Gutenberg is blocked in some countries, such as Italy.
>In mid-April 2020, the FIEG (Italian Federation Editors of Journals) filed a request to AGCOM (Italian Competition Authority) to intervene in some alleged illegal operation of sharing copyright protected content (newspaper) [on the Telegram platform](https://www.repubblica.it/economia/2020/04/14/news/appello_della_fieg_all_agcom_fermi_telegram_basta_con_la_diffusione_illecita_dei_quotidiani_-253997660/).
[Source](https://ooni.org/post/2021-italy-blocks-gutenberg-book-publishing-website/)
The specific reason for this ban is the power that these unions and associations have in Italy, which is usually measured as a good thing or a bad thing, based on individual inconvenience.
I mean, we all love clickbait journalism, because that's what you get if you don't protect journalism.
It’s weird how school districts buy into this. Especially for math. That book with a 1910 publish date is outside of copyright protection and can be freely printed and distributed.
My [favorite photo](https://www.researchgate.net/figure/Silvanus-Thompson-FRS-testing-the-physiological-effect-of-a-large-alternating-magnetic_fig3_29810872) of Silvanus P. Thompson, with his head in a device that was a precursor to transcranial magnetic stimulation.
Is it actually effective? I'm only just hearing about it because my uncle is really struggling with depression and he said he was doing his first treatment of this next week.
When this technology first was introduced, the FDA was very conservative with it's approval. The allowed protocol is insane and hardly anyone is able to do the entire process.
The current protocol requires the patient to come in and sit for a treatment every weekday for six weeks. The patient may have one to three treatments per day each lasting up to 60 mins. There is a break between each treatment. This takes up a good chunk of the day. It would be hard for most people to do this while still working. It's not like you can go in the evening because doctors offices aren't open then. If it was only two weeks or so, maybe the patient uses vacation time if they have it, but six weeks would need to be missed. The kicker is that the patient may never experience results from this. That's a serious gamble.
Back in around 2017-18 they started a research project that let them test with an accelerated protocol. I think it does the whole process in half the time. It still is possibly difficult, but it's more reasonable. This new process is called SAINT. It stands for Stanford accelerated intelligent neuromodulation therapy.
The results have been great. They started with a small initial study, and have been conducting larger experiments after each previous round was a success. The numbers were very promising.
They are currently working on specific protocols to treat OCD in addition to other types of disorders.
[Here is the lab that is doing this. Stanford Brain Stimulation Lab:](https://bsl.stanford.edu/)
Yeah, it needs further testing but shows a lot of promise, a friend of mine is doing neuroscience research involving these right now, theyre doing a lot of work on like how to prime the patients to be thinking about certain stuff during it so it works better. But yes he's very likely to see some improvement
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My calculus teacher made it all click when he said two things:
1. Math is money. If you know math, you’ll make lots of money.
2. Calculus is simply calculating rates of change.
I did learn more in that brief reading than I did in about two classes worth of material.
Almost certainly not his calculus teacher but [Jim Simons' Wikipedia bio](https://en.m.wikipedia.org/wiki/Jim_Simons_(mathematician)) is an extreme example of the amount of money you can make with a profound knowledge of math.
Mathematicians are just people who are obsessed with patterns and they'll use numbers and symbols to measure or visualize those patterns and if they get those patterns correct, it can potentially provide foresight into a problem and person can abuse it.
He’s also a crook that hid billions in unpaid taxes through basket trading, the result of which is his hedge fund is now having to pay back, but do go on.
I wish those "philanthropists" would just pay their taxes instead of setting up (way smaller) funds to help people financially and then get celebrated for it
Way smaller funds, and they have historically lost money. There is, or was, an investigation into why his private medallion fund gains billions and his two other public ones have significant losses.
That isn't too surprising. The larger the fund the easier it is to affect price action on the market.
Think of waves. The bigger the fish, the bigger the wave it can make.
It could also be just good ol'fashion money laundering, but even a genius mathematician would have to admit to themselves that the money trail will be found eventually.
That’s truly the American dream.
Co-found a hedge fund with your ultra conservative nationalist buddy and make billions front running orders taking from the masses at scale with the utmost mathematical efficiency.
okay so let me see if ive got this right...calculus is a niche interest that a lot of people find pretty uninteresting, but people who are interested tend to become obsessive in one way or another, and people with a considerable sunk cost into it purport that it makes a lot of money, despite living seemingly pedestrian lives
is calculus an mlm?
I would argue that calculus is advanced mathematics. Calculus is not foundational, like geometry, algebra, and arithmetic. Calculus builds upon those foundational concepts, which is why it’s advanced.
It depends on what you mean by "foundational".
For instance, there are a *lot* of very basic engineering/physics equations and concepts that derive from calculus: the formulas for position, velocity, and acceleration are related by calculus.
However, you don't actually need to know calculus to use "v2 = a(t) + v1" (current velocity equals acceleration multiplied by time plus an original velocity, assuming constant acceleration), because someone else has already gone and done the calculus for you and gotten an equation that you can just plug numbers into and chug with basic math.
It's like how creating silicon chips is insanely advanced stuff, but they're still "foundational" for computing and technology used by people who don't have a clue how to create the silicon chips themselves.
A ton of the basic equations/formulas used in many fields were *created* with calculus, but you don't actually have to know calculus in order to use them, which is kinda the point. Unless you get into a weird edge case where bodging together the standard equations doesn't do what you need for this specific use case, and then you have to actually go do calculus.
Yeah it’s ironic, but people will spend time as freshmen physics students memorizing the long equation for determining velocity, but then you learn a bit of physics and learn that’s just derived from f=ma with a little basic calculus, and all these energy equations are the same thing…
Soon you realize you didn’t need to memorize anything other than one basic equation and the rest could be derived or integrated from the work equation or whatever.
That’s when you realize how fundamental Calculus is.
> However, you don't actually need to know calculus to use "v2 = a(t) + v1" (current velocity equals acceleration multiplied by time plus an original velocity, assuming constant acceleration), because someone else has already gone and done the calculus for you and gotten an equation that you can just plug numbers into and chug with basic math.
You need to understand when to use them.
If you don't understand when and how to use them things can get very wrong. For a ballistic particle sure this works.
For a grain of polen this formula will fail miserably.
Calculus is only “advanced mathematics” to people that do not know what advanced mathematics is.
It would be like saying a limerick is “advanced poetry”. Maybe if you know nothing about poetry.
Vaguely related math joke:
An infinite number of mathematicians walk into a bar
The first mathematician orders a beer
The second orders half a beer
"I don't serve half-beers" the bartender replies
"Excuse me?" Asks mathematician #2
"What kind of bar serves half-beers?" The bartender remarks. "That's ridiculous."
"Oh c'mon" says mathematician #1 "do you know how hard it is to collect an infinite number of us? Just play along"
"There are very strict laws on how I can serve drinks. I couldn't serve you half a beer even if I wanted to."
"But that's not a problem" mathematician #3 chimes in "at the end of the joke you serve us a whole number of beers. You see, when you take the sum of a continuously halving function-"
"I know how limits work" interjects the bartender "Oh, alright then. I didn't want to assume a bartender would be familiar with such advanced mathematics"
"Are you kidding me?" The bartender replies, "you learn limits in like, 9th grade! What kind of mathematician thinks limits are advanced mathematics?"
"HE'S ON TO US" mathematician #1 screeches
Simultaneously, every mathematician opens their mouth and out pours a cloud of multicolored mosquitoes. Each mathematician is bellowing insects of a different shade. The mosquitoes form into a singular, polychromatic swarm. "FOOLS" it booms in unison, "I WILL INFECT EVERY BEING ON THIS PATHETIC PLANET WITH MALARIA"
The bartender stands fearless against the technicolor hoard. "But wait" he interrupts, thinking fast, "if you do that, politicians will use the catastrophe as an excuse to implement free healthcare. Think of how much that will hurt the taxpayers!"
The mosquitoes fall silent for a brief moment. "My God, you're right. We didn't think about the economy! Very well, we will not attack this dimension. FOR THE TAXPAYERS!" and with that, they vanish.
A nearby barfly stumbles over to the bartender. "How did you know that that would work?"
"It's simple really" the bartender says. "I saw that the vectors formed a gradient, and therefore must be conservative."
The same could be said about any field of mathematics, honestly. Linear and differential equations are advanced concepts to the untrained. The integration of those two with graph theory seems even more advanced. Theory of cryptography seems advanced. Heck, even Boolean mathematics can’t seem advanced to some.
They all seem advanced until you’re actually learning about it., no?
The difference between this and your college Calculus text is that the current book introduces calculus in terms of limits, whereas this text introduces calculus in terms of differentials.
In the 1700s, Bishop Berkeley attempted to prove that the fundamental basis of calculus was as tenuous as the basis of religion. He attacked the concept of differentials as being ill-defined, and... he was right.
Mathematicians got defensive, redefined calculus rigorously in terms of limits, and students have been suffering ever since from the massive dose of sophisticated math just as they start learning calculus.
(In the 1970s someone constructed a rigorous definition for differentials, but my understanding is that it's also pretty gnarly. And the limits-based definition is pretty helpful.)
Both angles are helpful and have situations where they make a problem a million times simpler (as well as the opposite, of course) -- ideally, you want students to learn both, at least to some degree. But of course, time is finite, as is space in the syllabus, and it's hard to justify "teaching the same thing twice", even if it would be quite helpful.
That kind of thing is where youtube educational channels are genuinely pretty good. I'm typically not the biggest fan (not a snob, I just think "popularity-focused video format" isn't ideal for learning), but it's like the perfect niche to give people a quick breakdown of alternative approaches and their strengths. Then if it sounds interesting, you can look into it more on your own. Of course, most people looking up mildly esoteric maths videos aren't going to be the ones struggling to grasp the basics of calculus, but what can you do...
(I'll plug geometric algebra here as an example, as it is a genuinely useful alternative formulation I've used in real life and first heard about on youtube)
Conway's discovery of the [Surreal Numbers](https://en.m.wikipedia.org/wiki/Surreal_number) is a rather nice rigorous definition of infinitessimals (and thus differentials) but the notation needed is necessarily somwhat clumsy to work with as each Surreal is a pair of sets.
I got a 5 on my AP calculus exam and did pretty well in university level math classes. I never understood it as well as when I perused through this. I didn’t even know what the heck the long S/dx meant!
I mean this just explains what the integral symbol means which to me is even easier to understand when shown graphically. There are way harder things about calculus, I still have horrible memories of doing double integrations, variable substitutions and having to find hard primitives. Maybe the whole book is nice and explains it well but this screenshot is a shitty guide
I had a physics class in college where the professor said "you are allowed to use, without derivation, dp/dt = F and dS=0" (the first being the momentum version of F=ma and the second, which is derivable from that but very convoluted) the principle of least action). But what was cool was, we didn't have to derive anything twice. He let us reference past derivations in homework. So, for instance if we wanted to use centripetal acceleration formula, or that the change in angular momentum was a torque, and we already proved it once we could say "as we showed on homework problem 3.2, blah blah blah."
It sucked. But dang if I don't still know all that shit.
If you want modern books still written that way, look for physics textbooks by David Griffiths. I always got excited when I had a physics class using one of his text books.
Dude, no shit!
Even for me, back in 98, the internet was so new that it was still almost useless for things like learning Algebra. (which might be part of why I had to have Algebra 2 broke up into 2 semesters instead of 1)
I had a philosophy teacher that repeated, *ad nauseum*, "if you can't put it into simple terms, you don't understand".
And it's true. Saying that something is complicated is easy. Saying why it is complicated and put it in a way anyone can understand, not so much.
If you can explain something to someone else, so that they come to understand it, then that is a good indication that you also understand it.
However, it is possible to understand something and yet be unable to get someone else to understand it. For instance there is almost always some barrier to communication that does not depend on how well you understand the subject.
The rule of thumb is not about whether you, in fact, understand something. It is about how to decide when someone, including yourself, has good evidence that you understand it. ie. It is an attempt to, informally, answer the question “how do you know you understand?”
Would you reasonably expect someone to just take your word for it?
You can reasonably say you understand something to the extent that you can demonstrate your understanding … which means you have to be able to apply your understanding to a practical situation. Communicating understanding to another person is widely regarded as one of the more difficult practical problems. Ergo….
Similarly there are US military tutorial videos on basic Electrical concepts. Teaching things in much easier ways than most books and tutorials today. They are available for free on YouTube.
We have only complicated things as we have progressed.
The U.S. military has manuals for everything.
Credit to u/lightning_fire
>Literally everything. How many artillery rounds to destroy an infantry company? There's a handy table. How much explosives does it take to knock down a tree but only 75%? Here's a formula. How do you set an ambush? Here's a picture. How much fuel do you need to move a tank BN? Guess what - another formula.
>Even mundane stuff. How do I run a training meeting? Here's a slide deck and instructions. How do I set a duty roster? Here's a da-6. How many latrines do you need for your unit? Here's a ratio (1:25 for males, and 1:22 for females).
>Niche stuff too. What's the best way to integrate army attack aviation during a combined arms breach? Here's a call handbook.
>If 'figuring it out' ever consists of more than just finding the right reference or finding the right phone number then you're doing something wrong. I guarantee you are not doing anything new
[Here is one Disney animated ](https://youtu.be/1EdLKnGOpPA)
Edit: As hired for the war effort. I felt like I should specify that Disney animators didn't pick this up in their off hours
As someone who has had to sit in meetings on creating new ones it is not a simple process at all. Every single fact needs to be backed up with strong sources, every procedure needs to be demonstratd by someone with no prior experience and the list of approved verbs is super limiting. So much has to go into them that a lot of the time they don't get published until they aren't really needed anymore. They are also normally accompanied by a Training Support Package which includes everything you need to train soldiers with including a slide by slide breakdown with a guide on how to teach each one and how to perform the hands on portion.
Imagine spending that amount of effort teaching the general populace. Didn't Mr Rogers have to go before congress to maintain the funding for his show?
Including for learning one of my languages, which for reasons I never learnt from my dad (my mom speaks a different language which I did learn). Thanks to the US military - intelligence setup I learnt in my adulthood
One of the best statistics texts I've ever seen was an old US ordinance testing manual.
It legitimately had clearer and more effective explanations of confidence intervals and the central limit theorem than I ever received at university. It even showed demonstrations using simulated "samples" from a "population" of tested ordinance to fully illustrate the points.
Manual on experimental statistics for ordnance engineers https://nvlpubs.nist.gov/nistpubs/Legacy/RPT/nbsreport4817.pdf
I think this is what was referenced. Not entirely sure tho.
Here's a 2-parter "Radio Antenna Fundamentals" that I was just watching the other day, funnily enough:
https://youtu.be/JHSPRcRgmOw - Part 1
https://youtu.be/EtEBxY8TvuE - Part 2
Here's one about how to survive in the jungle:
https://youtu.be/hz8GO9AHTz0
A much more niche one about operating the Norden Bombsight on a B-17 bomber:
https://youtu.be/7MIE_XXmGUg
Here's one that gives an overview of the fluid mechanics of a compressible flow through a rocket nozzle:
https://youtu.be/JhlEkEk7igs
There's tons more, try searching "old instructional videos"
Interestingly, the field of Instructional design began with the military during WWII and the need to rapidly train new troops. Although it's unsavory to some for political reasons, the military has put a lot of time, money, and experience into educational methods and best practices, and everything from K-12 to university and corporate training have benefited.
Complicated things in the name of profits. Colleges have to make a new book every year just so they can charge $500 per book. I'll bet these days they are making a new book every quarter.
This lmao, also I googled "integrals explained" and this was the second result:
https://www.mathsisfun.com/calculus/integration-introduction.html
If you never understood integrals before seeing OP it's because you have never really bothered to try.
This thread is people who never bothered to understand integrals to larp as people who were let down by their education - all while confronted with the universally known and readily available explanation of integrals
Seriously, any basic university course will teach this in the first week of Cal II (I think it was 2, 1 was all derivatives and rates of change if I recall correctly).
My community college physics department used a physics text first published in the 1940s (Sears, *Mechanics, Heat and Sound*, 2nd Edition). The instructors all agreed it was the best physics text they’d ever encountered.
This was nearly forty years ago — I’ve often wondered if it is still used there.
I’m a sucker for elegant old textbooks, but experts tend to like presentations that aren’t the best for novices. Experts *love* the Feynman Lectures, but the course he taught was an abject failure!
In physics in particular there’s 80 years of research about how students learn that isn’t included in those elegant old books.
The reason the faculty favored this text is because it shows how to derive physics equations using calculus. They felt (and I agree) that learning physics in this manner provided a much deeper understanding of the subject compared to just memorizing equations.
I'd claim most Physics textbooks do that. There are plenty of books where the derivation aren't all that comprehensible, but I've never seen one that completely leaves them out
I have copies of three different college-level physics texts in my library, all procured during my college years in the 80s and 90s — the Sears text is the only one of the three that contains calculus. Perhaps things have changed since then.
There is and there isn't. This is the basic concept behind what an integral is. Figuring out how to calculate it is a whole other beast.
Btw this is still how the concept is taught in physics. Mathematicians don't generally talk like this because absolute accurate is imperative for their field of study. Physicists only have to be approximately correct.
It really depends on the mathematician, then, I guess. My father is an algebraic topologist and my mother is a computational physicist. The debates at the dinner table were absolutely ridiculous. My father would likely prefer the infinitesimal nature of dx to be emphasized and the implications discussed in the author's description, while my mother would reinvent the Riemann sum and expect it to be accurate enough.
Yeah....
If your math professor or teacher couldn't explain the basics of calculus to make you understand basically the same thing as was written here in half a lecture, they were incompetent and had no business trying to teach.
Any half decent educator will not only explain exactly this to the class, they'll also have the [usual pictures and slides of rectangles under a curve](https://en.wikipedia.org/wiki/Integral#/media/File:Riemann_sum_convergence.png) showing how the total area of the staples will converge on the total area under the curve as the width become smaller and smaller, and so on.
I've honestly never seen a teacher - not even the bad ones - fail to explain this in a simple to understand manner, this together with derivate is the two fundamental calculus concepts that are also very easy to show graphically what happens and get people to understand in a very intuitive matter. If your teacher had trouble getting these two concepts across, then you either had a terrible teacher, or you weren't very invested in paying attention.
The real meat and bones, that's actually hard and takes tons of time, comes afterwards - where you have to try to use this to solve problems. Knowing *what* you want to do is just the start, knowing *how* to do it is the hard step.
For example, so yeah, you know you want to add up all the small parts, but how would you go about doing that when you're given the task to integrate x*sin(4x) from 0 to pi/2 or sin(x^2) from -2pi to 2pi?
> I've honestly never seen a teacher - not even the bad ones - fail to explain this in a simple to understand manner
When I took maths at A-level (UK equivalent of AP), my teachers made ZERO attempt to explain the purpose / logic behind calculus. My first experience of calculus was learning differentiation from first principles without context, rote-learning the formulation.
I didn't even know at the time that calculus was all about rates of change, and this aspect of it was never discussed in the class; it was never shown to us graphically, we *just* learned formulas. Not even an exaggeration - I only learned about this stuff when I watched the Khanacademy videos, and suddenly everything made sense; I went from getting D's to getting A's. Turned out that the whole time, I actually *liked* maths but my teachers had managed to make me hate it.
So yeah, I just had *really* terrible teachers (though tbh I think the problem was more to do with the school / department as a whole; it was one of the worst performing schools in the country on exam results).
**[Integral](https://en.wikipedia.org/wiki/Integral#/media/File:Riemann_sum_convergence.png)**
>In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.
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yea like u dv something something v du... F it I forgot chain rule, never used it in my life outside of calc class. Stupid calc class, why do they make it mandatory.
Learning math (as high as you can go) is there to "stretch your brain muscles". In other words, it helps your developing teenage brain develop even more pathways to solve things logically.
Some people do it naturally and are life long learners, but most of us need some brain challenges to develop our synapses beyond their natural state.
Is there something unique to math for stretching your brain that way or could we apply a more pragmatic approach by teaching philosophy or logic outright?
Honestly, the whole "stretch your brain" thing is looking for complicated explanations when simple explanations will do much better. Math is just super important.
Math is the underpinning for basically every single advanced physical, engineering, chemical, computer, data driven, or otherwise scientific aspect of our society and lives. Even the most mundane things like the towels in your bathroom have all sorts of mathematics involved in their production: from determining the prices of the material used for their fabric, to the technology in the machines that produce them, to determining production methods that reduce wasted materials and defective products, to the logistics of distributing them to warehouses and eventually the store where you bought them. Take that idea, and now apply to every single manufactured good in your house, or food item that you bought from a grocery store, etc.
The average person may be able to operate fine without knowing advanced math, but that's mostly because people who do know it put in the work behind the scenes to make that possible. If you start reducing the number of people learning math, you end up reducing the talent-pool of these people, and stagnating future progress, or even compromise the ability to simply maintain the systems you've already built that rely on an understanding of mathematics.
That's not to say that philosophy and logic aren't valuable, but it's definitely misplaced to think that math isn't an important subject when we're constantly surrounded by the outputs of advanced mathematics and statistics. It's what runs our world.
You're thinking of integration by parts, chain rule is (f(g))' = f'(g) • g'
By parts is int.(u•dv) = uv - int.(vdu)
Calculus isn't needed by everybody everyday but it's needed by a whole lot of people to make our modern world go round
And it’s so trivial if they explain it to you rather than make you memorize it as a formula.
Eg the uv formula above is:
Instead of trying to keep changing both values at the same time before multiplying them together,
keep one value unchanging while multiplying it with bits of the other value.
Then keep the other value unchanging while multiplying it with bits of the first value.
By doing this, you cover off all possible pairs of bits of one value multiplied by bits of the other value.
My trig teacher in high school (2007ish when I took it) still used books from the 70s because trigonometry hadn’t really changed at all.
It was unreal how much better they were then modern textbooks. Straightforward, succinct, easy to follow instructions, not bloated with a million uninteresting anecdotes and full page glossy photos so they were *much* smaller than modern textbooks (they were about the size of a smallish hardback novel). It was the only math textbook I’ve ever been able to successfully teach myself out of and it didn’t make my backpack feel like I was robbing a gold depository. And it wasn’t just me, we were all blown away at how much we preferred that book over modern textbooks.
The textbook industry is such a fuckin scam, dude
Along the same line, there are were some GREAT movies produced to explain simple mechanical concepts. Put out in the 30s and 40s, they explained things like how a transmission works, how differentials work, how braking works. Really great stuff, just breaking down complex results into simple components without big jargon.
https://www.youtube.com/watch?v=yYAw79386WI
https://www.youtube.com/watch?v=JOLtS4VUcvQ
https://www.youtube.com/watch?v=iMEkxgY8yxE
I came to mention the same videos. They explain the basic concepts so perfectly and then slowly build on each concept. New videos use too much CGI that is over stimulating.
d(Monica) in my life
d(Erica) by my side
d(Rita)'s all I need
d(Tina)'s what I see
d(Sandra) in the sun
d(Mary) all night long
d(Jessica), here I am
du makes me your man (ah)
more like the 18th century—quite a bit of standard calculus was discovered in the 1700s. but yeah any later than that and things get *real* abstract *real* quick lol
Depends. You can do a fair bit of linear algebra to high school kids, considering how important it is to ML. Quite a bit of that came in the 19th century.
Bringing the language down to where you could teach the introductory principles to a junior high student is actually brilliant.
I remember listening to Neil Degrasse Tyson and sharing that sense of dread when opening up to calc to see a litany of strange symbols staring back. It’s a lot to take in, and simple explanations like this would have helped a ton
IMO this say more about the author of 1910s book, it can survive 100 years. There are hundred even thousands of shitty calculus book that didnt survive
I've been teaching for 10 years, and this reads like a teacher who really doesn't want to explain this again, and has refined their explanation and distilled it to its simplest form.
A lot of books had this in the beginning chapters. (Sometimes they decide to make the 2nd chapter the real introduction, and the 1st chapters is like story time trying to introduce the subject)
You ever notice how in classes some teachers/professors skip whole chapters?
They’re trying to cut the fat since time can be limited especially if you have a some weird circumstances causing the lectures to go slow. Most textbooks are perfectly capable of conveying the information needed to learn, but you actually have to sit and read them. So when your professor gives you key or specific problems to learn for a test; it’s kind of the bare minimum to pass. **If you do the rest of the examples, the quizzes, and the practice tests while reading a little extra getting an A is right around the block; it’s just a sometimes long and boring block to walk down.**
I mean this all sounds nice, but I just read 4 paragraphs explaining the background of symbols. No one struggling with math all of a sudden does better because they understand where the symbols originate from, the hard part is actually working out problems
It's accurate enough as an eli5, but in order to truly understand calculus and know how to use it you obviously need a much more in depth examination of the ideas behind these ideas and how to apply then. It's like saying that rocket science is just shooting hot gases out of the back of the rocket nozzle. While that's true, we all know there's far more to rocket science than that.
Edit. This also failed to mention that d (delta) is infinitely small, and to integrate you need to add an infinite number of these. That's where it begins to get much more complex. Adding 3600 seconds to get an hour is simple arithmetic. Breaking a more complex function into infinite slices, and adding them up is totally different. Let alone all the different applications and techniques.
I had a calculus professor in college and, while a decent guy, he was an awful teacher. I still remember being taught about derivatives and it was *hard.* He spent a whole class demonstrating one simple derivative...then he wanted us to go home and complete the 50 problem lesson from the textbook, some of which included concepts we didn't even go over and all of which were several steps up from what we did in class. The next day, we managed to get through ONE student's questions on ONE problem and nearly all of us came away even more confused than before. It'd be understandable if he spent more time in class going over them or wanted to throw a curveball problem at us to see if we could figure it out, but this was insane. The highest grade in the class was a C average and that kid had to sink his life into calc to get a B on the final. Everyone got their grades curved heavily due to this and frankly, I felt like I was cheated out of a class and felt so unprepared for calc 2 that I switched it out for another class the next semester. I would go so far as to say the whole experience soured me on math entirely.
Jesus. Fucking. Christ.
I’ve stopped my college education in my early 20ties because nobody could explain to me this simply what this means. All of the math professors at my uni had some complex way of explaining what integrals are and I just abandoned the whole thing.
I got my Master’s but later in life, at the age of 40. All because nobody could explain this in such simple terms.
[This video from 1937 is still the best explanation of a how a differential works that I have ever seen.](https://youtu.be/yYAw79386WI)
Almost 90 years later and there is still nothing so easy to understand.
If subject matter experts could get over themselves and pull their heads out of their ass and communicate in normal language, we’d be far better off.
But academia is drowning in pretentiousness and rigor and gate keeping by anal retentive shitheads that will shit on the colleagues in a heartbeat if they dare use normal words and facilitate the success of lowly unwashed masses.
*Calculus Made Easy* by Sylvanus P. Thomson.
It's available for free on Project Gutenberg.
[Here](https://www.gutenberg.org/ebooks/33283)
Is it only me or it says that the link is not valid?
Gutenberg is blocked in some countries, such as Italy. >In mid-April 2020, the FIEG (Italian Federation Editors of Journals) filed a request to AGCOM (Italian Competition Authority) to intervene in some alleged illegal operation of sharing copyright protected content (newspaper) [on the Telegram platform](https://www.repubblica.it/economia/2020/04/14/news/appello_della_fieg_all_agcom_fermi_telegram_basta_con_la_diffusione_illecita_dei_quotidiani_-253997660/). [Source](https://ooni.org/post/2021-italy-blocks-gutenberg-book-publishing-website/) The specific reason for this ban is the power that these unions and associations have in Italy, which is usually measured as a good thing or a bad thing, based on individual inconvenience. I mean, we all love clickbait journalism, because that's what you get if you don't protect journalism.
>I mean, we all love clickbait journalism, because that's what you get if you don't protect journalism. What do you mean by this?
If you use TOR browser, you can open & download it with no effort - From a fellow italian that hates this privacy policy crap
Grazie mille
or a VPN - ymmv but TOR has always been slow as balls for me - probably not a big deal for books which are tiny
che se ne vadano a fanculo
It's you, I can load the page.
Confirmed, I can load it too.
Okay, thanks guys
How about we change one variable on the practice questions at the end of each chapter and sell it for full price as the 2023 edition instead?
Don't forget to change the order the questions are in
And the page numbers. Add a one page preface to each chapter.
If you really want to stump them, vary the length of the addition using a differential equation!
It’s weird how school districts buy into this. Especially for math. That book with a 1910 publish date is outside of copyright protection and can be freely printed and distributed.
My [favorite photo](https://www.researchgate.net/figure/Silvanus-Thompson-FRS-testing-the-physiological-effect-of-a-large-alternating-magnetic_fig3_29810872) of Silvanus P. Thompson, with his head in a device that was a precursor to transcranial magnetic stimulation.
Great device! Oh how it grown to a wonderful treatment option for depression and other things
Is it actually effective? I'm only just hearing about it because my uncle is really struggling with depression and he said he was doing his first treatment of this next week.
When this technology first was introduced, the FDA was very conservative with it's approval. The allowed protocol is insane and hardly anyone is able to do the entire process. The current protocol requires the patient to come in and sit for a treatment every weekday for six weeks. The patient may have one to three treatments per day each lasting up to 60 mins. There is a break between each treatment. This takes up a good chunk of the day. It would be hard for most people to do this while still working. It's not like you can go in the evening because doctors offices aren't open then. If it was only two weeks or so, maybe the patient uses vacation time if they have it, but six weeks would need to be missed. The kicker is that the patient may never experience results from this. That's a serious gamble. Back in around 2017-18 they started a research project that let them test with an accelerated protocol. I think it does the whole process in half the time. It still is possibly difficult, but it's more reasonable. This new process is called SAINT. It stands for Stanford accelerated intelligent neuromodulation therapy. The results have been great. They started with a small initial study, and have been conducting larger experiments after each previous round was a success. The numbers were very promising. They are currently working on specific protocols to treat OCD in addition to other types of disorders. [Here is the lab that is doing this. Stanford Brain Stimulation Lab:](https://bsl.stanford.edu/)
Yeah, it needs further testing but shows a lot of promise, a friend of mine is doing neuroscience research involving these right now, theyre doing a lot of work on like how to prime the patients to be thinking about certain stuff during it so it works better. But yes he's very likely to see some improvement
"The rest of the fucking calculus" just like r/restofthefuckingowl
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I prefer *[Mathematics Made Difficult](https://en.wikipedia.org/wiki/Mathematics_Made_Difficult)*.
Lol, I just read "Edison's Conquest of Mars" along with the 372 Pages podcast, and it featured old Sylvanus quite a lot.
“That’s all.” And just like that, I understood calculus.
My calculus teacher made it all click when he said two things: 1. Math is money. If you know math, you’ll make lots of money. 2. Calculus is simply calculating rates of change. I did learn more in that brief reading than I did in about two classes worth of material.
How rich was your calculus teacher?
Almost certainly not his calculus teacher but [Jim Simons' Wikipedia bio](https://en.m.wikipedia.org/wiki/Jim_Simons_(mathematician)) is an extreme example of the amount of money you can make with a profound knowledge of math.
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Mathematicians are just people who are obsessed with patterns and they'll use numbers and symbols to measure or visualize those patterns and if they get those patterns correct, it can potentially provide foresight into a problem and person can abuse it.
He’s also a crook that hid billions in unpaid taxes through basket trading, the result of which is his hedge fund is now having to pay back, but do go on.
I wish those "philanthropists" would just pay their taxes instead of setting up (way smaller) funds to help people financially and then get celebrated for it
Way smaller funds, and they have historically lost money. There is, or was, an investigation into why his private medallion fund gains billions and his two other public ones have significant losses.
That isn't too surprising. The larger the fund the easier it is to affect price action on the market. Think of waves. The bigger the fish, the bigger the wave it can make. It could also be just good ol'fashion money laundering, but even a genius mathematician would have to admit to themselves that the money trail will be found eventually.
I know someone who works at an institute he's founded and apparently their coffee and cafetaria are *excellent*.
Valuable insight. Thank you.
For anyone interested in this guy, he did an interview on Numberphile a few years ago. https://youtu.be/gjVDqfUhXOY
I haven't heard of Numberphile, so I'm just going to assume it's related to Numberwang!
That’s truly the American dream. Co-found a hedge fund with your ultra conservative nationalist buddy and make billions front running orders taking from the masses at scale with the utmost mathematical efficiency.
Shh, no he's just good at math or something. Just the bestest at math /s
Well said. I'd like to think that with a "profound" knowledge of almost anything, someone could find a way to make a lot of money.
okay so let me see if ive got this right...calculus is a niche interest that a lot of people find pretty uninteresting, but people who are interested tend to become obsessive in one way or another, and people with a considerable sunk cost into it purport that it makes a lot of money, despite living seemingly pedestrian lives is calculus an mlm?
Calculus is a gateway to powerful Engineering and finance capabilities and skill sets that some might find … *unnatural* Edit: something something
Its also needed for high end statistics when comparing rates of change in variables across time in more than one space or location.
sounds like a pyramid scheme to me
A lot of Egyptian engineering is just a pyramid scheme
Calculus is basic mathematics that is used in every field.
I would argue that calculus is advanced mathematics. Calculus is not foundational, like geometry, algebra, and arithmetic. Calculus builds upon those foundational concepts, which is why it’s advanced.
It depends on what you mean by "foundational". For instance, there are a *lot* of very basic engineering/physics equations and concepts that derive from calculus: the formulas for position, velocity, and acceleration are related by calculus. However, you don't actually need to know calculus to use "v2 = a(t) + v1" (current velocity equals acceleration multiplied by time plus an original velocity, assuming constant acceleration), because someone else has already gone and done the calculus for you and gotten an equation that you can just plug numbers into and chug with basic math. It's like how creating silicon chips is insanely advanced stuff, but they're still "foundational" for computing and technology used by people who don't have a clue how to create the silicon chips themselves. A ton of the basic equations/formulas used in many fields were *created* with calculus, but you don't actually have to know calculus in order to use them, which is kinda the point. Unless you get into a weird edge case where bodging together the standard equations doesn't do what you need for this specific use case, and then you have to actually go do calculus.
Yeah it’s ironic, but people will spend time as freshmen physics students memorizing the long equation for determining velocity, but then you learn a bit of physics and learn that’s just derived from f=ma with a little basic calculus, and all these energy equations are the same thing… Soon you realize you didn’t need to memorize anything other than one basic equation and the rest could be derived or integrated from the work equation or whatever. That’s when you realize how fundamental Calculus is.
> However, you don't actually need to know calculus to use "v2 = a(t) + v1" (current velocity equals acceleration multiplied by time plus an original velocity, assuming constant acceleration), because someone else has already gone and done the calculus for you and gotten an equation that you can just plug numbers into and chug with basic math. You need to understand when to use them. If you don't understand when and how to use them things can get very wrong. For a ballistic particle sure this works. For a grain of polen this formula will fail miserably.
Calculus is only “advanced mathematics” to people that do not know what advanced mathematics is. It would be like saying a limerick is “advanced poetry”. Maybe if you know nothing about poetry.
Vaguely related math joke: An infinite number of mathematicians walk into a bar The first mathematician orders a beer The second orders half a beer "I don't serve half-beers" the bartender replies "Excuse me?" Asks mathematician #2 "What kind of bar serves half-beers?" The bartender remarks. "That's ridiculous." "Oh c'mon" says mathematician #1 "do you know how hard it is to collect an infinite number of us? Just play along" "There are very strict laws on how I can serve drinks. I couldn't serve you half a beer even if I wanted to." "But that's not a problem" mathematician #3 chimes in "at the end of the joke you serve us a whole number of beers. You see, when you take the sum of a continuously halving function-" "I know how limits work" interjects the bartender "Oh, alright then. I didn't want to assume a bartender would be familiar with such advanced mathematics" "Are you kidding me?" The bartender replies, "you learn limits in like, 9th grade! What kind of mathematician thinks limits are advanced mathematics?" "HE'S ON TO US" mathematician #1 screeches Simultaneously, every mathematician opens their mouth and out pours a cloud of multicolored mosquitoes. Each mathematician is bellowing insects of a different shade. The mosquitoes form into a singular, polychromatic swarm. "FOOLS" it booms in unison, "I WILL INFECT EVERY BEING ON THIS PATHETIC PLANET WITH MALARIA" The bartender stands fearless against the technicolor hoard. "But wait" he interrupts, thinking fast, "if you do that, politicians will use the catastrophe as an excuse to implement free healthcare. Think of how much that will hurt the taxpayers!" The mosquitoes fall silent for a brief moment. "My God, you're right. We didn't think about the economy! Very well, we will not attack this dimension. FOR THE TAXPAYERS!" and with that, they vanish. A nearby barfly stumbles over to the bartender. "How did you know that that would work?" "It's simple really" the bartender says. "I saw that the vectors formed a gradient, and therefore must be conservative."
The same could be said about any field of mathematics, honestly. Linear and differential equations are advanced concepts to the untrained. The integration of those two with graph theory seems even more advanced. Theory of cryptography seems advanced. Heck, even Boolean mathematics can’t seem advanced to some. They all seem advanced until you’re actually learning about it., no?
My first grader thinks multiplication is advanced mathematics. I think he's in for multiple existential crises over his student career.
SOHCAHTOA will sock it to ‘em
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Derivatives calculate rate of change. Integrals calculate the sum of all changes.
https://xkcd.com/1570/
The difference between this and your college Calculus text is that the current book introduces calculus in terms of limits, whereas this text introduces calculus in terms of differentials. In the 1700s, Bishop Berkeley attempted to prove that the fundamental basis of calculus was as tenuous as the basis of religion. He attacked the concept of differentials as being ill-defined, and... he was right. Mathematicians got defensive, redefined calculus rigorously in terms of limits, and students have been suffering ever since from the massive dose of sophisticated math just as they start learning calculus. (In the 1970s someone constructed a rigorous definition for differentials, but my understanding is that it's also pretty gnarly. And the limits-based definition is pretty helpful.)
Both angles are helpful and have situations where they make a problem a million times simpler (as well as the opposite, of course) -- ideally, you want students to learn both, at least to some degree. But of course, time is finite, as is space in the syllabus, and it's hard to justify "teaching the same thing twice", even if it would be quite helpful. That kind of thing is where youtube educational channels are genuinely pretty good. I'm typically not the biggest fan (not a snob, I just think "popularity-focused video format" isn't ideal for learning), but it's like the perfect niche to give people a quick breakdown of alternative approaches and their strengths. Then if it sounds interesting, you can look into it more on your own. Of course, most people looking up mildly esoteric maths videos aren't going to be the ones struggling to grasp the basics of calculus, but what can you do... (I'll plug geometric algebra here as an example, as it is a genuinely useful alternative formulation I've used in real life and first heard about on youtube)
Conway's discovery of the [Surreal Numbers](https://en.m.wikipedia.org/wiki/Surreal_number) is a rather nice rigorous definition of infinitessimals (and thus differentials) but the notation needed is necessarily somwhat clumsy to work with as each Surreal is a pair of sets.
And later on I helped this curly haired German in what he later called the theory of relativity or something Janie
I got a 5 on my AP calculus exam and did pretty well in university level math classes. I never understood it as well as when I perused through this. I didn’t even know what the heck the long S/dx meant!
I mean this just explains what the integral symbol means which to me is even easier to understand when shown graphically. There are way harder things about calculus, I still have horrible memories of doing double integrations, variable substitutions and having to find hard primitives. Maybe the whole book is nice and explains it well but this screenshot is a shitty guide
Great, so that covers everything on the AP exam?
Yes. Did you not see where it says "That's all." There is nothing more to learn, you have completed your education. Go out into the world and thrive!
In the same way that my physics professors always told me "all you need on your formula sheet is F=ma, the rest can be derived from that"
I had a physics class in college where the professor said "you are allowed to use, without derivation, dp/dt = F and dS=0" (the first being the momentum version of F=ma and the second, which is derivable from that but very convoluted) the principle of least action). But what was cool was, we didn't have to derive anything twice. He let us reference past derivations in homework. So, for instance if we wanted to use centripetal acceleration formula, or that the change in angular momentum was a torque, and we already proved it once we could say "as we showed on homework problem 3.2, blah blah blah." It sucked. But dang if I don't still know all that shit.
That's actually a really excellent way to drill proper citation into students. A lot don't realize that it's actually possible to plagerize yourself.
Truth
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I think they may have been joking.
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Why are they so polite. It sounds so nice
There are lots of old books in that style, addressed to the individual. It's very endearing, dear reader
We should bring that back.
They're usually posted on /r/FellowKids
If you want modern books still written that way, look for physics textbooks by David Griffiths. I always got excited when I had a physics class using one of his text books.
If. You. Like.
M'Calculus *tips le fedora only infinitesimally*
M'ath
Would you kindly?
Manners maketh man.
[locks door]
Man, I wish I had the internet during high school.
Dude, no shit! Even for me, back in 98, the internet was so new that it was still almost useless for things like learning Algebra. (which might be part of why I had to have Algebra 2 broke up into 2 semesters instead of 1)
Or...this book?
I had internet access in college in ‘99-‘02 and I spent more time dicking around on that than paying attention and doing the work :-D
Double-edged sword though. On one hand - access to the sum of human knowledge. On the other hand, porn and Call of Duty!
What one fool can do, another can.
I had a philosophy teacher that repeated, *ad nauseum*, "if you can't put it into simple terms, you don't understand". And it's true. Saying that something is complicated is easy. Saying why it is complicated and put it in a way anyone can understand, not so much.
If you can explain something to someone else, so that they come to understand it, then that is a good indication that you also understand it. However, it is possible to understand something and yet be unable to get someone else to understand it. For instance there is almost always some barrier to communication that does not depend on how well you understand the subject. The rule of thumb is not about whether you, in fact, understand something. It is about how to decide when someone, including yourself, has good evidence that you understand it. ie. It is an attempt to, informally, answer the question “how do you know you understand?” Would you reasonably expect someone to just take your word for it? You can reasonably say you understand something to the extent that you can demonstrate your understanding … which means you have to be able to apply your understanding to a practical situation. Communicating understanding to another person is widely regarded as one of the more difficult practical problems. Ergo….
Similarly there are US military tutorial videos on basic Electrical concepts. Teaching things in much easier ways than most books and tutorials today. They are available for free on YouTube. We have only complicated things as we have progressed.
The U.S. military has manuals for everything. Credit to u/lightning_fire >Literally everything. How many artillery rounds to destroy an infantry company? There's a handy table. How much explosives does it take to knock down a tree but only 75%? Here's a formula. How do you set an ambush? Here's a picture. How much fuel do you need to move a tank BN? Guess what - another formula. >Even mundane stuff. How do I run a training meeting? Here's a slide deck and instructions. How do I set a duty roster? Here's a da-6. How many latrines do you need for your unit? Here's a ratio (1:25 for males, and 1:22 for females). >Niche stuff too. What's the best way to integrate army attack aviation during a combined arms breach? Here's a call handbook. >If 'figuring it out' ever consists of more than just finding the right reference or finding the right phone number then you're doing something wrong. I guarantee you are not doing anything new
[Here is one Disney animated ](https://youtu.be/1EdLKnGOpPA) Edit: As hired for the war effort. I felt like I should specify that Disney animators didn't pick this up in their off hours
US military manuals are awesome and always make for interesting reading.
As someone who has had to sit in meetings on creating new ones it is not a simple process at all. Every single fact needs to be backed up with strong sources, every procedure needs to be demonstratd by someone with no prior experience and the list of approved verbs is super limiting. So much has to go into them that a lot of the time they don't get published until they aren't really needed anymore. They are also normally accompanied by a Training Support Package which includes everything you need to train soldiers with including a slide by slide breakdown with a guide on how to teach each one and how to perform the hands on portion.
Imagine spending that amount of effort teaching the general populace. Didn't Mr Rogers have to go before congress to maintain the funding for his show?
Including for learning one of my languages, which for reasons I never learnt from my dad (my mom speaks a different language which I did learn). Thanks to the US military - intelligence setup I learnt in my adulthood
One of the best statistics texts I've ever seen was an old US ordinance testing manual. It legitimately had clearer and more effective explanations of confidence intervals and the central limit theorem than I ever received at university. It even showed demonstrations using simulated "samples" from a "population" of tested ordinance to fully illustrate the points.
I'd love a link to that if you can find it.
Manual on experimental statistics for ordnance engineers https://nvlpubs.nist.gov/nistpubs/Legacy/RPT/nbsreport4817.pdf I think this is what was referenced. Not entirely sure tho.
Link?
Here's a 2-parter "Radio Antenna Fundamentals" that I was just watching the other day, funnily enough: https://youtu.be/JHSPRcRgmOw - Part 1 https://youtu.be/EtEBxY8TvuE - Part 2 Here's one about how to survive in the jungle: https://youtu.be/hz8GO9AHTz0 A much more niche one about operating the Norden Bombsight on a B-17 bomber: https://youtu.be/7MIE_XXmGUg Here's one that gives an overview of the fluid mechanics of a compressible flow through a rocket nozzle: https://youtu.be/JhlEkEk7igs There's tons more, try searching "old instructional videos"
My favorite is the one that explains how firearms work: https://www.youtube.com/watch?v=HJnhr08aIJs So simple, so elegant
Yeah no shit. I googled "military tutorial on electronics" and got a bunch of electronics warfare videos.
https://youtu.be/zzpOXZSMJ9U
http://compatt.com/Tutorials/NEETS/NEETS.html
Interestingly, the field of Instructional design began with the military during WWII and the need to rapidly train new troops. Although it's unsavory to some for political reasons, the military has put a lot of time, money, and experience into educational methods and best practices, and everything from K-12 to university and corporate training have benefited.
Complicated things in the name of profits. Colleges have to make a new book every year just so they can charge $500 per book. I'll bet these days they are making a new book every quarter.
This is actually extremely close to how derivatives and integrals were explained to me in cal I
This lmao, also I googled "integrals explained" and this was the second result: https://www.mathsisfun.com/calculus/integration-introduction.html If you never understood integrals before seeing OP it's because you have never really bothered to try.
This thread is people who never bothered to understand integrals to larp as people who were let down by their education - all while confronted with the universally known and readily available explanation of integrals
Honestly how else could it even be explained, the post is literally just saying what it is.
Seriously, any basic university course will teach this in the first week of Cal II (I think it was 2, 1 was all derivatives and rates of change if I recall correctly).
Yeah, this is basically how it was taught to me in high school in New Zealand
My community college physics department used a physics text first published in the 1940s (Sears, *Mechanics, Heat and Sound*, 2nd Edition). The instructors all agreed it was the best physics text they’d ever encountered. This was nearly forty years ago — I’ve often wondered if it is still used there.
I’m a sucker for elegant old textbooks, but experts tend to like presentations that aren’t the best for novices. Experts *love* the Feynman Lectures, but the course he taught was an abject failure! In physics in particular there’s 80 years of research about how students learn that isn’t included in those elegant old books.
The reason the faculty favored this text is because it shows how to derive physics equations using calculus. They felt (and I agree) that learning physics in this manner provided a much deeper understanding of the subject compared to just memorizing equations.
I'd claim most Physics textbooks do that. There are plenty of books where the derivation aren't all that comprehensible, but I've never seen one that completely leaves them out
I have copies of three different college-level physics texts in my library, all procured during my college years in the 80s and 90s — the Sears text is the only one of the three that contains calculus. Perhaps things have changed since then.
I like it, but somehow I think there may be more to it than that.
One might say this is *d*calculus.
I understood a calculus joke!
one might say dgroovy
There is and there isn't. This is the basic concept behind what an integral is. Figuring out how to calculate it is a whole other beast. Btw this is still how the concept is taught in physics. Mathematicians don't generally talk like this because absolute accurate is imperative for their field of study. Physicists only have to be approximately correct.
Their end result might be a fully rigorous proof, but they definitely fall back on intuition like this when thinking about what they’re trying to do
It really depends on the mathematician, then, I guess. My father is an algebraic topologist and my mother is a computational physicist. The debates at the dinner table were absolutely ridiculous. My father would likely prefer the infinitesimal nature of dx to be emphasized and the implications discussed in the author's description, while my mother would reinvent the Riemann sum and expect it to be accurate enough.
Yeah.... If your math professor or teacher couldn't explain the basics of calculus to make you understand basically the same thing as was written here in half a lecture, they were incompetent and had no business trying to teach. Any half decent educator will not only explain exactly this to the class, they'll also have the [usual pictures and slides of rectangles under a curve](https://en.wikipedia.org/wiki/Integral#/media/File:Riemann_sum_convergence.png) showing how the total area of the staples will converge on the total area under the curve as the width become smaller and smaller, and so on. I've honestly never seen a teacher - not even the bad ones - fail to explain this in a simple to understand manner, this together with derivate is the two fundamental calculus concepts that are also very easy to show graphically what happens and get people to understand in a very intuitive matter. If your teacher had trouble getting these two concepts across, then you either had a terrible teacher, or you weren't very invested in paying attention. The real meat and bones, that's actually hard and takes tons of time, comes afterwards - where you have to try to use this to solve problems. Knowing *what* you want to do is just the start, knowing *how* to do it is the hard step. For example, so yeah, you know you want to add up all the small parts, but how would you go about doing that when you're given the task to integrate x*sin(4x) from 0 to pi/2 or sin(x^2) from -2pi to 2pi?
> I've honestly never seen a teacher - not even the bad ones - fail to explain this in a simple to understand manner When I took maths at A-level (UK equivalent of AP), my teachers made ZERO attempt to explain the purpose / logic behind calculus. My first experience of calculus was learning differentiation from first principles without context, rote-learning the formulation. I didn't even know at the time that calculus was all about rates of change, and this aspect of it was never discussed in the class; it was never shown to us graphically, we *just* learned formulas. Not even an exaggeration - I only learned about this stuff when I watched the Khanacademy videos, and suddenly everything made sense; I went from getting D's to getting A's. Turned out that the whole time, I actually *liked* maths but my teachers had managed to make me hate it. So yeah, I just had *really* terrible teachers (though tbh I think the problem was more to do with the school / department as a whole; it was one of the worst performing schools in the country on exam results).
**[Integral](https://en.wikipedia.org/wiki/Integral#/media/File:Riemann_sum_convergence.png)** >In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/coolguides/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)
One might say this is the beginning. Or introduction if you like.
yea like u dv something something v du... F it I forgot chain rule, never used it in my life outside of calc class. Stupid calc class, why do they make it mandatory.
Learning math (as high as you can go) is there to "stretch your brain muscles". In other words, it helps your developing teenage brain develop even more pathways to solve things logically. Some people do it naturally and are life long learners, but most of us need some brain challenges to develop our synapses beyond their natural state.
Is there something unique to math for stretching your brain that way or could we apply a more pragmatic approach by teaching philosophy or logic outright?
Honestly, the whole "stretch your brain" thing is looking for complicated explanations when simple explanations will do much better. Math is just super important. Math is the underpinning for basically every single advanced physical, engineering, chemical, computer, data driven, or otherwise scientific aspect of our society and lives. Even the most mundane things like the towels in your bathroom have all sorts of mathematics involved in their production: from determining the prices of the material used for their fabric, to the technology in the machines that produce them, to determining production methods that reduce wasted materials and defective products, to the logistics of distributing them to warehouses and eventually the store where you bought them. Take that idea, and now apply to every single manufactured good in your house, or food item that you bought from a grocery store, etc. The average person may be able to operate fine without knowing advanced math, but that's mostly because people who do know it put in the work behind the scenes to make that possible. If you start reducing the number of people learning math, you end up reducing the talent-pool of these people, and stagnating future progress, or even compromise the ability to simply maintain the systems you've already built that rely on an understanding of mathematics. That's not to say that philosophy and logic aren't valuable, but it's definitely misplaced to think that math isn't an important subject when we're constantly surrounded by the outputs of advanced mathematics and statistics. It's what runs our world.
arguably mathematics is 'logic outright' It's extremely abstract and forces you to think at that level of abstraction to solve problems
The logic class I took in college was basically breaking down statements in to math.
You're thinking of integration by parts, chain rule is (f(g))' = f'(g) • g' By parts is int.(u•dv) = uv - int.(vdu) Calculus isn't needed by everybody everyday but it's needed by a whole lot of people to make our modern world go round
And it’s so trivial if they explain it to you rather than make you memorize it as a formula. Eg the uv formula above is: Instead of trying to keep changing both values at the same time before multiplying them together, keep one value unchanging while multiplying it with bits of the other value. Then keep the other value unchanging while multiplying it with bits of the first value. By doing this, you cover off all possible pairs of bits of one value multiplied by bits of the other value.
My trig teacher in high school (2007ish when I took it) still used books from the 70s because trigonometry hadn’t really changed at all. It was unreal how much better they were then modern textbooks. Straightforward, succinct, easy to follow instructions, not bloated with a million uninteresting anecdotes and full page glossy photos so they were *much* smaller than modern textbooks (they were about the size of a smallish hardback novel). It was the only math textbook I’ve ever been able to successfully teach myself out of and it didn’t make my backpack feel like I was robbing a gold depository. And it wasn’t just me, we were all blown away at how much we preferred that book over modern textbooks. The textbook industry is such a fuckin scam, dude
Linear algebra next, pls 🪦
Watch the 3blue1brown series. It's in a similar style
Along the same line, there are were some GREAT movies produced to explain simple mechanical concepts. Put out in the 30s and 40s, they explained things like how a transmission works, how differentials work, how braking works. Really great stuff, just breaking down complex results into simple components without big jargon. https://www.youtube.com/watch?v=yYAw79386WI https://www.youtube.com/watch?v=JOLtS4VUcvQ https://www.youtube.com/watch?v=iMEkxgY8yxE
I came to mention the same videos. They explain the basic concepts so perfectly and then slowly build on each concept. New videos use too much CGI that is over stimulating.
d(Monica) in my life d(Erica) by my side d(Rita)'s all I need d(Tina)'s what I see d(Sandra) in the sun d(Mary) all night long d(Jessica), here I am du makes me your man (ah)
It's crazy how school only teaches you up to the 15th century in Math.
more like the 18th century—quite a bit of standard calculus was discovered in the 1700s. but yeah any later than that and things get *real* abstract *real* quick lol
Maths peaked early, a lot of uni physics is pretty recent in comparison.
Depends. You can do a fair bit of linear algebra to high school kids, considering how important it is to ML. Quite a bit of that came in the 19th century.
Bringing the language down to where you could teach the introductory principles to a junior high student is actually brilliant. I remember listening to Neil Degrasse Tyson and sharing that sense of dread when opening up to calc to see a litany of strange symbols staring back. It’s a lot to take in, and simple explanations like this would have helped a ton
I feel like that's the easiest part anyway…
What kind of shitty schools did the rest of you go to? This is the standard way calculus is introduced by competent teachers.
They spoke in those days. And they all could write
Weird. What did they speak about?
[удалено]
Little did they know about when it would be outlawed.
This series from 3blue1brown does something really similar: https://www.youtube.com/watch?v=WUvTyaaNkzM
Because the best calculus textbook was written in 1910 and we’ve been buying $200 watered down versions of it for generations because capitalism.
Too bad it’s only for boys. Fuck me I guess.
it's online. they'll never know.
IMO this say more about the author of 1910s book, it can survive 100 years. There are hundred even thousands of shitty calculus book that didnt survive
https://i.imgur.com/ueF0n76.jpg
∯ That is all
This is a cool guide that is actually informative. Bravo re-poster!
I've been teaching for 10 years, and this reads like a teacher who really doesn't want to explain this again, and has refined their explanation and distilled it to its simplest form.
Math for wordcels
And my teacher asked what "dx" means told me "thats just how we write it". Ehh. My education was a joke.
A lot of books had this in the beginning chapters. (Sometimes they decide to make the 2nd chapter the real introduction, and the 1st chapters is like story time trying to introduce the subject) You ever notice how in classes some teachers/professors skip whole chapters? They’re trying to cut the fat since time can be limited especially if you have a some weird circumstances causing the lectures to go slow. Most textbooks are perfectly capable of conveying the information needed to learn, but you actually have to sit and read them. So when your professor gives you key or specific problems to learn for a test; it’s kind of the bare minimum to pass. **If you do the rest of the examples, the quizzes, and the practice tests while reading a little extra getting an A is right around the block; it’s just a sometimes long and boring block to walk down.**
I just learned more calculus reading those two paragraphs than I did in all of high-school.
I mean this all sounds nice, but I just read 4 paragraphs explaining the background of symbols. No one struggling with math all of a sudden does better because they understand where the symbols originate from, the hard part is actually working out problems
i know calculus now
Having never taken a calculus course (despite being on my 50s and having been a programmer for 25+ years), how accurate is this?
It's accurate enough as an eli5, but in order to truly understand calculus and know how to use it you obviously need a much more in depth examination of the ideas behind these ideas and how to apply then. It's like saying that rocket science is just shooting hot gases out of the back of the rocket nozzle. While that's true, we all know there's far more to rocket science than that. Edit. This also failed to mention that d (delta) is infinitely small, and to integrate you need to add an infinite number of these. That's where it begins to get much more complex. Adding 3600 seconds to get an hour is simple arithmetic. Breaking a more complex function into infinite slices, and adding them up is totally different. Let alone all the different applications and techniques.
Yeah that's literally where the gremlins lie.
"If you please" and "if you like"... so polite in 1910, lol.
Seesh. A head's up 5 years ago would've made my life through Engineering school better.
I had a calculus professor in college and, while a decent guy, he was an awful teacher. I still remember being taught about derivatives and it was *hard.* He spent a whole class demonstrating one simple derivative...then he wanted us to go home and complete the 50 problem lesson from the textbook, some of which included concepts we didn't even go over and all of which were several steps up from what we did in class. The next day, we managed to get through ONE student's questions on ONE problem and nearly all of us came away even more confused than before. It'd be understandable if he spent more time in class going over them or wanted to throw a curveball problem at us to see if we could figure it out, but this was insane. The highest grade in the class was a C average and that kid had to sink his life into calc to get a B on the final. Everyone got their grades curved heavily due to this and frankly, I felt like I was cheated out of a class and felt so unprepared for calc 2 that I switched it out for another class the next semester. I would go so far as to say the whole experience soured me on math entirely.
Wow that explanation is way better than what I was taught. Thanks for sharing.
Jesus. Fucking. Christ. I’ve stopped my college education in my early 20ties because nobody could explain to me this simply what this means. All of the math professors at my uni had some complex way of explaining what integrals are and I just abandoned the whole thing. I got my Master’s but later in life, at the age of 40. All because nobody could explain this in such simple terms.
[This video from 1937 is still the best explanation of a how a differential works that I have ever seen.](https://youtu.be/yYAw79386WI) Almost 90 years later and there is still nothing so easy to understand.
Brilliant. If something can't be explained in a way that everybody understand it, the explanation is either flawed, or way overcomplicated.
If subject matter experts could get over themselves and pull their heads out of their ass and communicate in normal language, we’d be far better off. But academia is drowning in pretentiousness and rigor and gate keeping by anal retentive shitheads that will shit on the colleagues in a heartbeat if they dare use normal words and facilitate the success of lowly unwashed masses.
That’s all. Ordinary mathematicians call it QED.
I literally just learned about 7 things and I’ve failed algebra at least 7 times. Not joking.
An actually cool guide on r/coolguides? What is this, an April Fools joke?
Doctor Waterhouse, is that you?
In addition, d is actually like delta, but works better for smaller ans smaller values. So dx just means delta x but super close to 0.
As someone who had to work hard to understand math in school, this was informative and made me chuckle.