User talk:GFauxPas/Archive1

Questions
Question please to PW veterans, is it necessary to state here that I used derivatives of a constant multiple, chain rule, and derivative of logarithmic function? How much do I need to spell out? It seems that the house style is to say everything out but that seems very wordy. I assume that the correct thing to do is to include them.


 * As it stands, the results you've cited are precisely those which are expected. Good job.
 * Might be worth looking at the eqn template. When that's used, the reasons behind each step can be included in the c column, so the implications flow from one line to the next without the justifications getting in the way.


 * I am not such a veteran on PW that I know what need, and what needn't be spelled out. Mind though, your proof only works for positive real $x$, as $y$ needs to be in the domain of the natural logarithm. Handling the cases $x=0$ and, in particular, $x$ negative, can be hard. The proof appears correct for positive real $x$, though.
 * Concerning the derivative theorems, I think it might be justified to create a page which gives the standard derivatives of some functions, and the chain rule (and formulae implied like product and quotient rule). But then, maybe this page already exists. --Lord_Farin 04:02, 3 November 2011 (CDT)
 * Good idea. It doesn't and it should. The work on differentiation work was done before we properly developed the technique of transclusion for bagging up related results into one page. As you suggest: Chain Rule, Product Rule, Quotient Rule, perhaps even Leibniz' Rule. Diffs of some standard functions in another page (or series of pages) again, perhaps bagging up trig functions in one page and inverse trig functions on another page. We can categorise and nest as much as we like. I won't do it now as I'm supposed to be at work (just waiting for a long download to finish).--prime mover 04:24, 3 November 2011 (CDT)

Ah silly me, I missed that restriction to positive x. Thank you very much for pointing that out, when I realize what kind of mistakes I make I'm less likely to make them in the future. I'll have to look at the eqn template. In any event, from looking at my syntax, I think I'm getting better at house style. --GFauxPas 05:54, 3 November 2011 (CDT)

Induction
I need more time to work on this and it's getting to unwieldy, so I'm taking it off for the time being while I learn more about induction etc. Thanks for your help Lord_Farin. --GFauxPas 16:42, 3 November 2011 (CDT)


 * Please take time, by the way, to study the format in which other pages have been written. It would be instructional for you to note the changes that have been made so far to what you have entered, and to take note for future entries. --prime mover 00:18, 18 October 2011 (CDT)

Okay thank you for the advice, I'm very new at this but I hope I can learn it --GFauxPas 06:04, 18 October 2011 (CDT)
 * So are many others. You'll pick it up. You'll be expected to. You're a mathematician. --prime mover 11:28, 18 October 2011 (CDT)

It seems a bit presumptuous to call myself a mathematician, I'm just a freshman math student D: but thanks for the compliment, I think? Question please: I have my own proof that came about in physics, an object travelling in a parabolic path will have the same magnitude of velocity at any two points with the same elevation, but will differ in sign. Mathematically, given a second order polynomial and a horizontal line intersecting it at two points, the derivatives at the two points of intersection will be equal in magnitude but differ in sign. Is this theorem-worthy? What is the name of this theorem if I want to type up a proof? What do I search for to see if it's already in the wiki? --GFauxPas 11:37, 18 October 2011 (CDT)
 * See those links on the left? See where it says "Proof Index"? I invite you to press it and explore. --prime mover 12:32, 18 October 2011 (CDT)

sin x over x
Your edits in Sandbox look promising! Graphic is superb. But we might want to rename the graphic once it has been finished to match the name of the proof it goes with. --prime mover 14:42, 23 September 2011 (CDT)


 * No problems with replying to my comment here by putting a response on my own talk page - but it is better to keep the conversation in one place by replying on the same page it starts. I make sure that all pages I edit are on my watchlist, so I am notified whenever a response is made on the same page I made the comment being replied to.


 * There's a mention of this issue on the main talk page, and I can see why it's been raised - if the conversation goes on a long time and spreads over several talk pages it gets difficult to follow it. --prime mover 16:51, 23 September 2011 (CDT)

How do I rename an uploaded file, do I have to reupload it with a new name?--GFauxPas 22:54, 24 September 2011 (CDT)


 * I believe you don't have the authorisation to rename pages. I've done the renaming of the file in question - see what's in the sandbox. --prime mover 03:19, 25 September 2011 (CDT)

Primemover, I can't find the accepted template for how to cite sources. How do I do it? Is there a standard way?--GFauxPas 17:19, 25 September 2011 (CDT)


 * There isn't a standard way of doing it. There's BookReference which references an entry in the Books section (but for that you need to have set up a page for the book in question), or there's just the technique of describing the entity and someone will go through and tidy it. For a link on the web just link to it. There are citation links to Planetmath and a couple of others. --prime mover 18:07, 25 September 2011 (CDT)

Convergence and other principles of analysis
Currently I am a teaching assistant for an advanced analysis course, so if you have any questions regarding real (multidimensional or not) analysis, feel free to drop a note on my talk page. When you eventually get there, I might be able to help out on Complex Analysis as well. --Lord_Farin 14:27, 23 October 2011 (CDT)

Awesome, thanks a lot --GFauxPas 14:33, 23 October 2011 (CDT)

Notation
I note from your front page you're setting up some copypasta for yourself. Before you go too far down that route, pls note the following:

1. The raw symbols ≡, · and Δ and so on are never used on ProofWiki. The $\LaTeX$ code is always used: $\equiv, \cdot, \Delta$ (or when appropriate $\triangle$ and its variants).

2. For "defined as" we use $:=$ as this is a specific symbol meaning "is defined as". The $\equiv$ symbol has plenty of other meanings and it is best kept for those.

Hope this is OK. --prime mover 16:29, 24 October 2011 (CDT)

Thank you for your insight prime.mover. Point 1 I was aware of and I just put it there for my own reference and for copy pasting when I don't have latex format available.For point 2, a notation is just a notation so I'm glad you pointed out that $:=$ is the house style. $\equiv$ is just what I've been using in my math notes. PW uses $\iff$ for the biconditional so that means that $\equiv$ is used for congruence in modular arithmetic? Thanks for the correction. --GFauxPas 20:04, 24 October 2011 (CDT)

Prime.mover, you expressed your concern elsewhere that there's not enough of a foundation on PW to have rigorous proofs of statement in pred.calc. Feel free to voice your suggestions here, this is the kind of proof I can make at my level of knowledge and I won't be offended at all if you want me to leave it in my user talk page for the time being. --GFauxPas 17:23, 3 November 2011 (CDT)
 * I'm going to have to do some research, but be aware that a) on this site we don't call $p \land p$ "multiplication" any more, that was a terminology that was introduced by Boole and we have since evolved from that stage. It's Definition:Conjunction now. b) What you call "tautology" is called the Rule of Idempotence on this site, as a taugology is usually referred to nowadays as something completely different. --prime mover 01:37, 4 November 2011 (CDT)


 * As for Leibniz's Law and the proof of equality being an equivalence relation, then I do believe there's a point to including it, if only for historical reasons. Having said that, I haven't a clue whether "Leibniz's Law" itself should be in the definition, axiom or proof namespace. Not having read the Tarski book (itself somewhat historical now) I'm not in a position to judge that one. --prime mover 17:24, 4 November 2011 (CDT)

A while back I wrote a proof that all integers are even or odd. The reason I wrote it was just to practice mathematical induction. The theorem itself seems rather unimportant, and so I don't see a reason to make a page for it, but at someone's request I can make it. --GFauxPas 15:56, 4 November 2011 (CDT)
 * I have a feeling that one might already be up, but I don't think it was proved by induction. Can't remember and I don't feel like looking at the moment. --prime mover 17:24, 4 November 2011 (CDT)

Lord_Farin, just wanted to let you know that I figured out your explanation and PW's approach here Talk:Fundamental_Theorem_of_Calculus/Alternative_Second_Part_Proof, thank you! --GFauxPas 22:19, 5 November 2011 (CDT)


 * Glad I put up sensible stuff. HTH --Lord_Farin 09:01, 6 November 2011 (CST)

Mean Value Theorem for Integrals
(proof has since been moved to here)

... and I forgot the rest of the proof. I have to think about it.

--GFauxPas 08:52, 6 November 2011 (CST)
 * How about Intermediate Value Theorem applied to $f(m)\le \dfrac1{b-a}\int \le f(M)$? Still, then, I think the proof can be shorter by leaving a lot of unnecessary stuff out. Also, personally I think Axiom:Law of Excluded Middle needn't be mentioned in any non-(first-order)-logic proof. --Lord_Farin 09:01, 6 November 2011 (CST)

Is $\delta x$ the same as $\Delta x$, only (arbitrarily) smaller in magnitude? --GFauxPas 16:25, 7 November 2011 (CST)
 * Both are a sloppy, IMO old-fashioned way of dealing with limit concepts. They are what one refers to as 'differentials' or 'variations' (compare the $\mathrm d x$). AFAIK mostly applied in mathematical physics these days. I prefer the rigorous $\epsilon$-$\delta$ approach vastly above this treatise. It can be made rigorous if I remember correctly, but such requires a firm basis in $\epsilon$-$\delta$ approach (or at least I had that when I heard of this, and still found it tough). --Lord_Farin 16:29, 7 November 2011 (CST)

Oh, they're epsilons\differentials. Well $\Delta x$ isn't a problem, it's just $x_2 - x_1$. I just saw $\delta x$ in the difference quotient and I've only been accustomed to $\Delta x$ := change in x, and $\mathrm dx$ = infinitely small change (informally, I know that doesn't make sense in $\R$). --GFauxPas 16:43, 7 November 2011 (CST)

Seeing as how we're using an uncommon notation for intervals to avoid ambiguity, what's the defining criterion for whether we use an atypical "better" notation or not? For example, the though the standard notation for function inverse is $f^{-1}$, that notation is the same as that of the multiplicative inverse, very different things. There's $f^\gets$ and $\breve f$ which aren't ambiguous, but they're not common. --GFauxPas 16:41, 8 November 2011 (CST)


 * Good question. There are many notations for intervals, and none of them are very good because it's easy to mistake two numbers separated by a comma for all sorts of other usages. $[a..b]$ is not common but it's completely unambiguous and has a precedent in computer languages, so I'm sort of expecting it to catch on. Getting mathematicians to change their notation, though, is not easy.
 * As for the inverse function notation, "generally speaking" you don't mistake $f^{-1}$ for a multiplicative inverse because the contexts are different. The $f^{\gets}$ notation has been noted on the page for inverse mapping so I suppose we could start using it, if you particularly like it. --prime mover 17:05, 8 November 2011 (CST)


 * I think I oppose to that. The notation for the interval isn't really ambiguous, even if you never saw it before, the meaning is clear. With $f^{\gets}$ I am having the hunch that it will create unnecessary fuss. But that's me, and probably an instantiation of the notation change thing... --Lord_Farin 17:09, 8 November 2011 (CST)

Lord_Farin, why did you change $\Delta x$ to $h$ and $\centerdot$ to $\times$ on the proof for Derivative of Cosine Function? I'm not objecting, but I'd like to know what the difference is. --GFauxPas 07:56, 10 November 2011 (CST)
 * I just liked to enforce consistency with Derivative of Sine Function. There is no conceptual difference; it's merely notation. I chose this notation as I have grown allergic to the use of $\Delta x$ in physics classes, where it almost surely meant some argument wasn't rigorous. Lastly, $\Delta x$ is independent of $x$, and when calling it $h$ instead, that fact is more obvious. Small other point: When asking me questions, the page User talk:Lord_Farin seems appropriate ;) --Lord_Farin 08:03, 10 November 2011 (CST)

Natural Logarithm
Hi. Just wanted to point out that in Derivative of Natural Logarithm Function, your equivalence statement doesn't hold. Compare $x_n = (-1)^n/n$. I think you'd best work with $u = \left|{ \dfrac {\Delta x}{x} }\right|$ instead. (This arises as $\Delta x$ isn't necessarily positive; if it is, one says a function is right-continuous). --Lord_Farin 10:28, 11 November 2011 (CST)


 * Thank you Lord_Farin, I'll work on it more, it's giving me a hard time. --GFauxPas 11:04, 11 November 2011 (CST)

Talk page etiquette
I hate having to keep calling you out like this, I sort of hope you will pick up the conventions as you see them being used.

However, it's probably worth mentioning the usual protocol in talk pages, when there's a conversation going on.

You may have noticed that when you start a line with a colon it acts like a tabulator, that is, indents the line by a fixed and noticeable amount. Starting with more colons indents the line further.

Anyway, in a talk page the convention is to start each reply with one colon more at the start of the line. In that way it can be easily worked out who said what, and when they said it, and what they are replying to. Only when the indentation becomes too much (usually about 7 or 8 colons in) is the conversation reverted to the far left hand side of the page again. Usually this does not happen on ProofWiki because issues are usually solved within two or three postings. --prime mover 12:34, 11 November 2011 (CST)
 * Noted. Sorry to disappoint. I'll cop out and play the "not sleeping enough" card. --GFauxPas 12:41, 11 November 2011 (CST)

Book referencing
What I said the other day about learning from examples.

Can you do the same with your book referencing? No doubt you have a copy of Tarski. Rather than just citing it the [terse] way you're doing it, can you stick it inside a BookReference template? Then it will lead you to perhaps put an entry together in the Books namespace. I can't do this myself because I don't have that Tarski book. --prime mover 13:55, 14 November 2011 (CST)
 * I know, I was planning on doing that as soon as I got home, when I put that in I didn't have the book on me. Next time I'll wait until I can do it all at once. Sorry I didn't know doing that "placeholder" method was frowned upon. --GFauxPas 14:02, 14 November 2011 (CST)


 * Nope you're all right. Nice one. What you might want to do, so as to save misunderstanding, is putting a call to the template in place, then we'd all know it's planned. --prime mover 15:41, 14 November 2011 (CST)

Real Number Line
Good job! A couple of things:

a) Tarski may well have been making unwarranted assumptions when he wrote this proof, and/or mathematics has moved on since his day. However, I believe it's not enough to just state that there exist mappings between the two sets for there to be a bijection between them - we need to show they are injective. After all:
 * $A = \{0, 1\}, B = \{2\}, f: A \to B: f(0) = 2, f(1) = 2, g: B \to A: g(2) = 0$

fits the conditions: $f$ and $g$ are both mappings, but $A$ and $B$ are definitely not equinumerous.

b) From the words put forward by Qedetc in the talk page, it appears that Tarski may have defined some axioms in order for assumptions on this page to be valid. Qedetc was insistent that no such mapping could possibly exist without such, but I was not able to pin him down on exactly what needed to be done. Can you read that book carefully (or do further research) to see whether there's something there about that? I don't have the information to hand.

c) When citing an entry in a book, try not to use page numbers, as different printings may cause these to change between editions. Please refer to the section / chapter instead. See the existing examples for a sample of how this works. --prime mover 04:30, 20 November 2011 (CST)


 * Understood, thank you prime.mover. I had a feeling that something like this was the case, but I gave up trying to follow the conversation on the talk page, I felt it was way above my head. I'll see if I can find something, and I'll try the talk page again. --GFauxPas 07:11, 20 November 2011 (CST)


 * I was in over my head too. In fact, I'm fairly sure Qedetc was in over his head as well. Probably so far over his head you couldn't see the bubbles. --prime mover 07:21, 20 November 2011 (CST)


 * I want to add that I refuse to believe all of analytic geometry is based on the axiom of choice. --GFauxPas 08:36, 20 November 2011 (CST)

Sizing of Images
When you are creating images, there is no need to create different versions depending on what size you want it. All you do in the code is add " px" to tell the browser how many pixels (high? wide? Don't know) to render the image.

In order to keep the images folder tidy (it can't be organized easily) I will update the cone page to use File:ConeVolumeProof.png and size it appropriately, then delete File:ConeVolumeProof2.png.

--prime mover 04:45, 20 November 2011 (CST)

Grammatical note
I see you starting lines with a capital letter where it does not need one. Here is an example:


 * The hyperbolic tangent function is defined on the complex numbers as:


 * $\tanh: X \to \C$:


 * $\displaystyle \tanh z := \dfrac {\sinh z}{\cosh z}$


 * Where $\sinh$ is the hyperbolic sine, $\cosh$ is the hyperbolic cosine, and $X = \{ z : z \in \C, \ \cosh z \ne 0 \}$."

The word "where" should not have a capital letter. The above is all (technically) one sentence, like:


 * "The best food is:
 * FISH AND CHIPS
 * where chips are made of deep-fried potato."

See? As "where" is part of the same sentence, it does not start with a capital letter.

I have been changing them consistently where I've seen them, hoping you'll pick it up by following examples, but now I see you changing one in the other direction, I have to mention it.

I understand that Microsoft make things complicated by automatically making the first letter after every new line / return start with a capital, but Microsoft are cracked.

Oh yes, and you either need "displaystyle" and "frac" in a LaTeX group, or "dfrac", you don't need both, as "dfrac" means "display style fraction". --prime mover 13:22, 23 November 2011 (CST)


 * Sorry to disappoint, for some things I learn quickly, for some things I'm a slow learner. Noted. --GFauxPas 13:25, 23 November 2011 (CST)