Tropers that do NOT hate math

[understood about 60% of that]

[thought he was good at math]

[feels shame]

[desire to know more intensifies]

[also totally made up that percentage out of nowhere]

edited 13th Oct '16 8:45:58 AM by SolipSchism

I also didn't understand all of that. I assume that's partly because I'm not thoroughly used to English Mathematical language. I've always seen integral and antiderivative as synonyms, and one of my Calculus teachers has expressed as dislike of the term antiderivative. Or, at the very least, their literal translations in Portuguese. We did see indefinite integrals, but again, going by literal translation, that was simply integrating integrating a function f without specifying the integration bounds in order to find F such that the derivative of F regarding (some variable) is f.

... Should we request La Te X support in the forums?

Ummm so basically the nature of these discontinuous functions are constant and their difference is represented by the equations with the letter thingies?

(Please forgive this troper for sounding like an idiot. He simply wishes to learn.)

edited 14th Oct '16 10:14:10 AM by Jamiester

Okay, to explain in a bit more detail, "indefinite integral" and "antiderivative" are not the same thing. There are theorems that show they're closed related in many circumstances, but their definitions are quite different.

First, let's define indefinite integrals. Given an interval [a, b] in the real line R, an integrable function g: [a, b] -> R (if you haven't seen this notation before, it just means the domain of g is [a, b] and the values of g are real numbers), and a number c in the interval [a, b], the indefinite integral of g based at c is the function G: [a, b] -> R defined by G(x) = (integral from c to x of g(t) dt) for each x in [a, b].

Note that the above definition makes sense for any integrable function g defined on an interval; an indefinite integral of g might not even be differentiable (for example, if g has a jump discontinuity). If we choose a different number d in place of c to start with, we get a different indefinite integral, but the two indefinite integrals only differ by a constant (more precisely, they differ by the integral from c to d of g(t) dt).

Now let's talk about antiderivatives. Given any subset S of R and a function g: S -> R, an antiderivative of g is a differentiable function G: S -> R such that g is the derivative of G. That's it!

Note that S doesn't have to be an interval, and g doesn't have to be integrable. If G_1 and G_2 are two antiderivatives of the same function g, then (G_1 - G_2)' = G_1' - G_2' = g - g = 0, so G_1 - G_2 has derivative zero. But this doesn't mean G_1 - G_2 has the same value everywhere — it just has to be constant locally, i.e., sufficiently close to each point in S. So if S is, say, the set of all real numbers other than zero, then G_1 - G_2 could have one value on all negative reals and a different value on all positive reals.

Here's how these concepts are related: by the fundamental theorem of calculus, if g: [a, b] -> R is a continuous function, then every indefinite integral of g is also an antiderivative of g. (But this isn't always true for discontinuous functions or functions whose domain isn't an interval.)

edited 14th Oct '16 10:21:09 PM by Enthryn

Oooh, I see it now. Can't forget the fundamental theorem of Calculus only applies to continuous functions.

By the way, in your notation, [a,b] also includes (-inf, +inf), right? Or R, I suppose. I'd guess (-inf, +inf) can be used to denote R and vice-versa, within context.

@Unrelated: I was checking the Dota 2 subreddit and found a Microsoft report on their True Skill algorithm. I haven't studied Probability yet, so I don't understand the paper, but it could still be fun for someone who has. Also, I get the feeling I might like probability when I get to study it.

Oh, yeah, you can glue together the statement for bounded intervals to also get unbounded intervals (such as all of R). The key point is that, for indefinite integrals, the domain has to be path-connected, i.e., there's a continuous path connecting any two points in the domain.

So, convex?

"Convex" and "path-connected" are the same in R, but not in higher dimensions — the paths connecting two points can be curved, not just line segments. For example, the interior of a non-convex polygon in R^2 is path-connected, but not convex. To give another example, every individual letter in the word "convex" is path-connected, but not convex. However, what I said earlier was all just for the 1-dimensional case anyway. (Remember, we're talking about what the domain looks like, not the graph of the function.)

Integration is a bit more subtle in higher dimensions: path integrals can depend on the path, not just on the endpoints, so you can't naively define "indefinite integrals" in the same way. However, trying to do so in a certain setting — Riemann surfaces, which are certain 2-dimensional surfaces arising from the theory of functions on the complex numbers — leads to the theory of Jacobian varieties, which is a beautiful and complex (pun totally intended) subject in its own right. As a special case of this, trying to define a good notion of "indefinite integral" for cubic equations in the plane leads to the theory of elliptic curves, a major area of study in modern number theory and the subject of many famous conjectures, including the Birch and Swinnerton-Dyer conjecture, which has a million dollar prize attached to it.

edited 15th Oct '16 1:57:16 PM by Enthryn

Posted this in the "Things you didn't know until very recently" thread, but yesterday I learned about Kepler's First Law of Planetary Motion. It wasn't actually subject material for my Physics class; the discussion question for the week was about the difference between a scientific law, theory, and hypothesis, and we had to do a bit of research, pick a law or theory or hypothesis, describe it, and say whether we thought the term law or theory or hypothesis was appropriate and why.

Since Ohm's Law, conservational laws, and the laws of thermodynamics were already taken, I just read Wikipedia's list of scientific laws and picked the first one that seemed interesting.

I already knew that orbits are elliptical, which is stated by the law, but I did not know the other part of it, which is that the center of gravity for an orbit will be at one of the ellipse's two foci.

(I also didn't consciously realize, although I would have if I'd thought about it, that the center of gravity for an object orbiting a much more massive object (like the Earth around the Sun) is not in fact the center of the larger object—it's a point somewhere between the centers of those two objects. Which makes sense; a binary star system is orbiting a point between the two of them, so if you gradually shrink or grow one of the objects, that point is going to slide toward the center of the more massive one, while never quite reaching the center. So the center of gravity for the Earth is inside the Sun, but it's not actually the center of the Sun. Earth doesn't technically orbit the Sun, per se—it and the Sun both orbit a point slightly off of the Sun's center. Which means the Sun wobbles around a bit (a very, very small bit) as a result of the planets orbiting it.)

God, the universe is fascinating.

One of my teachers last semester wanted us to prove that, except I had no idea what he was doing for the most part of that "minicourse" (the final one month and a half of classes). Now, in Calculus II, our teacher finally explained the deduction of Kepler's First Law of Motion... Except he explained it before explaining differential equations @_@ And I was very sleepy during those classes >_< At least I understood it slightly better now, and have most of it written down.

Related: I'm seeing center of mass in Physics currently. Though I think we'll only study it regarding finite sets of points. That also was covered by the book of that "minicourse" I mentioned in the last paragraph, except I don't even remember my teacher mentioning it. My Calculus I teacher did mention in one class that for... uh, perhaps continuous objects, the center of mass is the intersection on n hyperplanes (where n is the dimension the object is in). Each hyperplane only needs to divide the object into two halves with the same mass. He didn't prove it because that wasn't his goal, but said that you could try doing the following: suppose that you found the point that is the intersection of n hyperplanes. Suppose that you can obtain an additional hyperplane that also divides the object in two halves with the same mass but doesn't contain that point. And from that you try to obtain a proof ad absurdum.

Well, I mean, the basic idea of Kepler's First Law is comprehensible with a fairly basic knowledge of geometry (you just have to know the definition of an ellipse, i.e., what foci are). That level of understanding doesn't allow you to do much with it, but then, that wasn't the point of me learning about it, so. :p

Kepler's Second Law actually kind of seems even more interesting: That a planet's area velocity is always constant. That is, for any equivalent period of time, the area bounded by a line from the planet to its center of gravity at the beginning and end of that period of time is constant. I had never even heard the term "area velocity" before but that's fucking fascinating.

Edit: Wait, Second. Kepler's Second Law.

edited 20th Oct '16 8:39:14 AM by SolipSchism

Actually, the Second Law also follows from the deduction I mentioned in my last post. It's an integration exercise, basically. Polar integration ;)

It's been a year since I finished Calculus...

I wasn't prepared.

I'm suffering through Calculus 3 currently. I gotta admit, I haven't been studying nearly enough to properly learn it, but apparently I don't need to "learn" learn everything my teacher is talking about because you'd need the tools of Analysis for that. So I was kind of focusing on the wrong topics, and feeling unmotivated to study them >_<

In the past three classes he covered differential forms, and I think he finished proving "Stoke's" Theorem for alternating (k - 1)-linear forms in R^n today.

Fun stuff. Stokes' theorem is nice, especially how it unifies and generalizes all those weird special cases like Green's theorem and the divergence theorem. Out of curiosity, what book are you using?

At the very least I learned how to prove those theorems using the (one dimensional) Fundamental Theorem of Calculus. Once I finally understood that they were all corollaries of the FTC.,I found that so cool X3

But now he's using other tools to talk about unifying those theorems, and I'm lost. It doesn't help I haven't been understanding the physical applications he was talking about in the past month or so. Usually I don't care too much about the physical intuition, but I had none of it, so they were hard to follow. But again, maybe I wasn't supposed to understand everything he was talking about because one'd need the tools of Analysis to actually do so (rigorously), and I haven't officially studied Analysis yet.

The book we're using is, well, the professor's. Análise Vetorial Clássica (Classical Vectorial Analysis). He also, as usual, recommended Courant, as his bible. Or, if you're a physicist, uh, Apostol, I think? Stewart too, probably. Tromba, at least for the first month, I think.

By the way, I don't see any mention of coffee in your Troper Page

It's not specifically math, but I just finished BA 212, Financial Accounting 2, with a 98.26%, which just feels like further confirmation that I've finally found the right major. Finished PH 203, Intro to Physics (which is harder than a class with "Intro" in its name would suggest), with an A as well, which is a pretty math-heavy class, mostly algebra with a bit of trig and geometry, but very little calculus. Feels good.

Where has this thread been all my life? I love math! I lean more towards the pure and theoretical because I love how it ties into art and philosophy

Oh, I haven't posted here in forever, have I? It was one of the first forum threads I ever did, though, I think.

I still really like math, although for Reasons I've been out of school and thus math classes for a while. I've still been poking into maths-y things for fun, though, videos and stuff.

And occasionally messing around with numbers on paper for fun. Nothing very difficult or clever, but not quite on the "Working out the statistics of rolling four six-sided dice and adding the three highest, by hand" level of tedium, either.

I'm still kinda proud of having done that, even though it was just busy work with no real grasp of statistics involved. (I worked out each group of similar combinations, wrote the number of combinations in the group, and multiplied that by the number of permutations per combination, and tallied those all up under their result. So, 18 broke down to 6666 [1 x 1] and 666(5-1) [5 x 4] for 21/1296, and so on.)

I'm certain there are much better ways of doing that by far, but I found some strange enjoyment in doing it like a spreadsheet. Still, if I could change what I took in high school, I would add Statistics, among other things.

(Also, hello there and welcome. Things here tend to be quiet, in my experience. Also also, nice avatar.)

Math was always one of my favorite subjects because it made objective sense.

@Cailleach: What do you find most beautiful in math?

@Raichu KFM: I've learned the quicker way of doing that in Prob class this semester. In fact, it was one of the questions in the final exam (not about six-sided dice particularly, but the theory covers that particular case). I've also been thinking about making my own RPG system, so I've been using an online tool to do what you did with a spreadsheet XD It doesn't do everything I wish it did (namely vector-valued random variables) but it does make graphs for the stuff it does. Not the prettiest graphs, but unless I stop being lazy and actually do something myself they'll have to make do :P

@Nith_Strike: *insert joke about Gödel's Incompleteness Theorem here*

edited 28th Jun '17 7:49:04 PM by Victin

i have become quite fascinated with Googology and large numbers lately

some of my favorites are Loader's Number, TREE(3), SCG(13), Marioplex, Minecraftplex and the Poincaré Recurrence Time for the universe

there's just something fascinating about numbers so large they can't fit in our universe

It's ME

Mr. Shuri hiimself

I like algebra but I hate geometry. I’m currently in geometry right now and the last time I got an B- on a test was in January when we went remote for two weeks because of the omicron. Last semester, I got an A-, B-, C-, D, and C-. This semester, I got a B-, F, and F. I suspect the bad grades came from this student assistant the teacher has who I tried to date last year and was mad at me because I called her boyfriend, so she intentionally gave me bad grades.

I have a feeling that this thread will stay inactive, but I have to say this.

After over more than a decade of getting frustrated by and giving up on mathematics...I actually fell back in love with math and decided to self-study it.

My fascination and appreciation of mathematics was resparked by three factors.

One is the AI craze that has been going on for a few years by now, and the fact it being pretty much one of the pinnacles of applied mathematics.

The second is my realization/acceptance that in current world full of uncertainty and vagueness, firmness and certainty of mathematics (unless we are talking about super advanced and abstract math, that is) is rather comforting and admirable.

Lastly, my desire to understand the natural world and sciences in general have sparked in a massive explosion...and we all know how math is the foundation of all sciences.

At the beginning of this year, I still had no idea exactly where I need to start. Thankfully, now I know that I should focus on algebra, geometry, statistics, and most important of them all, the goddamn calculus.

I will probably spend about five years self-studying mathematics until I can grasp basic undergraduate level (and not even senior, just around freshman/sophomore level) of the major fields, before moving on to the subject of physics, which is more or less a form of applied mathematics.