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Diff. Geometry Question 
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#1
Sep2408, 06:43 AM

P: 76

Hi everyone,
Perhaps this should go in the career guidance section, but I had a question directly related to differential geometry. I'm in Year 10 in Australia, and I'm very, very passionate about theoretical physics (perhaps mathematical physics too). String theory interests me, and I'm aware of such things as CalabiYau manifolds, but I wanted to ask about the subject of differential geometry itself. What is the subject like? How enjoyable is it and how does it compare to all other areas of mathematics that people have studied (whether it be pure or applied)? It looks interesting from what I've seen, but I have only heard about it from a physics point of view. Just to give everyone a background, I do a topcourse year 10 mathematics course, which I am doing very well in, and I also do Mathematics Methods 4B (this is a 100hour, pure maths course one step down from pretertiary level pure mathematics), however I didn't do so well with cubics (made up lost ground with exponential functions and logs though, no test on those yet, however...). Also for note, my least favourite aspect of mathematics is statistics, and my absolute favourite would have to be geometry (I've done quite well here too, with marks like B+'s etc). So, what is diff. geometry like as a subject (perhaps, even if I don't understand it, an example problem might be useful just to showcase it)? I am guessing the required subjects to study diff. geometry would be topics like multivariable calculus, partial diff. equations (?) and some abstract and linear algebra? Is the path leading to diff. geometry more fun that diff. geometry itself? And how fun is it as a subject? I'm really deeply curious about many physics concepts relating to the universe (like particle physics, string theory and the like), and so, is this the mathematics underlying many of these models (if it's just string theory, that is fine too)? What about algebraic geometry? How does this compare? Is this fun, or perhaps more fun that diff. geometry? I know one of the above is more useful for relativity and one for string theory, but I can never really remember which...! Cheers Davin 


#2
Sep2508, 03:38 PM

Sci Advisor
HW Helper
P: 9,495

at the website
http://www.math.uga.edu/dept_members/faculty.html there are free differential geometry notes on shifrin's page, and free algebraic geometry notes on smith's page. 


#3
Sep2708, 04:07 AM

P: 76

I like the idea of algebraic/differential geometry, as geometry is my favourite branch of mathematics. How close are these subjects to traditional geometry that is studied at undergraduate level, or, in particular, at an earlier stage than this (say, years 1012)? As for which is more fun/more elegant/better to study for theoretical (or mathematical) physics, which would this be? I really love physics, I just need to work really hard on the mathematics, although, admittedly, I'm not doing badly at all (but, conversely, not the best either). I am only human, not some hugely talented genius or prodigy! Also, if I do an Analysis course (covering Hilbert spaces, Banach spaces etc) when I'm in my final year of undergraduate study, would this be algebraic geometry or different? I'd like to see someone pour their soul into differential geometry or algebraic geometry on these pages, if this is what they find to be their passion. Do express it, because I honestly want to hear the opinions of the mathematical community. Who enjoys what, and why? How pure or elegant is the work you do? How difficult? How much fun is the subject? How deeply linked to the structure of mathematics/physics is the subject you study (or a subcomponent of it)? Does the subject have relevance outside of mathematics, or is it truly pure? How about the subject's relationship with physics? Where is it used, and how? How deeply tied with fundamental theories of the universe (although incomplete, things like string theory and quantum field theory, despite them being unproven) is your subject? Do these theories inspire you in your work, or are you only interested in the subject in a purely mathematical sense? Also, what's your favourite phenomena or relationship in these subjects? Why is the particular subject or branch of mathematics more appealing to you than another branch of mathematics? Also, how about your interest in pure vs applied mathematics? These types of questions I'd like to hear answered, so let's see some passion flowing through these pages! Davin 


#4
Sep2708, 04:38 AM

P: 76

Diff. Geometry Question
Sorry for the double post, but I had a few more questions:
How rich (and deep) is the history of the branch of mathematics you are interested in? What advice could people give (be it regarding mathematics, or physics) for someone interested in studying theoretical/mathematical physics? I will be doing a combined degree at the University of Tasmania (in Australia) in Computing and Science. My majors will likely be Physics and Applied (or Pure) Mathematics, and I'll be doing mainly software engineering for the computing component. This gives me a "safety net" for careers, and also would be pretty useful. I might be interested in doing some experimental physics on the side, but I am not yet sure. Which type of mathematics is more relevant for theoretical physics? Any inspiring words for either type of mathematics? How do they pan out at college/university (I refer here to the American meaning rather than the Australian one, where a college is often year 11/12)? I'd love to be able to do Honours in Theoretical Physics (here in Australia, that is a further year of study, after completion of a bachelor's degree, that has a focus on both research and coursework  so it becomes more specialised). The requirement is achieving at a certain standard in the bachelor's degree, and then achieving well enough in a honours course means that you can likely pursue a PhD (if you have the funding and academic ability etc). What I'm asking is not for an fight about the best branch of mathematics (here I'm mainly interested in differential/algebraic geometry), but an intelligent, reasoned, respectful and passionate debate about these fields of mathematics! Let's bring it on! Davin 


#5
Sep2808, 12:46 PM

P: 78

I'm a computer programmer who plays with math and physics as a hobby, so take this response with as much salt as you like.
If you want a very simple "sound bite" characterization of that which differential geometry most closely resembles, it's calculus on manifolds (but there is more to it than that). I would claim its "feel" is much closer to calculus and analysis than it is to geometry. A manifold is a generalization of ordinary Euclidean space to something which only "looks like" Euclidean space in small regions. Globally, a manifold may be "bent" or perforated or stuck to itself in odd ways, but "locally" it resembles R^{n}. This leads to a problem, which is that in general it is impossible to define a single coordinate system on an entire manifold. Under conditions where you cannot define a global coordinate system, and hence any nonlocal operation must be "pieced together" from sections done with different coordinate systems, how do you define an integral or a derivative? One might claim that that is the fundamental problem of calculus on manifolds, and I would go on to claim that calculus on manifolds is the "starting point" for differential geometry. The general solution to the problems presented by manifolds, which have many over lapping coordinate systems, is to find objects which are invariant under a change of coordinate system. Such objects are tensors. Thus, differential geometry also includes tensor calculus. I will now go even farther out on a limb by claiming that algebraic geometry (which I have not studied) brings techniques from advanced algebra to bear on geometric problems through use of category theory, in which one maps objects from one field (geometry) onto objects in another field (algebra) in an effort to answer questions about the first. But I may very well get shot down on this. In all areas of physics, differential equations, partial and total, are endemic. General relativity is based firmly on differential geometry; you cannot get anywhere in GR with a decent grounding in the basics of tensor calculus. Special relativity makes heavy use of algebra but isn't so strongly tied to general tensor calculus. Quantum mechanics is off in another part of the forest; tensors come into it but a good grasp of techniques from analysis is also called for. Fourier transforms might be said to form the bedrock of QM. Solid state physics, which is what ultimately makes your computer go, uses a little of everything, as I recall; it includes E&M, QM, a little relativity, and of course Fourier transforms. From your message it's hard to tell what you're studying. Nobody outside Australia is going to have a clue what a "year 10 mathematics course" is down there, and I sure don't  what's it cover? Anyhow my guess it that you are currently precalculus. So, the things you probably need to study hardest in are a) Calculus (including basic analysis and the basics of Fourier theory) b) Algebra  I mean college algebra, aka "Calculus III", not high school algebra. This includes linear algebra, group theory, rings, fields, symmetry, bilinear forms, and various other items. Effort invested in really understanding algebra will pay off no matter what field you end up in; it forms the underpinnings of an astonishing array of areas. Many parts of it are every bit as "visual" and comprehensible as basic calculus and plane geometry, but in general it takes more effort to obtain a level of understanding which allows you to visualize results in algebra than it does in some other areas. c) If and when you're solid on algebra and you've mastered multivariate calculus, that will be time to start thinking seriously about tensors. But tensor calculus combines algebra in N dimensions with calculus, so you need (a) and (b) before you can understand (c). d) Differential geometry comes *here* on the list. 


#6
Sep2908, 12:00 AM

P: 76

Hi Sal,
I wrote a lengthy reply to this, but unfortunately the server lost it for me... grr! It is interesting to hear that diff. geometry is almost unrelated to high school geometry, and is closer to calculus of manifolds and analysis. I am very ignorant of analysis, and so I would love to hear what it is all about (all I know is that the subject has a lot of relevance to theoretical physics). As for this fundamental problem of calculus on manifolds, that sounds interesting. I was certainly not aware that this is where tensors are from. Now, regarding tensors, are they as scary as many people say? Or is this a percieved image with no realism to it? Mathwonk states in his paper on algebraic geometry (that I briefly skimmed through) that this subject is related to the study of geometry of polynomials, or as he now likes to think, the study of geometry of rings (or was it groups?). This sounds interesting, but as with a large chunk of higher mathematics, I am again very ignorant of the subject. I am aware that groups and rings are important concepts in abstract algebra, but that is all. This shows my ignorance of mathematics! I will have a closer look at solid state physics, as it is an area of theoretical physics that I have not looked at. I think, at my age, it would be better for me to see the "big picture", perhaps, something that I haven't been doing. I am guessing, because higherlevel geometry and analysis are "pure" branches of mathematics, they are on a completely different level, than say, partial diff. eqn's? Is it not Fast Fourier Transforms (or FFTs) that govern the rendering of computer graphics from the graphics pipeline? Is it correct that Fourier transforms are essentially methods to solve differential equations? Now, as for my mathematics courses, I wrote a huge list of the topics covered in the post that I lost, so if you are after further details (as I wrote in my earlier post) then just reply to me. I do two courses, my topcourse year ten mathematics course, and my "mathematics methods 4b" course, which I will refer to as "extended maths", is a course on pure mathematics, one step below pretertiary level (the 4 indicates it is one step below, as 5 is for pretertiary, and the "b" refers to the length [100 hours], whereas a standard pretertiary subject here is denoted as 5C [150 hours long]). In my regular mathematics course, these are the subjects (that I have studied, or in the process of studying, so far): Trigonometry Financial Mathematics Geometry Statistics Trigonometry covered nonrightangled triangles and problems dealing with bearings. Financial Mathematics is simply simple and compound interest. Geometry covered threedimensional shapes, with concepts such as volume and total surface area (it also included questions such as: "Given these dimensions of a wine glass [cylinderical, with a hemispherical bottom], find to what height, you must pour the wine, to fill the wineglass to exactly half of it's volume"). It was these types of questions that I liked solving, and as a sidenote, geometry is my favourite branch of mathematics so far. As for my extended maths class, the following is a list of subjects that I have covered, or yet to cover: Algebraic Manipulation Linear Equations Quadratic Equations Cubic Equations Division of Polynomials Simultaneous Equations Indicial Equations Exponential and Logarithmic Functions Differential Calculus Theoretical, and Experimental Probability I am currently working with logarithmic functions, and so calculus is up next! I have found I am doing much better currently, with logarithms, than I did with cubics (probably my weakest mathematics topic). I really hope to master the mathematics I am taught, as often I find, say the algebra topics (like cubics and simultaneous equations) to be less intuitive to my thinking than geometry, for example. So, at the moment, I am dealing with precalculus topics, but it won't be for too much longer (I'm going to do pretertiary Mathematics Methods next year). This is essentially a calculus course (this is all that's required for all of the maths/physics units at the University I will be attending, excepting for the advanced calculus and applications units offered in firstyear of an undergrad programme). If I do well enough with this, I will do the pretertiary Mathematics Specialised course in Year 12 (the hardest high school mathematics course available). This course covers calculus, series, sequences, convergence and divergence of functions and linear algebra, and so this course would give me a huge advantage for my degree. Take note that these courses are generally taught by people who have a really solid mathematics grounding (ie a degree in it, most commonly), as opposed to what I've heard about the American AP Calculus courses (I have heard rumours that they are often taught by incompetent teachers. Is this true?). You mention that it generally takes more effort to visualise concepts in algebra than it does in calculus and geometry. I have (clearly) no experience with calculus yet, and so I can't talk about that, but I feel that, as I stated earlier in the post, I find geometry concepts more intuitive to my style of thinking than algebra concepts. I always like to visualise concepts, and this is far more difficult with algebra. What do people on this forum find about their intuition when dealing with higher mathematics? Do some of you grasp algebraic concepts better than calculus concepts? Or geometry concepts? Do you have no real, *deep* intuition regarding the topic you spend the most time with? To succeed in an area of higher mathematics, how good must your level of intuition be? I would like to hear about this. I am looking forward to the journey through this material, and particularly looking forward to becoming an undergraduate (in a little more than 2 years)! Also, a few questions regarding analysis, before I finish writing this post. How intuitive is this field? Does it throw you in the deep end? And how elegant is the field? At my university, as an undergraduate I can study these two courses: "Analysis 3" and "Topics in Advanced Mathematics". The "Topics in Advanced Mathematics" course covers various topics such as geometry, set theory, number theory, history of mathematics and a few others. If I'm not mistaken, you are able to choose the area of the list of topics provided that you wish to study. The course itself only requires that you have studied a secondyear mathematics unit. Perhaps if I can choose, I wold be wise to choose to study algebraic and differential geometry? As background, I will have had grounding in secondyear abstract algebra, second year calculus, second year differential equations, and secondyear linear algebra. Differential equations and linear algebra are covered in the same course, so I would be guessing that it is structured as half/half. The university/college I plan to attend is the University of Tasmania (aka UTAS). I am particularly looking forward to hearing about a mathematician's intuition, and would love to hear my other questions answered! Thanks for reading my post, and for your time! Cheers, Davin 


#7
Sep2908, 12:02 PM

P: 78

I talk too much, and this post has gone 'way overboard. I won't be responding in this thread again, unless some very specific question (with a short answer!) comes up.
"done over again, right". But I suspect the term may cover quite a bit more than that depending on where you are and who's using it. Real (as opposed to complex) analysis is calculus in R^{n}, with everything proven rigorously as you go; epsilonics are the order of the day. A few things are a little hard to grasp at first, such as the definition of continuity which may seem to have no apparent relation to what "continuity" means in calculus1 (a function is continuous if and only if the inverse image of any open set is open). Typically one doesn't go nearly as far into techniques for solving particular integrals and such in an analysis class, as one does in basic calculus; the issue is more with definitions than with solutions to numeric problems. (Some schools kind of merge calculus and analysis. I think Harvard may do it that way.) calculus has a clue what a tensor is should be aware of. 1) You should get through calculus, linear algebra, and all the other things which come up in a good twosemester postcalculus algebra course before you worry much about tensors. (A halfsemester of linear algebra is just an introduction, please note.) 2) Very few people who start out doublemajoring in math and physics actually get two degrees, one in math, and one in physics. Don't bet the (psychological) farm on getting a double degree; odds are it won't work out that way. The reason (2) is true is mostly because it's a wide world and it's hard to see what the possibilities are from the vantage point of a high school classroom. Math and physics stand out, everybody knows about them, they're really "techie", they have a certain caché, and a lot of people think it would be really "cool" to be in both fields. Maybe they seem like "ritzier" fields than some pedestrian old engineering discipline, for instance. And so lots of folks enter university thinking they're going to be math/physics majors all the way. But when you get to university you find, first, that there are an awful lot of fields with interesting work, and with extremely challenging work  theoretical ("pure") math and physics have no monopoly on being difficult to master or having cool, sophisticated mathematical underpinnings! And, second, you learn a fact about the world: A degree in theoretical math or theoretical physics is excellend preparation for ... well, teaching other people to do it, at the university level, if you happen to be brilliant and if you find a tenure slot somewhere. If you can't find a tenure track post at a university, you can always, um, paint houses or something. To put it bluntly, it's nice work ... if you can get it. difficult, complex ... and not what I would call theoretical physics, at least not in the same sense that, say, astrophysics is "theoretical". In fact, I would describe it as "Applied" with a capital A. As I said, it's solid state physics which makes your computer go. If you don't know what I mean when I say that, then you definitely have something to learn about solid state physics, and about electrical engineering, as well. Entering an applied field is not "death", and what's more there's more demand for people who work on things which have applications than for people who don't. answer that. same thing. Fourier transforms move a function from configuration space (where it has, like, values and stuff) into phase space (where you get to see the frequency spectrum). They are used in quantum mechanics, where each quantum state could be said to be one component of the Fourier transform, and they're used in signal processing, because the transform of an acoustic signal is exactly its frequency spectrum, and the transform of an image is its spactial frequency spectrum. But AFAIK polygons get blasted into the frame buffer in your graphics card with no use of Fourier transforms. advance. I'll start with a little background, and I expect it'll rapidly become obvious what I'm leading up to. When I was an undergrad I took a semester of organic chem, and I really loved the "carbohydrate game", as they called it in that course, which was looking at how carbohydrate molecules could be mushed around from one form to another. And when we finished that unit of the course, I asked about further information or opportunities to study that sort of thing, because it was far and away the coolest bit of chemistry I'd ever encountered ... and they told me "There isn't any more. That was all of it. That field is dead, it's been mined out, nobody's studying it any more." I took a semester of point set topology, and I thought it was just the coolest thing ever. The textbook (Munkres) was more fun to read than a dime store novel, and the "mental pictures" it all produced were fabulous. Talka about "visualizeable"  it practically defined that word! At the end of the course, I asked about the next level of point set topology  what was the followon course? For surely, I wanted to take it! ... and they told me, "There isn't any more. That was all of it. That field is dead, it's been mined out, nobody's studying it any more." The "followon" to point set topology was algebraic topology, which bears about as much resemblance to point set topology as Mandarin Chinese bears to Fortran. Now, geometry.... Let me put it this way. I majored in math at a good school, and in all my college years, I never took a geometry course, in the sense that you mean it; what's more I'm not sure any were offered at my school (aside from the differential/algebraic sort). In fact, the only geometry class I've ever taken in my life was a high school class called "coordinate geometry", and it didn't bear much resemblance to the plane or solid geometry classes of yesteryear. I started to read a bit about plane and solid geometry last year, just for fun, from a book (published by Dover) in which the author talks about the stuff "they never taught you in school" because, for the most part ... geometry isn't even taught any more. Calculus has almost entirely superseded it. On the bright side, there are a lot of other fields in math which are just as visual, and just as pleasing. The downside is that, for the most part, you must work harder to visualize things when you're not talking about the actual geometry of the universe we (appear to) live in. faculty in my high school was variable, some better than others; however, the factulty were generally a whole lot better at college. Since you've probably never encountered much in the way of serious proofs before, and you've almost certainly never been called upon to prove much of anything, it's likely to be difficult to start with, but an introductory analysis class is geared to people who are all going through the same struggle, so it's not really a problem. The first analysis test I recall taking had ten questions, and each one was just a statement, no questionmark, no "please do...", no nothing. Just a statement. And for each one, we were expected to either prove it or find a counterexample. the time you pick. It's too early to try to make a decision like that; wait until you've gotten through calculus and algebra! Quelle bizarre! 


#8
Sep2908, 11:43 PM

P: 76

Hi Sal,
This is fair enough. Just wanted to thank you for the great (and informative!) reply. You note that there aren't many jobs in theoretical physics  part of the reason why I'd want to do a computer science degree also (a combined degree with science [maths/physics] if that is what I end up doing!). So by the gist of it, analysis is essentially a course of proofs? Are proofs tedious or enjoyable (no, I admit, the only sort of "proofs" I have seen or dealt with is, say, the derivation of the quadratic formula)? You say  don't bet the (psychological) farm on getting a double degree. What do you mean by this? What is a psychological farm??? I think, where you say I am getting too far ahead of myself, I think there is reason. I always like to know about, I suppose, the end goal (and the journey itself) because if I think I'm not going to enjoy it (or possibly hate it), then I'm not likely going to pursue that subject, if I can avoid it! That's my reasoning anyway. I'll look into solid state physics, as I said earlier, but perhaps I had better wrap up here before I start rambling on! Cheers, Davin 


#9
Sep3008, 08:13 AM

P: 78

of arguments that start out being "visual"  proofbypicture. A good proof is often called "elegant", "beautiful", or "lovely", which gives you a good idea of how mathematicians feel about them. The derivation of the quadratic formula actually isn't all that nice a proof, in my opinion, because it's not visual (at least, I can't visualize it). Many far more abstract theorems have far more easily visualized proofs. As another example of a proof of something rather basic, have you seen the "Aha" proof of Pythagoras's theorem? I've got a bit on it, here: http://physicsinsights.org/pythagoras2.html But if it turns out later that what really grabs your attention is, say, material science, or computer programming, or some other field you didn't plan on, go with it, and don't feel like you've "sold out". In other words, don't be disappointed in yourself if you end up going in some other direction later on. ******************** I said I would shut up, but I can't resist talking some more. Here's an example of something simple which can be shown with geometry, as I learned on my own, or can be done with calculus, as I learned in school. It may also give you some idea of why I might claim calculus is "geometry on steroids". The example is the volume of a sphere. Using simple geometry, and in particular Pythagoras's theorem, we can derive a reasonable value for pi, as we show here: http://physicsinsights.org/pi_from_pythagoras1.html By packing three skew pyramids into a cube, and using a simple and intuitive (but not rigorous) argument to show that the shape of the base of the pyramid doesn't affect the volume we can show the volume of a pyramid  or a cone  is [tex]{1 \over 3}Base \cdot Height[/tex]: http://physicsinsights.org/pyramids1.html Finally, we can use a very simple visual argument to find the area of a circle, and by subtracting a sphere from a cube to obtain a cone and using the formula for the volume of a cone which we just found (above), we can obtain the formula for the volume of a sphere: http://physicsinsights.org/spherevolume1.html Now, let's do that over again using calculus. We start by observing that the circumference of a circle is presumably going to be a constant multiple of its diameter, and we call the ratio of the radius to the diameter "pi". We'll find out what "pi" is shortly. Arrg No we won't. The Tex code got too thick and blew the server's mind when I tried to preview it  "Database error, please reload and try again...". There is apparently a rather small limit on the number of "tex" sections one can have in a post, and blabber that I am I run into it from time to time. All attempts at entering the calculus formula have failed this morning, so I'll skip it. Let it suffice to say that, using calculus, we can solve exactly the same problem far more succinctly, though not quite so visually. 


#10
Sep3008, 10:16 AM

P: 152

Hey! I'm sixteen too. Differential Geometry would also require a bit of Topology, Set Theory and Tensor Analysis. These were enjoyable for me but what interests me more is Complex Analysis and Fourier Analysis. The subjects in mathematics are interrelated, so you should start from the bottom and go up from there (except for Calculus; I learned Calculus before I learned Algebra). And you should also diversify your interests.
And tensors aren't all that bad if you know what you are doing and understand what they mean. 


#11
Sep3008, 06:51 PM

P: 76

Wow, Thunder. You are my age and you are studying complex analysis?? Impressive.
How did you manage to learn calculus at age 12, and without understanding algebra first? I'm only just about to start calculus... What about your other subjects? How balanced is your study? And are you some kind of genius, or have you just applied yourself from a young age and pursued your interest? It's also nice to hear Sal saying that it is the good proofs in mathematics that are often regarded as "elegant", as loosely defined as that term may be. A question: does string theory and quantum field theory for example, revolve around proofs? Or are they less "proofintensive" than this? Cheers, Davin 


#12
Oct308, 12:15 AM

P: 152

Answering your questions:
Yes, I am studying Complex Analysis (reading my second book on the subject). I don't think it's that impressive. Well, when I was 11, I saw a Calculus text book in our county library. I didn't know what Caculus was, so I checked it out. I read through it and others by the time I was 12 and I learned the "advanced" Algebra along the way (I already had a faint knowledge of Algebra at this time, solving linear equations only). I had also read several books on both Collegelevel Physics and Chemistry when I was twelve. I also like History, English, and Biology, though not as much as the others. And now I am starting to learn a bit about quantum mechanics, relativistic physics, and string theory. I wouldn't call myself a genius. I'm not really at all motivated to do anything. I just feel like learning all I can in math and science because I enjoy it; the school and career stuff are just bonuses. Reid 


#13
Oct308, 01:45 AM

P: 76

Heh, ok then.
If you started out curious about mathematics, then sure, I can believe that. Must take a lot of effort then, to study college level physics, chemistry and mathematics at your age (and of course you were at high school level with calculus). Chemistry can be tangly; in elementary school (primary school) did you do any science in your classes? Frustratingly, I didn't, although I played with chemistry kits and electronics kits at that age. So, how did you manage to just pick up these topics at that level, having probably missed everything to come? Davin 


#14
Oct408, 04:19 PM

P: 152

I learned Differential and Integral Calculus when I was in the sixth grade (1112 years old) and I learned Ordinary Differential Equations, Chemistry and Physics in the seventh grade (12 years old). I read my first two books on Complex Analysis in ninth grade (14 years old). I don't know what you mean by "high school" Calculus.
It wasn't effort because I really enjoyed it. I didn't have any REAL science teachers until I was in the ninth grade. I've never messed around with those kits; they were too expensive for me. I mostly just read A LOT of books and picked up the stuff that I should have already known along the way. It was easier to understand Chemistry and Physics because the books started with Classical Mechanics and went up through electromagnetism and Modern Physics. 


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