout square" and "adjoint functor" on their own pages, if they were ever written, but I'm not sure they belong here.

- You're right on about contravariant functors. I probably should've mentioned it. Sometimes these days when people write "functor" they just mean covariant functor, and a controvariant functor from A is just regarded as a covariant functor from A^op (A with all arrows reversed). It's definitely more illuminating this way, and cohomology is indeed an example.

- The other example I'm thinking about adding is Hom(-,V) and Hom(V,-), both functors from Vect_k to Vect_k. Given a fixed vector space V, Hom(-,V) is the functor that sends an object W in Vect_k to the set of linear maps from W to V, which is itself a vector space. The first of these is contravariant and the second is covariant (or the other way, I always get it wrong), and it's easy to write down why. It's a nice pair of examples in an easy-to-understand category. --MarkGall 17:00, 1 October 2009 (EDT)

## Awesome work!

Great job on some of these math articles, especially all that content to diffeomorphism! JacobB 17:57, 1 October 2009 (EDT)

- Thanks! I'm not really sure what to do with that one -- the notion and examples are obvious to anyone who knows what a smooth manifold is, but not very useful to anyone else. It's a hard thing to motivate. But hopefully the exotic spheres will be news to some. Jump in if you have any ideas to make it easier to read for people with less background!

If you like the geometric style argument we talked about on Talk:Essay:Quantifying Openmindedness, you should check out "Visual Complex Analysis" by Tristan Needham, a doctoral student of Roger Penrose. It's full of cool geometric arguments for complex analysis and Riemann manifolds that I haven't seen anywhere else. JacobB 00:11, 2 October 2009 (EDT)

- I'm a big fan of that book! I was a teaching assistant in a complex analysis course a couple years ago, and almost all of my review sessions (an hour a week) consisted of presenting the arguments from Needham to try to convince the students the theorems were plausible! My intuitions in physics are really bad, I ought to work on seeing things like that. Do you know any physics books along those lines? --MarkGall 00:43, 2 October 2009 (EDT)

- Oh, that's awesome! I had the privilege of studying under Dr. Needham at USF, in private sessions my senior year, and I know for a fact that he is working on a similar book for differential geometry. He based the methods of the book off of Newton's Principia, which uses tons and tons of geometric arguments (and a similar lack of rigor!), and Misner Thorne Wheeler is another book recommended by Dr. Needham, and riddled with geometric arguments. Even if you're not interested in GR, it's an excellent resource for differential geometry. JacobB 01:05, 2 October 2009 (EDT)

- Actually, I had a Fields medalist professor recommend once that we all read Needham's book, that's where I heard about it in the first place. I wish the differential geometry one had come out earlier -- my text had almost no geometry at all. GR is something that I'm hoping to understand better -- I'll definitely give Misner Thorne Wheeler a look. Thanks! --MarkGall 01:16, 2 October 2009 (EDT)

- A Fields medalist professor? That's so cool, who is it? JacobB 01:20, 2 October 2009 (EDT)

## Typo

Good work MarkGall, you have become a real contributor here.--Jpatt 15:29, 3 October 2009 (EDT)

I've learned a great deal from you Mark. Your catching the typo was one of many examples. Thanks.--Andy Schlafly 17:31, 3 October 2009 (EDT)

## Blocking and editing

Mark, you've received well-deserved blocking and night editing. Congratulations!--Andy Schlafly 18:52, 6 October 2009 (EDT)

- Thanks, I appreciate it! Yet another distraction from those late-night problem sets. Is there a page where I can learn how to block correctly? --MarkGall 19:19, 6 October 2009 (EDT)

## Gravity

I finally got around to reading the gravity material in Talk:Essay:Quantifying_Openmindedness, and there are a lot of correct things, a few things that may miss the point slightly, and a few instances of people talking past each other unnecessarily.

First, you are absolutely right that the divergence theorem establishes an exponent of exactly 2 for divergenceless vector fields in flat Euclidean space. But there's no reason to require that the gravitational "field" be divergenceless. In fact, it could be argued that "dark energy" creates the nonzero divergence, if people used a pure classical gravitational "field", which they don't.

Remember that Newton's formula was a mathematically exact consequence of Kepler's laws (ellipses, focus, equal-area law, etc.), which were an exact geometrical formulation from Brahe's not-completely-exact observations. And that all this was done hundreds of years before the divergence theorem. Newton was exactly matching Kepler's laws, and therefore set the exponent to exactly 2. That was exactly the correct thing to do, in terms of the observations available at the time.

But I think the reasoning in terms of geometry and divergences and "the force spreading out over a greater area" isn't right. The notion of forces being inversely proportional to the square of the distance is something that comes from optics and electromagnetism. We know that light power must diminish exactly as the square of the distance because it contains energy, energy is conserved, and the vacuum is completely transparent. But there is no such requirement for the gravitational "field" of the Sun. It doesn't emit energy. If gravity worked differently, Kepler's geometrical laws might have been different, and Newton very well might have made the force inversely proportional to r^{3}. He was fitting data; he knew nothing about divergences or the flatness of space.

Andy also makes the very valid point that a mathematical theory must obey physical observations, and, if it doesn't, that theory needs to be re-examined. So the question might well have been worded as:

- Do you think that is impossible that the correct behavior of gravity is that of Newtonian gravity, but with the gravitational force proportional to 1/r
^{2.00000001}, rather than 1/r^{2}?

We could make other similar questions:

- Do you think that is impossible that the formula F = ma (or E = mc
^{2}, or E = hν, or any other famous formula) is not correct?

But formulating the gravity question in terms of an exponent of 2.00000001 may not be the right way to formulate the question. It is well known that alternative exponents don't work. This was tried around 1900, to explain the anomalous perihelion shift of Mercury, but it didn't work. There is no one exponent that works for both Mercury and Earth. General Relativity is the accepted better explanation, though it is known that there are problems with that -- quantum gravity, cosmological constant, dark energy, and a peculiar behavior of the Voyager spacecraft.

Now one *could* ask whether the person is openminded enough to question the "experts" that have analyzed the Mercury data, and understand General Relativity, and have crunched the numbers. After all, very few people have crunched the numbers; they depend on perceived "experts". So, in view of question 13, this may be a valid question to ask:

- Do you think it impossible that General Relativity is not correct?

But I doubt that many people who doubt General Relativity believe that going back to Newton's theory with a different exponent is the way to go. Unless, of course, they have crunched those numbers.

PatrickD 00:17, 7 October 2009 (EDT)

- You raise some interesting points. To your question "Do you think that is impossible that the correct behavior of gravity is that of Newtonian gravity, but with the gravitational force proportional to 1/r
^{2.00000001}, rather than 1/r^{2}?", I might still answer no, on the grounds that if there is divergence in the gravitational field, then I wouldn't consider the force to be Newtonian gravity: you need dark energy, or an object creating a force from outside itself (possibly I'm wrong about this -- would you agree with this statement at least?) - I agree that a theory must fit observation, and some question along these lines would be fine -- it strikes me as odd to ask things about gravity being an inverse square law, since (at least as I understand physics), outside of the Newtonian context we don't really want think about gravity as a force with such and such strength at such and such distance anyway. And if we're talking about gravity not being an inverse-square law, then to me, we're no longer talking about Newtonian physics anyway. Some of your other suggested formulae I don't know the derivations of -- I bet there's another one that I'd prefer to use for this question.
- I also like your point about trusting experts in science. To some extent, this is necessary -- I'm not sure I have any personal reason to believe in the germ theory of disease, much less general relativity: I'm just taking someone's word for it. Obviously how much we should trust the experts on a particular subject depends on a lot of things. I bet some philosopher of science has thought hard about this -- I'd be interested to read the results! --MarkGall 00:33, 7 October 2009 (EDT) (hope I didn't forget to respond to anything!)
- Fantastic job on complex number, by the way.

## Pictures, etc.

(Now that things have quieted down to the point where I don't have a 75% failure rate when I try to view a page.)

Warning -- long, rambling message!

About gravity:

- ... if we're talking about gravity not being an inverse-square law, then to me, we're no longer talking about Newtonian physics anyway.

Yes, someone else had mentioned the point that Newton's law is Newton's law, and Newton said 2.0; case closed. You would have to call it something else. That's why my "straw man" question had been carefully worded "... the correct behavior of gravity is that of Newtonian gravity, but with ...".

The issue of trusting experts (I'm not about to reconstruct a few thousand years of science experiments in my basement; the Hubble telescope won't fit) is central to the whole question, and is what the "quantifying openmindedness" question should be aimed at. But I have no idea how to phrase that question in the right way.

About complex numbers: Thanks for the kind words. I need all the public support I can get.

I see that my picture request has gotten noticed by an admin (Jpatt). I had been about to ask you, here, to do what you could to get some action, since I'm staying strictly off of admins' talk pages. But Jpatt's "If he provides the pic link, no problem" comment shows that either he doesn't appreciate the difficulty, or else I don't appreciate what "pic link" means. I need to either mail the jpeg to someone (and the normal Conservapedia "email this user" feature doesn't take attachments) or I need to be given access to some web site, somewhere on the planet, that I can upload it to. And no, I'm not going to create a facebook account :-(

About the Riemann mapping theorem: Conformal map displays, that is, graphs of horizontal and vertical lines as mapped under analytic functions, are very pretty to look at. Can you outline for me what the square/circle function would be? I may have known it at one time, but I sure can't remember it now. It sort of goes crazy in the "corners", I believe.

As far as software to create it, I recall that William Beason had once named "winplot", something that I had never heard of. But I looked at the WP page for a random diagram (Argand diagrams from their complex number article, naturally!), and they have some nice pictures, with the usual explanation of how they got them. Answer: Matlab! For example, go to the "geometric interpretation of the operations" section of their "complex number" page. (I won't give the URL; I can't stand the captcha.) Click on a picture; look at the properties. You'll see the actual Matlab code. So we could do this sort of thing. But I don't have Matlab; it's hugely expensive. Further looking around shows that there's a free program from the FSF called GNU Octave. This might be worth checking out. That is, if either of us finds ourselves with an enormous excess of spare time :-) If this works out, we should be able to put something on the Tips for writing math and science articles page.

There's also a little graphing program I wrote for my own use that might be able to be pressed into service.

About the "imaginary number" fiasco. This guy needs to be watched carefully. Not with a "bot", but with human eyeballs. I think he's a fairly obvious parodist trying to get his jollies by singlehandedly creating another Bible translation / Stephen Colbert phenomenon. He has a zero chance of succeeding in any case; he won't get 2 million Google hits with that kind of garbage. I had planned to revert with an edit comment along the lines of "We have a higher standard of logic here; proving that there is no real number satisfying X doesn't show that there is no complex number satisfying X." His "proof" makes no sense anyway, even though it's a proof of something that can be shown in 1 or 2 lines.

Then I saw that you had jumped in, and figured that I ought to jump in with a redirect.

Oh, I see he's at it again, at 13:00 EDT. I'd rather not engage him personally; I'm keeping a low profile, especially on talk pages.

- Thanks for the tip about graphing. The usual way to construct conformal maps between polygons and a circle is using Schwartz-Christoffel mappings. Unfortunately this is a bit of a pain to compute, but I think the square <-> circle example is a particularly nice one for people who have seen this before, and might not be too impressed by circle <-> upper half plane or other such. I'll take a look at your suggestions when I have a chance -- I think I may have tried to do this in mathematica once as well. Unfortunately I don't have upload rights to help you, but I'll keep at that talk page.
- I've got my eye on Mr. Conservative Math Project too. I do not think he will be long for this site; I'll ban him if he posts any more nonsense. --MarkGall 13:38, 12 October 2009 (EDT)

## Userbox

I thought you might like this userbox. Template:User weapon--Duncan^{Channel 16} 21:37, 14 October 2009 (EDT)

- Thanks! It's a bit conflicted with the beliefs expressed in my other userbox, but I can appreciate the sentiment (: --MarkGall 21:56, 14 October 2009 (EDT)

## Great block

Great block and reversions this evening, Mark! Meanwhile, I'm going to see how Calculus is covered in your superb Majoring in Mathematics.--Andy Schlafly 23:04, 17 October 2009 (EDT)

- Thanks! I think calculus might not be in there at all yet, but it probably ought to be. Maybe there can be a section called "basic tools" or something at the top, dealing with calculus and linear algebra. The outline of the article is growing faster than I have time to fill it in! Please do add your new suggestions to the page, and I'll keep working whenever I have time. It's the middle of the semester now, and things are getting busy again. --MarkGall 23:14, 17 October 2009 (EDT)

## Good Bezout's theorem addition

Great addition of Bezout's theorem!

I wonder if we have enough on advanced calculus, though admittedly that is often handled in engineering rather than math departments. Perhaps it is folded into differential geometry in most math departments.--Andy Schlafly 15:37, 24 October 2009 (EDT)