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Calculus Made Easy

_virtu
65 replies
12h13m

I’ve been craving some Physics courses since it’s been about a decade since I was in school. I picked up a Classical Mechanics book to get back into the swing of things and of course it went through some basic linear algebra. It’s been a while since I’ve thought about the dot product of two vectors.

You know what blew me away though? Not one textbook I looked in mentioned “why” the dot product is important; that is it’s useful for determining the similarity of two vectors. They all focused on the mechanical details of computing the dot product, but never spelled out the reason it can be useful. I went through a few other resources before I broke down and had a little chat with ChatGPT to discuss the meaning behind it and it makes perfect sense after that.

In comparison to when I was in college, things are much slower paced so I can take the time I need to ensure I have a full grasp of a concept before moving forward. I guess all of this is to say that as I’ve continued forward through more concepts I keep finding that the books I’m reading offer a mechanical view instead of a holistic view of the material. This feels like the biggest issue with most math books I’ve read and it makes me wonder where books that offer more semantic meaning of concepts instead of recipes exist.

liammclennan
28 replies
12h1m

It's not just the books it is the whole method of teaching. I remember learning the steps to calculate an eigenvector without a single comment on why one would ever want to do that. I think it is done so that the educator can claim "this course teaches all of calculus and linear algebra and quantum mechanics". To actually explain things properly would require more modest course goals.

ozim
20 replies
10h14m

I still would argue that method of teaching is perfectly fine.

You cannot simply explain to someone complex stuff - best way is to let people grind through to build their own understanding.

Parent poster wrote that "it’s useful for determining the similarity of two vectors" - now I would ask why do I need to determine similarity of two vectors as it does not mean much to me - if I would be grinding through math problems I would most likely find out why, but there is no way I could understand and retain it when someone would just tell me.

vasco
6 replies
9h36m

You cannot simply explain to someone complex stuff

This is absolutely not true. Many times I've experienced the moment of something complex "clicking" after hearing or reading an appropriate explanation for a phenomenon - finally seeing the right visual or appropriate example or comparison. I find it hard to believe you've never experienced this other than through grinding problem sets.

ozim
5 replies
9h4m

I don't believe something can "click" - without grinding first.

I believe something "clicked" only because you were grinding or already quite familiar with the topic.

There is no way to simply explain complex topic so someone would just get it or someone would "click" on after reading one book on the topic.

vasco
4 replies
8h19m

Like I said, I've had many of those moments. It is strange you haven't. You telling me I just wasn't aware of my own grinding is a bit strange. I've literally had examples where person A explains X and I don't get it, then 2 hours later person B explains X again and I get it. There's no grinding in the middle. There's good and bad ways to explain complex things, and the success rate will vary and the amount of "grinding" you need to do I think also varies depending on the quality of the explanations you get. Maybe I'm missing the core of your point.

tr8798
0 replies
2h59m

Just the fact that there are bad explanations for things immediately proves there are ones relatively better, i.e. good.

olddustytrail
0 replies
1h35m

Well I've definitely had those moments but I happen to have an alternative explanation.

Person B explains X and I don't get it, then 2 hours later person A explains X again and I get it. There's no grinding in the middle. I just needed both perspectives to make sense of X.

jawilson2
0 replies
6h8m

I agree and disagree. I've definitely experienced this, but I've come to the conclusion that the problem is bouncing around in my head for those 2 hours, quietly grinding away in the background. Then, when person B explains it, my brain is more receptive to it. At work as a developer, I'll often encounter a difficult problem, and walk away for a few minutes. Then, if still stuck, I'll go to the gym. Usually when I come back I'll have the answer coded in 5 minutes.

However, I've definitely had the case where person B just explains it better (for me). I still can't completely discount that my brain was primed for it by person A.

hobs
0 replies
5h13m

The only charitable reading of the comment is that the GP is a purely sequential learner - they've dont have epiphanies and gasps of insight like a partially global learner. The learning style stuff is semi-debunked since usually people dont just fit in one category, but they are classically divided as:

Sequential learners prefer to organize information in a linear, orderly fashion. They learn in logically sequenced steps and work with information in an organized and systematic way.

Global learners prefer to organize information more holistically and in a seemingly random manner without seeing connections. They often appear scattered and disorganised in their thinking yet often arrive at a creative or correct end product.

codingrightnow
4 replies
8h23m

Knowing why and when to use math is equally as important as knowing it. One of the reasons I lost my love for learning it was this missing information.

sumtechguy
3 replies
6h6m

What I found very annoying with calculus specifically was the previous 15 years I had been memorizing formulas. Formula to get the area of something remember this thing. Formula to get the volume of something remember this formula. Formula to get the angle of something remember this formula. But if I had known the way of calculus and derivatives I could make those formulas. I now have the ability to have a formula factory instead of devoting tons of mental space to keeping those formulas. I feel I wasted 15 years rote memorizing things instead of understanding the N spaces things live in and how to get the formulas.

vundercind
0 replies
3h3m

Illustrative point for the “the value isn’t explained” issue: I’ve spent two years on calculus in school plus some more time on my own and don’t know how or why I’d e.g. use it to derive the formula for the area of an oval (I think that’s the kind of thing you’re getting at?)

Actually, I can count the times I’ve applied math from later than 6th or 7th grade on one hand. I’m almost 40 and have been writing code for pay since I was 15. I struggle with this with my own kids and dread their reaching those later classes because I have no compelling answer for “why do I have to learn this boring shit?”

ghaff
0 replies
5h23m

But you--or at least most of your classmates--probably weren't in a place to just learn calculus before taking high school physics or even simple geometry. And this happens at a lot of different levels with math, physics, chemistry, etc. There are a lot of inter-relationships and often moving forward requires taking some things on faith (for now).

financltravsty
0 replies
2h33m

How did you learn to make those formulas?

z3phyr
3 replies
9h47m

Ask why do I need to determine similarity of two vectors

Simple:

Start with

a) Suppose you are making a video game..

b) Suppose you are determining ballistic trajectory of your missile system based on model rockets

c) Suppose you are running a fighter robot group..

Or any of the stuff children are supposed to *actually* do and then take these classes with determination to do the actual creative things that they wanna do all life.

There is an aspect of jest in the above comment, but it also contains some likely truth. Children love doing stuff, and these are the things that may enable them.

kyykky
1 replies
8h9m

Take a random group of students from the general population and one of those examples (Edit: or any single given example whatsoeve). Turns out 95% are not really interested.

Edit 2: The teacher probably gave some example from biology or something that you didn't care about and therefore forgot about it.

z3phyr
0 replies
5h34m

The core skill of the teacher lies in recognizing the interests of the pupil and then working on refining those skills so that the pupil can use those skills for at least their betterment, if not the society.

And that is one of the toughest things to get right. Children are extremely curious, that's how they learn and master absolutely anything including arts, dancing, music, history, skating, catching insects, street smarts etc. It's on us as teachers to not let that curiosity wither into nothingness.

xandrius
0 replies
8h8m

This is exactly how I would teach SO many subjects in school: take what kids are actually interested about and make them see the connection.

The worst thing was to learn something just because the teacher said so. If I hadn't had the motivation not to fail, I would definitely not have gone this far in my studies and in life.

The_Colonel
1 replies
5h53m

You're making a false dichotomy. Learning is a combination of guidance and own hard work.

Maybe you prefer to figure out everything yourself, but you have just one lifetime, and having access to guidance while grinding will allow you to learn things faster (and thus more).

pk-protect-ai
0 replies
54m

This is quite an old book. I wish I had an access to or even knew about this book during my school days 40 years ago. When I discovered this book, I was already a middle-aged engineer and was just looking for books for my kids. The first two pages blew my mind. If only I had this book back then... However, I remember now the several sleepless days and nights in sequence when I tried to make sense of finite fields at university. I could not sleep; I could not rest... And then I understood that you can create your own algebra whenever you want; you just need to follow the rules. This was so mind-blowing, and at the same time, no single teacher even tried to point out this wondrous fact that actually changed my mind in such a significant way. It could literally have saved a couple of years of my life if I had read this book back in high school. I'm not a mathematician; math exists for me only when it is applicable to what I have at hand. But you realize that you missed a HUGE AMOUNT OF TOOLS, too late in your life.

jampekka
0 replies
9h56m

I think more intuitive/holistic ways of teaching would be a lot better. But it's hard to do, especially in dead tree format.

To get someone understand something holistically, as in link to their previous knowledge base, requires knowledge of what their knowledge base is. Traditionally this has been done with structuring the teaching with prerequisites etc and hoping it works.

I struggle with this quite a bit when I teach students with heterogeneous background. To be effective, one has to first probe what the students already knows to be able to relate the new stuff to that, and this requires interaction. Hypertext is/would be helpful for self-learning, but it's sadly very underutilized. LLMs may be better. But probably even those can't at least in the current form replace interactive human teaching as they don't really form/retain a model of what the user knows.

hobs
0 replies
5h15m

And you'd be dead wrong by all methods of pedagogy that we find useful! People learn best with stories and meaning, its why the ancients were able to reproduce stories of almost inhuman length via memory.

Grinding through to build your own understanding when someone can just give you useful meaning and context to connect to your other parts of learning is a core teaching skill, and anyone avoiding that because its "too hard" is doing a deep disservice to their students.

aerhardt
4 replies
11h44m

I was taught linear algebra and multivariate calculus as a business major. They could hardly justify why they were teaching it in that context - they were weeder courses - but I always wished they had at least tried to give us a hint of applications. Nothing, it was all algebra for the sake of algebra. Atrocious.

klysm
3 replies
10h38m

I think the concepts of linear systems and multivaribale calculus are important for just understanding systems in general. Even without applying them all the time you can think about dynamics with them

aerhardt
2 replies
9h18m

Multivariate calculus is also useful in probability, which in my degree was rigorous too, and is broadly useful in business, so perhaps I’m being unfair about all that math not being useful in business management. I’m grateful because later I got a degree in software engineering… But the point about math being taught like shit stands; if calculus and algebra can be useful in thinking about systems they should make an effort to show it.

klysm
1 replies
9h5m

100% agree that mathematical pedagogy in the USA is in terrible shape. It’s a hard problem, but we can do so much better

aerhardt
0 replies
7h21m

This was in Spain, but I have no problem believing it's roughly the same everywhere in the world, with few exceptions.

vasco
1 replies
11h38m

I think I only learned linear algebra about 3 or 4 years after I graduated. I learned how to do the computations during the course but the teacher was so bad I had no idea what anything was for. Could've been an IQ test course for all it mattered. Here transpose this matrix now. Ok.

klysm
0 replies
10h39m

Pretty common with lin alg and diffeq unfortunately. Many schools teach it as a toolbox instead of for understanding.

Nevermark
11 replies
9h31m

In high school trigonometry I am sure I was clear that sine and cosine formed the circle. How could I not?

But that fact’s significance and too obvious simplicity, with all its ramifications, only hit me deeply and profoundly when a year later I realized I could use those functions to draw a circle on a screen.

Before that they were abstractions related to other abstractions that I had to memorize to pass a course.

To this day I am frustrated when reading papers about abstract algebraic relations and other such concepts, without even a sentence or two discussing any intuitive way to think about them. Just their symbolic relations.

I appreciate that in the game of math that view becomes natural. But most of us learn math with additional motivations and are interested in any perspective that highlights potential usefulness or connection to the real world. Many of us mentally organize our knowledge teleologically.

Yet even when usefulness is known to exist, it is often neither mentioned or referenced. Or even considered relevant.

Edit: the same goes for not showing a single concrete example of an abstract concept. A kind of communication that would unlock many mathematical papers to a much larger audience of intelligent and relevant readers.

skhunted
5 replies
7h6m

I’ve been teaching mathematics for over 30 years at the community college level. Most people at the time of taking a course don’t have a sophisticated enough understanding of math to really appreciate “intuitive explanations” because they don’t have intuition.

Take parametric curves. I explain that they generalize the concept of a function. Every function can be parametrized in a trivial way. They don’t really understand this concept. They have a hard time parametrizing a function and do so only becuase of a formula.

The fact is most people need to go through the mechanical process of doin g before they can get to a point of understanding. It takes almost the entire semester for me to convince beginning algebra students that the reason that 2x + 3x is 5x is because of the distributive property. And when they do understand it they don’t understand why that is important.

Later on when things click for someone they will often say things like, “Why didn’t they just tell this when we took the course?” Usually we did. You just didn’t have a sophisticated enough understanding of things to grok it at the time you took the course.

Nevermark
4 replies
6h20m

I am sure you are right, and a little convo here isn't going to do the topic justice. So many aspects to how people understand and learn things.

And I know picking on your example isn't in the league of a general solution.

But if 2x (which is x + x) is two apples in a box, and 3x (which is x+x+x) is three apples in another box, then you put those two boxes in a bigger box (another +), people already intuitively can see the distributed property of scalar multiplication vs. addition of some unit, they just didn't have a name for it.

Likewise, a 3x4 square of paper next to a 7x4 piece of paper can be easily seen to be a 10x4 piece of paper. Multiplication of numbers over added numbers.

So one way to introduce distribution is to start by showing examples of several places where people already understand the concept of multiplication distributing, and use it every day, but just didn't know it was one concept with a name.

Once people can recognize distribution as an already familiar relationship in everyday life, then the symbols can be visited as the way we write down the already known and useful concept so we can be very clear and general about it.

Anyway, that's just a reaction to one example, which may not mean much.

skhunted
3 replies
6h3m

Yes, almost everyone gets that. Then you need to explain that x is really 1 x. Then you need to explain that -x is really (-1)x. Everything is great. We all understand. Now simplify (1/2) x + 3x. You’ll lose most people at this step. Then explain (1/2) x - (2/3)x. More confusion. Now explain that ax+x is (a+1)x. You lost a lot of people at this step. Now explain that xy+ x^2y is (x+x^2)y and that this is just the distributive property “in reverse”.

Sometime later a person will really grok all this and then say, “why didn’t they just tell us this is all just the distributive property?”

Nevermark
2 replies
5h55m

Ah, I get it.

A different approach I have thought about, which would really tear the textbooks apart, is introducing every concept in its simplest form as early as possible.

Then when it is eventually expanded on, its familiarity will aid in taking further steps more quickly and intuitively.

For instance, something as simple as adding up the area of a fence of varying heights, or the area of multi-height wall to be painted, being referred to as integrating the area, in early grade arithmetic, creates a conceptual link for down the road.

Systematically going over K-12 materials, just making similar small adjustments to terminology and concepts to be highlighted, would be interesting.

skhunted
0 replies
5h10m

As I see it the issue with your integrating example is that area is the correct word for finding “area”. Integrating is not finding the area. The indefinite integral is not about area. The definite integral in dimension 1 has to do with signed area. I don’t think having people ingrained to think finding area is “integration” would be a good thing. Especially since most people don’t take calculus.

To your point, people do constantly try to tweak things to make subjects easier to understand and more intuitive.

lupire
0 replies
2h25m

That's how Riemann integrals are defined in class. Vertical slices.

Tossing the word "integral" at younger children won't make that easier or harder.

xanderlewis
2 replies
6h34m

To this day I am frustrated when reading papers about abstract algebraic relations and other such concepts, without even a sentence or two discussing any intuitive way to think about them. Just their symbolic relations.

If you’re talking about research papers, that’s just because they’re written for domain experts and aren’t really for giving you intuition. They’re written in a deliberately terse (one might say elegant) style to convey the research findings in formal mathematical language and nothing much else. If you want to gain an intuitive grasp of things, read a proper textbook in detail or play around with the ideas on paper. Or both!

I guess the reason is that once you’ve acquired the intuition, having the literature cluttered up with the same explanations again and again becomes clunky and increases the volume of material to be sifted through when you’re just looking for a result you need in your research and don’t need all the extra chatter. It’s just cleaner that way. But to an outsider it does look more opaque. It’s a trade off.

Nevermark
1 replies
6h6m

having the literature cluttered up with the same explanations again and again becomes clunky

I think that really is the best reason for not being more accessible. Along with less work - given a good paper already can take a lot of work to write clearly even for the immediate audience.

But there is tremendous value in reaching a wider audience, for readers, writers, and the very real serendipity of cross pollinating ideas. So an easily skipped concise titled section, that gave a little context or example for the non-inside crowd, would be a nice tradition. Even an appendix - although that might strike the established culture as too quirky.

Some papers manage to do something like that, a colorful example or perspective adding levity as well as clarity. So it is not breaking any barriers. Just not standard or prescribed.

Or maybe it wouldn't have much impact. I tend to find reasons to dive into many different new topics, so it is a prevalent need for one!

xanderlewis
0 replies
2h10m

an easily skipped concise titled section, that gave a little context or example for the non-inside crowd, would be a nice tradition.

I completely agree — especially in the modern era where extra pages cost nothing.

WillAdams
1 replies
3h25m

I've been trying to work through this in the context of programming a CNC using a recent trilogy of books:

_Make: Geometry: Learn by coding, 3D printing and building_ https://www.goodreads.com/book/show/58059196-make

_Make: Trigonometry: Build your way from triangles to analytic geometry_ https://www.goodreads.com/book/show/123127774-make

_Make: Calculus: Build models to learn, visualize, and explore_ https://www.goodreads.com/book/show/61739368-make

(oddly the Calculus book was published second, so I guess I'll need to re-read it after I finish the trigonometry book)

Hopefully, this will provide me with a sufficient grounding in conic sections that I can solve my next CNC project with a reasonably efficient set of calculations (trying to do it using my rudimentary understanding of triangles from trigonometry had me 4 or 5 triangles deep, barely half-way to the final point I needed, and OpenSCAD badly bogged down performance-wise).

Nevermark
0 replies
1h2m

Thanks for that list, I just ordered them! I hadn't heard of that series before

elric
3 replies
7h17m

A long time ago, when I was in high school, we had an introductory course to differentials and integral calculus. When I asked what the purpose of integration was, the teacher shouted that I should save the stupid questions for my parents ... She was a shit teacher for various reasons, but that was the day that I lost my drive for maths.

It wasn't until years later that I found that it was all about "the area under the curve" and why that would be useful. At no point in those high school classes did we ever work a practical example. I was pissed off all over again when I found out how useful that stuff could be, and how much I'd missed out on.

I'm sure most teachers mean well, and I'm sure most of them try. But by god there are some truly awful twats out there who should never set foot in a classroom again.

lo_zamoyski
0 replies
5h49m

When I asked what the purpose of integration was, the teacher shouted that I should save the stupid questions for my parents

What an awful person. Chances are she was getting defensive and covering for her own lack of understanding. If I were a parent, I would confront her about that, not least of all her contempt for students and for learning, but toward parents.

Teachers don't know everything, and when they don't know, they should be able to admit that without hesitation or defensiveness. This sets a good example in general, of humility, instead of inculcating the notion that life is about having all the answers, or rather, pretending to have all the answers. All this does is set up people to become imposters. Of course, if you're teaching calculus, you should have at least a basic grasp of the material, and if you don't, you should say so, so that you've not put in a position where you have to teach it.

I'm sure most teachers mean well, and I'm sure most of them try.

I think it is generally accepted that primary education isn't exactly packed with the best candidates, both from the point of view of pedagogical ability as well as mastery of the material.

Moru
0 replies
7h11m

We had a good math teacher. There was a formula he just told us to memorize, the class asked how it worked but he just said we don't need to know why or how, just like we don't know how a calculator works. What he didn't know was that the class last week in electronics was about how calculators work.

He had to confess he didn't know why or how either of them works, he just uses them :-)

ndriscoll
2 replies
3h40m

That seems pretty surprising to me. The lower level/physics books I've seen introduce the dot product with both a geometric and algebraic definition, and show they're equivalent in 2-3 dimensions. The "how" is the algebraic definition and the "why" is the geometric definition.

It's not really that it measures similarity. Physics isn't interested in that. It's that it tells you lengths and angles, which you need in all sorts of calculations. In more advanced settings, a dot product is generally taken as the definition of lengths and angles in more abstract spaces (e.g. the angle between two functions).

In machine learning applications, you want a definition of similarity, and one that you could use is that the angle between them is small, so that's where that notion comes in. A more traditional measure of similarity would be the length of the difference (i.e. the distance), which is also calculated using a dot product.

lupire
1 replies
3h26m

Angle is a measure of similarity (well, distance/nearness).

In physics, the dot product is used to losslessly project a vector onto an orthonormal basis, and the angle measures how much of the vector's magnitude is distributed to each bases vector.

The angle can be defined in terms of the dot product, because you don't need the angle (as in a uniform measure of rotation) in order to compute important physical results.

ndriscoll
0 replies
2h50m

Angle is one way to measure similarity. A more natural one in most settings is distance. In any case, physics isn't really concerned with similarity.

The angle can be defined in terms of dot product because |a|=sqrt(a•a) can be shown to be a norm, and because a•b/|a||b| can be shown to always be between -1 and 1, and because those things agree with length and cosine of the angle for Euclidean spaces. It's not that you don't need the angle. It's that the dot product gives a good definition of angle in settings where it's otherwise not clear what it would be (e.g. what's the angle between two polynomials `x` and `x^2`)

In programming terms, there are interfaces for things like length and angle (properties that those things should satisfy). If you implement the dot product interface, you get implementations of those other ones automatically. The "autogenerated" implementations agree with the ones we'd normally use in Euclidean geometry.

nathan_compton
2 replies
8h46m

The sheer amount of material a student needs to digest in order to become conversant as even a pseudo-professional is enormous, which I think excuses, to some degree, the strange style of text books. I personally find that education is a process of emanations: first one digests the jargon and the mechanical activity of some subject (taking a dot product, in this case) and then one revisits the concepts with the distracting unfamiliarity of the technical accoutrements diminished by previous exposure. Thus able to digest the concepts better, the student can revisit the technical material again with a deeper appreciation of what is happening. The process repeats ad-infinitum until you ask yourself "what even IS quantum field theory?"

wholinator2
1 replies
6h49m

Is there someone now that can explain the intuition behind QFT?

nathan_compton
0 replies
3h56m

The intuition behind QFT isn't the problem. I'd argue its quite intuitive: write a classical field, assume some plausible commutation relations, turn the crank. To add interactions pretend that you observe the results at infinity or whatever and take some terms of a power series representing the amplitudes, adding a cut off which you calibrate with an experiment. All fine and dandy. Just sucks that the machinery doesn't quite pass a combination of mathematical rigor and philosophical substance.

mdavidn
1 replies
5h49m

I remember being shocked in the first year of college that introductory physics and introductory derivatives and integrations were not taught together. The calculus class never explains why these methods are useful, and the physics class expects rote memorization of the final algebraic equations.

zehaeva
0 replies
5h41m

It might be because you weren't in a Physics or an Engineering program.

Colleges tend to have two tracks for physics, one that's closer to high school physics, which is as you described. A collection of algebraic equations that you have to either remember or, if your professor was kind, given a crib sheet of.

The other is the "Engineering" or "Calculus" based physics track where, as you can imagine, you're taking Calc 1 and Physics 1 at the same time.

I have seen some, kinder, programs where you take Calc 1 in your first semester and start the Physics classes in your second semester.

jahnu
1 replies
11h12m

Not teaching the Why is such a sin! I didn't understand calculus properly at all until I read Steven Strogatz' brilliant book Inifinte Powers, which not only explained the why but the history of why. 10/10 book for me.

https://www.stevenstrogatz.com/books/infinite-powers

lo_zamoyski
0 replies
4h37m

Modern education is grounded in a different worldview than the classical liberal arts[0]. The classical liberal arts are so-called because they are freed from the burden of having to be practical or economic in nature (which is not to say they could not or did not incidentally have practical application), intended to produce a free man. Here, too, by "free" we mean free to be good, that is, more fully human, not what we mean by freedom today as doing whatever you happen to feel like doing, a recipe for enslavement, misery, and despair, and therefore directly opposed to the good and to becoming more human.

Opposed to the liberal arts were the illiberal or servile arts. These are necessary and good, of course, but necessarily inferior to the liberal arts because their end is not truth or formation; they are instead practical, concerned with effecting some kind of economic end. The point here is not to disparage, but to understand how all of these are related and ranked according to a "for the sake of" relation. A human being doesn't exist to eat, he eats to exist, for instance.

Modern education is very much oriented toward the servile arts, and what passes for the liberal arts today is anything but the classical notion.

The point is that modern education is less interested in leading to understanding, realizing virtuous habits, and leading to freedom, and more interested in churning out workers. Workers don't ask "why" (though we can agree that those who do can, guided by prudence, contribute more economically). Indeed, that is perhaps the key difference between classical science and modern science: the emphasis of the former is truth, while that of the latter is control of nature. Of course, it isn't that you must choose absolutely between understanding and effectiveness, and the classical tradition does not claim either that study precludes work. Study often requires work, for sake of preparing the way for truth. Rather, it is that the end of the modern educational tradition is different from that of classical education, and this end determines the form of the pedagogical methodology. It is a difference in anthropology, of the vision of man.

All men work, but what do they work for? Do they work for work's sake, or perhaps to make money to satiate their base appetites (modern view)? Or do they work in order to be free to pursue higher ends[1]?

[0] https://www.newadvent.org/cathen/01760a.htm

[1] https://a.co/d/hE5830i

wodenokoto
0 replies
8h53m

This is why 1b3b is so popular. Instead of teaching the mechanics he teaches the intuition.

With that being said, I do remember my math and physics teachers in high school spend lots of time talking about the why and intuitions and let the books state the how.

treflop
0 replies
4h4m

I find the math portions in physics books are just basic refreshers.

I guess if you want to learn math, only a math textbook will actually care.

threatofrain
0 replies
11h7m

Colleges often have multiple classes on the same math subject, one made for physics and ME/EE people, one made for psych people, and one for CS. Some people don't realize that they accidentally picked up a textbook meant for a specific college pathway they don't care about.

Understandably college courses & textbooks meant for CS people will be more focused on computation, while a math major who is taking Linear Algebra will get a more theoretically motivated course. Gilbert Strang is an example of an engineering-focused text while Sheldon Axler or Katznelson & Katznelson is an example of what a math major would experience.

jampekka
0 replies
10h4m

Maybe the books assume that the geometrical interpretations of the dot product are already known by the reader? I think they (both the projection interpretation and relation to angle between vectors) were taught in high school at the latest. There's also a lot of interpretations and uses for the dot product, some of which aren't necessarily that useful for classical mechanics.

But in general, literature using and/or teaching mathematics does tend to be too algebraic/mechanistic. Languge models can be a very good aide here!

diffeomorphism
0 replies
8h58m

Not one textbook I looked in mentioned “why” the dot product is important; that is it’s useful for determining the similarity of two vectors.

Most textbooks motivate it by the angle between the vectors or as projections (e.g., for hyperplanes). Numerics-focused ones will further emphasize how great it it is that you can compute this information so efficiently, parallelizable etc.. Later on it will be about Hilbert space theory or Riemannian geometry and how having a scalar product available gives you lots of structure.

This feels like the biggest issue with most math books I’ve read and it makes me wonder where books that offer more semantic meaning of concepts instead of recipes exist.

All of the good ones do both. They first give the motivation and intuition and then make matters precise (because intuition can be wrong).

analog31
0 replies
6h4m

I've seen a lot of comments, in this thread and others, to the effect of: "I didn't get math until I looked at it in a different way, with a lengthy span of time in between." Maybe just the two different looks and the time span by themselves are beneficial.

ajkjk
0 replies
6h0m

There are two ways to see every operation: a mathematical way and a physical way. The mathematical view of the dot product is an operation on vectors that adds their multiplied components, a·b = a_x b_x + a_y b_y + a_z b_z. The physical view of the dot product is what you said, comparing two vectors for similarity, or, in alternatively, multiplying their parallel components like scalars. The difference between these perspectives is in what is regarded as the defining property of the operation, which affects what you keep "fixed" as you vary aspects of the theory you're working in. For instance, when switching to spherical coordinates, the mathematical version of the dot product could still look the same, but the physical version has to change to preserve the underlying concept, which means its form becomes quite messy: (a_r, a_θ, a_φ)·(b_r, b_θ, b_φ) = a_r b_r (sin (a_θ) sin (b_θ) cos (a_φ - b_φ) + cos a_θ cos b_θ.

The difference in pedagogy seems to be which of these perspectives is treated as fundamental. Math education tends to treat the mathematical operations as fundamental. Physics treats the concept as fundamental and regards the operation as an implementation detail. It is very similar to how in software development you (for the most part) treat an API's interface as more fundamental than its implementation.

Unfortunately even physics books don't go over the intuition for underlying math very well, to their detriment. They seem to just assume everyone already perfectly understands multivariable calculus and linear algebra. I think it's because by the time you've gotten through a physics PhD you have to be completely fluent in those and the authors forget what it was like to find them confusing.

adhamsalama
0 replies
10h21m

Try 3blue1brown. You'll love it.

abrookewood
0 replies
10h19m

So don't leave us hanging ...

019341097
0 replies
5h5m

You might really enjoy working through the Art of Problem solving series. They’re early math -> calc books for kids that are getting into math competitions, and they explain so much in so much detail and really get to the root of why while also developing intuition. Get the e-book version. The explainers are incredible.

conwy
34 replies
13h42m

As someone who's taking a university entrance course in Calculus I find these kind of "calculus made easy" pamphlets irritatingly trite.

The hard part isn't the highest level concepts, which are actually fairly easy to grasp and somewhat intuitive.

The hard part is all the foundational knowledge required to solve actual math problems with Calculus.

The most difficult parts of Calculus (for me at least) are:

1. Having a very thorough grasp of the groundwork / assumed knowledge. Good enough that you can correctly solve an unexpected problem, from completing the square to long division of polynomials to an equation involving differentials.

2. Understanding and correctly applying the notation and graphing techniques, from Leibniz notation to sketching curves.

This is why large books and courses exist covering only introductory Calculus, not even beginning to scrape the surface of more advanced math.

getcrunk
17 replies
13h32m

Most of your first point is … algebra? Yes if your algebra is weak you will not be able to cope with solving calc equations. The solution to that problem is not to be found in a calculus made easy. It would be found in algebra made easy.

conwy
13 replies
13h12m

Yes, it's algebra.

Math isn't like programming. In programming you can often solve a problem using a library, framework, language facility, etc. without entirely understanding why it works all the way down to the binary level.

In math you can't often solve a more advanced problem such as Calculus problem without understanding the more foundational math such as algebra, fractions, etc.

If "information hiding" / layers of abstraction was possible in math, I would have completed my university entrance course months ago, but here I am still struggling.

Sure, we could have Algebra made easy and also Trigonometry made easy, Fractions made easy, Functions made easy, etc. etc.

I just find it personally irritating that all this foundational knowledge is brushed aside when it's really core to someone's actual competence dealing with actual math problems.

Maybe it's just assumed that people went to a good high school or had a private math tutor and already learned the foundations very well, but I think at least that assumption would be coming from a place of privilege.

It's similar to telling someone to take a Bootcamp in React and that will be enough for them to succeed as a software engineer. But to solve the kind of problems they are going to face in reality they will eventually have to learn at least some foundational Javascript and maybe a little about algorithms and data structures.

goosejuice
4 replies
12h40m

Do you find it surprising though?

Certainly not everyone is in the same place in their learning journey as you. Material on calc, at a university level, is typically going to focus on calc. Yes it is assumed that you have learned the fundamentals before taking that course.

I was in a similar situation as you. If you really want to learn it there's no substitute for skipping over the fundamentals. I did that and did fairly well but it's all long forgotten. Never use the stuff :)

conwy
3 replies
12h13m

Never use the stuff :)

So many people tell me this that it's become cliche at this point.

I find it demotivating, but unfortunately I have to press through, as there is literally no other way I'm going to gain entry to my university's bachelors program.

A part of me wonders if this kind of fundamental knowledge could be actually useful, similar to being able to cook your own food instead of takeaway.

Kind of like how "first principles" thinking can apparently lead to new discoveries because you're not just mimicking / re-using the same structures that were already built.

vnce
0 replies
10h35m

I’m a product manager and I use the concepts to read and understand new algos, research papers, etc. you’re right that you won’t be calculating (that builds problem solving) but grasping the principles will help you proceed to more advanced concepts in other fields

Good luck you’ll get there.

goosejuice
0 replies
29m

My experience certainly isn't representative! I just happen to build things where university level maths rarely comes up. Stats comes up more than anything and sadly only had to take one course in that area.

Since you bring up food. As a former professional baker it would also take me some time to make croissants professionally at the level I used to. At least for me personally, if I don't use it I lose it. But I can certainly pick up faster than someone seeing it for the first time if I needed to.

Along the way you'll pick up some intuition that you can use elsewhere that's hard to quantify. Outside of the loans I don't regret taking any of the maths required for my CS degree.

Personally, I found the calculus lifesaver by Adrian Baker to be helpful in my studies as someone that was missing some fundamentals

RF_Savage
0 replies
11h24m

Being able to check the numbers software and manufacturers provide is a good use.

lll-o-lll
2 replies
12h24m

In math you can't often solve a more advanced problem such as Calculus problem without understanding the more foundational math such as algebra, fractions, etc.

Yep, this is the number one reason people think they aren’t suited for math. Everything is built on everything else, and if you missed anything you’re screwed. It takes a while to realise you are screwed, you can get by on rote for a surprising distance.

Ultimately, “there is no royal road”, but a good tutor will help you find those gaps and build out the missing bricks.

philwelch
0 replies
8h55m

This does depend on the curriculum to some degree, and whether you’re just trying to grasp a concept firmly enough to move onto a more advanced concept or whether you’re trying to build a practical skill in solving problems. For instance, it’s entirely possible to understand higher level mathematics without having much skill at all in pencil-and-paper arithmetic. I know this because one of my best friends in college got straight A’s in upper level mathematics and EE classes but, due to his unusual background, only bothered learning arithmetic when he needed to prepare for the GRE.

I didn’t enjoy math as a child, and I used to be a lot more bitter about this when I first started to grasp what mathematics actually was. As a child, mathematics seemed like a small amount of “learn and understand a new abstract concept” (which I was pretty good at) bogged down with a huge amount of “okay now you have to solve a a bunch of problems based on that concept over and over again before we’ll trust you with another concept”. Eventually I figured out that mathematics itself really is the concepts, and that the concepts eventually build up to a level of complexity where it was increasingly challenging and fun to grasp them.

Maybe the reason it’s taught this way is because the vast majority of people aren’t mathematicians and aren’t really attracted to mathematics out of an abstract intellectual appreciation for the beauty of mathematical concepts; they just want to solve problems. And this is perfectly reasonable. But if I had it to do over again, I probably would have put more effort into mathematics and study more of it, at much higher levels, if I knew it would eventually get a lot more interesting.

And eventually things do start to branch out a bit. The standard K-12 curriculum up through calculus mostly builds up like a single tower where everything is built on everything else, but there are parts of mathematics where you can just sort of go in a different direction for awhile.

conwy
0 replies
12h9m

Yep, this is the number one reason people think they aren’t suited for math. Everything is built on everything else, and if you missed anything you’re screwed. It takes a while to realise you are screwed, you can get by on rote for a surprising distance.

That's exactly what happened to me!

This is why I'm learning about differentiation yet struggling to factor simple fractions with a surd.

It's similar to the "expert beginner" problem described by Erick Dietrich (https://daedtech.com/how-developers-stop-learning-rise-of-th...).

bakuninsbart
1 replies
9h39m

In math you can't often solve a more advanced problem such as Calculus problem without understanding the more foundational math such as algebra, fractions, etc.

This is true to a good degree, but maybe a bit less so than you believe. Trigonometry is a topic that only clicked for me after finishing my uni calculus curriculum, I didn't get a great grade, but got by with a technique similar to how we handle complex numbers: Instead of giving up after being unable to solve an eg. weird chain of sin, arccos etc. functions, just declare it to be u(x) and do the calculus bits around it. In the last step substitute the actual function back in and you have an incomplete, yet technically correct solution.

conwy
0 replies
7h46m

I know what you mean, in fact I remember earlier on when I started the course, I had wanted to use these kinds of substitution techniques, etc. and thought I could finish the course in a few days. Boy was I wrong!

These techniques definitely won't work in a tough online multiple-choice test (of the kind I'm getting) where they deliberately sprinkle in subtle quirks to deceive you, which would require very disciplined algebra, fractions, powers, etc. to identify.

jddj
0 replies
10h58m

It was a while ago now but I remember our university mathematics required passing a small algebra module that covered essentially all of highschool algebra.

enugu
0 replies
4h10m

Reusing and blackboxes do appear a lot in higher level mathematics. Indeed, the idea behind abstract algebra is to hide 'implementation' details. The concept of abstract data type in programming is similar to structures studied in algebra.

  It is common for mathematicians to rely on theorems as black boxes(ex: classification of surfaces) even without knowing the proof. Secondly, people can even write research papers without knowing how to work with some object covered in the paper, by working with collaborators who are experts on a different topic.

  It would be helpful to isolate the essence of calculus itself from the symbolic techniques, for ex to actually calculate integrals(especially  magical seeming substitutions and nontrivial factorizations) as many of these symbolic techniques will appear in different topics even outside calclus.

Here's a criterion for testing this core understanding calculus - Can somebody given a problem (say optimization, or finding volumes) convert it into a standard type of differentiation or integral, then use symbolic software like Mathematica to do the computation and then get the right answer. Often, calculus students memorize standard recipes for problems and get confused by a problem which is not hard symbolically, but requires some thought to set up correctly.

bee_rider
0 replies
3h9m

It’s almost always algebra in the early calculus classes I think. I tutored an “into to calc for non-STEM majors” class for a couple years, and it was always algebra. If you have teaching assistants for the class, and you go to them with: I think I understand the calculus, but I’m struggling to simplify things in algebra, they might be able to help you out.

Math classes build up, and at some point unfortunately they do have to start assuming that your previous classes were solid. Calculus is where algebra and trigonometry gets some of that treatment. It is extremely common for a calculus class to reveal some shaky algebra foundations though, so I’d hope your school has some help there…

hintymad
2 replies
12h50m

Algebra is the simple part. I’d say it’s more about math maturity. At least 1/3rd of my classmates had a hard time grasping the epsilon-delta definition of limit, let alone the deeper definitions like Cauchy sequence or those used in the proof that R is dense(and we were in an elite university’s competitive program). Among the survivors of single-variable calculus, at least 1/3 could barely get by the multi-variable calculus. I saw too many of my friends struggle with different integrals, and got massacred by Green’s equation.

My guess is that most people hit a wall of abstraction at certain point.

conwy
1 replies
12h2m

My guess is that most people hit a wall of abstraction at certain point.

I don't think it's a limit to their abstraction, I think it's that they didn't work properly on the fundamentals, so they had a superficial understanding of the abstractions.

To give a fitness analogy it's like trying to do heavy barbell presses before you can even do 10 pushups in a row.

My experience with programming is that once you get really really good with fundamentals you suddenly leap ahead and pick up new languages, paradigms, etc. incredibly fast.

Maybe this partly explains the 10x phenomenon - it's because they worked very hard on the fundamentals.

globalnode
0 replies
10h46m

my view is software by comparison is like a single surface of knowledge; once you know the basics, thats it, nothings too hard to learn. maths on the other hand is more like a volume of knowledge.

svat
5 replies
12h16m

these kind of "calculus made easy" pamphlets

The link is not a pamphlet (unless you read only the linked HTML page). It is an entire book, published in 1910 by Silvanus P. Thompson, and sufficiently well-regarded that it was re-edited in 1998 by Martin Gardner, and (independently) lovingly re-typeset in TeX by volunteers (and also turned into this website). Clearly it serves a need, and is not merely a “trite” pamphlet.

(The edition by Gardner is actually recommended against by some, who see in it a clash of two strong personalities, individually delightful.)

conwy
3 replies
12h7m

It serves a need just not my need!

(As someone who's surprisingly bad at math and trying to undertake a Calculus pre-req to get into university!)

radicalbyte
1 replies
11h35m

I read Calculus Made Easy about 15 years after my last math lesson, and had forgotten a lot of the mechanics of algebra. I went through Algebra and Algebra II for Dummies before reading it. They're really concise, very easy to read and absolutely did the job for me.

Calculus Made Easy is an amazing book btw, by far the best introduction and much better than the way I was taught at school as it actually builds your intuition.

conwy
0 replies
10h20m

Ok well thanks for the Algebra book recommendations, will see about working through those first.

latexr
0 replies
10h18m

It serves a need just not my need!

Which is fair, but if you believe that you shouldn’t have insulted the work itself by dismissing the value of its content and calling it a trite pamphlet.

weebull
0 replies
10h10m

It's a great book, and one my father recommended to me to get me through the concepts when I was having trouble with the standardised teaching of the day.

It comes down to Leibnitz Vs Newton, and the world has standardised on the notation of one (I forget which). However the notation is a destination when learning it all, and the foundational ideas behind calculus were best explained taking ideas from both of them.

That's what this book does. It takes you through with every simple jumps in logic allowing you to discover calculus yourself and you therefore have the foundations to reason about it yourself. You don't just have to learn the final answers by rote.

gofreddygo
4 replies
12h46m

So you want to do calculus ? You need algebra. What parts of algebra ? Go figure !

this is one big hurdle in learning math backwards. You discover new missing pieces at every corner. Each missing piece leading to another missing piece.

Learning math from the basics to advanced (as recommended by most) is very frustrating at how slowly you actually develop the math muscle.

At a deeper level, conceptual grasp does not make you good at math, its not enough. You may fool yourself into thinking you "get it" till you try to solve a few exercises. You need to repeat the lower levels enough to make it into muscle memory (which some people refer to as math intuition or groundwork) before embarking onto higher levels that build on it.

So working your way bottom up is slow and frustrating, top down is slow and frustrating. What do you do?

Just keep at it. One key observation for me was that at some point the misery and rabbit hole nature diminishes, quite rapidly. The groundwork of solving all those exercises repeatedly pays off and the next set becomes a little easier. Getting to calculus after spending ridiculous amount of time on algebra is the only way I have known to work.

And this is true for learning progamming too. knowing the concept of loops is essential but, you still can't write efficient code to sort an array. You need to get the syntax and write enough loops and then progress to exercising writing specific sorting algorithms repeatedly to get them into muscle memory.

But there is an inflection point beyond which the same concepts repeat but in different variations and they take progressively lesser time to get a grasp on.

thats just how I've learned math and programming. Also why a large percentage of people just give up hope and accept they just don't have the math gene. Meh.

conwy
2 replies
11h53m

this is one big hurdle in learning math backwards. You discover new missing pieces at every corner. Each missing piece leading to another missing piece.

Yes this is exactly what happening to me.

E.g. I got up to Week 2 of the course and suddenly made the big (to me) discovery that sqrt(a/b) = sqrt(a)/sqrt(b).

It seems trivial I know when you see it written like that, but the problem is to recognise and apply that principle in the context of a broader problem such as factoring.

Just keep at it.

Thanks, this gives me confidence that I'm not wasting my time haha

I am beginning to get better at it, to the point that I can often work out why I got a question wrong on my own without referring to the answer.

gofreddygo
0 replies
1h35m

It's really frustrating how every single person I know who got (really) good at math or programming got there the same way, but never even hinted about it to me till I saw them use the same techniques. The clever ones figured out the important parts faster and spent more time on repeating the common idioms, theorems and required prior knowledge (e.g. the sqrt(a/b) = sqrt(a)/sqrt(b) piece for you) instead of the problem or spending too much time on conceptual understanding

The really important part for me was to rip these small but critical parts out and form somewhat like mental workout routine that I kept repeating multiple times per week. By week 5/6 I could solve the same/similar/related problems which weeks ago took me several minutes with ease and I had more brain power left to think about higher level and related concepts and techniques that formed more connections, making the experience a lot more fruitful, productive and faster. Without that mindless, disciplined mental routine to get the basic and critical stuff in muscle memory, I do not believe I could have made it through.

Good luck.

barrenko
0 replies
7h38m

what parts?

all of it really
beltsazar
2 replies
11h0m

I feel the opposite. In high school I was pretty good at solving calculus problems but had little understanding what "limit" actually is. When in college I finally understood the definition of limit and all the foundational theorems arised from it, I was blown away.

For most people—who won't solve complex math problems daily at work—the takeaway from learning math is not their mechanical ability at solving math problems. The takeaway is their understanding of math concepts and ideas, which will shape their thinking skills in general.

jasomill
0 replies
9h32m

For me, the biggest stumbling block in understanding the usual ε/δ limit definition in high school was teachers reading |x - a| as "the absolute value of x minus a" rather than "the distance between x and a".

The later reading suggests a more intuitive (to me) definition: a limit f(x)→q as xp exists if, for every open interval Y containing q, an open interval X containing p exists such that f(x)∊Y for all xp in X (and then if f(p)=q, f is also continuous at p).

Another nice property of the above definition: replace "interval" with "ball" or "neighborhood" for analogous definitions for functions between metric and topological spaces, respectively.

conwy
0 replies
7h44m

For most people—who won't solve complex math problems daily at work—the takeaway from learning math is not their mechanical ability at solving math problems. The takeaway is their understanding of math concepts and ideas, which will shape their thinking skills in general.

Agree, but for some others, there are real world consequences, e.g. whether they get accepted into a university or whether they can read and properly understand an academic paper.

ecshafer
1 replies
4h56m

Your issues seem to be algebra. I recommend Khan academy personally and just working through all of the highschool math that he goes over. I found his stuff when he was still just a guy on youtube back when I was in the same position as you. Studying calculus, did fine in high school, but my school was not good and totally unprepared me for actually studying math, skipping over a lot of those fundamentals. So often I would have a professor or TA take a complex equation, show an "obvious trick" that we "knew from algebra" and it would be the first time I ever saw that in my life. There is really no other solution than to study and relearn algebra, geometry and trig yourself as you learn calculus.

Yhippa
0 replies
4h38m

I remember taking a graduate level cryptography course several years ago and relying on Khan Academy to grok the fundamental concepts.

wackget
17 replies
16h24m

Great if this works for you but honestly I don't find this "easy" at all. The writing style is annoyingly formal and already on page 2 it jumps into "draw the rest of the fucking owl" territory:

> A very simple example will serve as illustration.

> Let us think of xx as a quantity that can grow by a small amount so as to become x+dxx+dx, where dxdx is the small increment added by growth. The square of this is x2+2x⋅dx+(dx)2x2+2x·dx+(dx)2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x2x2.
nextaccountic
4 replies
15h59m

I actually love the intuition built with geometric arguments using infinitesimals.

You need something like smooth infinitesimal analysis [0] to make this rigorous, but it's much better than anything involving limits.

[0] https://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf

gmane
3 replies
15h34m

I'm of the opinion that there's a reason why a subset (myself included) of people who when initially exposed to infinitesimals, and specifically the part where you start just disregarding terms, reject them (it's one of the oldest arguments related to calculus! [0]). Those geometric arguments are essentially less rigorous versions of limits! And that lack of rigor hurts those arguments until you have a rigorous justification for them (that didn't appear until the 1960's if my memory is right).

I've come around to infinitesimals, but mostly through exposure to the large hyper-reals. (for context for someone who doesn't know, the idea is to define a number, k which is greater than all real numbers. If you take 1/k, you have a very small number and you can fit an infinite number of 1/k's between 0 and the "next" real number. This concept is what sold me on infinitesimals.)

[0] https://en.wikipedia.org/wiki/The_Analyst

akira2501
1 replies
15h9m

I wonder if it's working with floating point numbers that made me less uncomfortable when first discovering infinitesimals. The idea that something just falls out of our current representable scope under certain operations seemed fine to me. I've always had a soft spot for infinitesimals and a slight dislike for epsilon-delta limits.

floxy
0 replies
15h0m

I've always wondered if infinitesimals are really just an algebra of epsilon-delta proofs.

nextaccountic
0 replies
13h22m

Those geometric arguments are essentially less rigorous versions of limits! And that lack of rigor (...)

Yes it's equivalent to limits, but limits are a very cumbersome machinery, specially if you use the epsilon delta definition (there exists .. such that all ..).

But note that I just linked you a PDF that does fully 100% rigorous calculus using only infinitesimals with no limits. Yhey aren't disregarding small terms willy nilly (like it was done in the early history of calculus)

The only catch about SIA is that it requires you to use intuitionistic logic rather than classical logic in your mathematical arguments (which I admit is a barrier, but it also buys you some things). And what it offers is much simpler proofs that support intuitive reasoning.

There is also this book, "A Primer of Infinitesimal Analysis" [0], which develops a big chunk of calculus and classical mechanics using only infinitesimals, and is fully rigorous.

[0] https://www.cambridge.org/br/universitypress/subjects/mathem...

johnkizer
3 replies
15h46m

IMO "made easy" would involve connecting everything single concept in calculus immediately to the whole reason it exists - physics.

I made the mistake of taking algebra-based physics, then calculus, and only after the calculus course did I realize how much harder I made my life by not starting with calculus (and learning it as the mathematical language of physics).

math_dandy
0 replies
15h24m

I’ve always found it kind of badass that physics students are just expected to pick up the math they need as they study the physics.

chrisweekly
0 replies
15h34m

See https://betterexplained.com for that kind of "made easy" intuition-building / common-sense -oriented material.

bigger_cheese
0 replies
14h32m

When you mention linking to physics are you talking about Parametric equations?

i.e something like, Distance, Velocity and Acceleration with respect to Time.

Velocity is rate of change of Distance in respect to time (ds/dt) Acceleration is rate of change of velocity in respect to time (dv/dt)

You can derive the equations of motion v^2 = u^2 + 2as etc.

Things like the Bernoulli equation from Fluid Dynamics and a lot of other engineering principles can be derived this way.

skulk
1 replies
15h56m

I get the feeling that if someone understands how to square x + dx and can also follow the similar triangle argument on the next page, they'll be totally fine studying calculus starting with limits instead of this.

ramblenode
0 replies
15h20m

I find the geometric proof of the product rule using differentials much more intuitive than the difference quotient proof. The difference quotient proof is a clever algebraic trick, but it doesn't (at least for me) give any deep insight about why the product rule works.

floxy
1 replies
15h30m

Looks like something went wrong with your cut-and-paste:

Let us think of x as a quantity that can grow by a small amount so as to become x+dx, where dx is the small increment added by growth. The square of this is x²+2xdx+(dx)². The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x².
billforsternz
0 replies
12h46m

I always enjoyed calculus, I thought it was kind of magical and didn't worry too much about where the magic came from. I've known the derivative of x^2 is 2x for nearly 50 years but just found out why due to your formatting correction here. Thanks! Also gratitude for the poster who explained that the fundamental theorem of calculus (i.e. reverse differentiation is integration) is essentially just the calculation involved in going from an odometer reading to a speedometer reading then back again!

vixen99
0 replies
12h6m

You make a very reasonable point but I'm sure you would have been rather more accommodating if you'd known (via an acknowledgement) that the original author wrote exactly these words in 1910 in a style which was undoubtedly a lot more readable and welcoming for kids learning calculus than other texts available at the time.

teo_zero
0 replies
12h52m

I don't find this "easy" at all. The writing style is annoyingly formal

... and you need to know what a shilling and a farthing are to understand some of the examples.

taylorius
0 replies
4h53m

Funny, I find the language most engaging and enjoyable to read. And unambiguous, which is much to be desired when new concepts are being explained.

Razengan
0 replies
15h37m

It's from a book written in 1910.

causality0
10 replies
15h46m

I liked math up until my pre-calc teacher told me I couldn't go any further until I memorized two dozen different trigonometric identities and was able to immediately identify which to use.

dboreham
4 replies
15h24m

There are stupid people everywhere, unfortunately. Also sadly except in unusual circumstances (e.g. they have a trust fund to live off), pre-calc teachers are not going to know much about mathematics.

syndicatedjelly
2 replies
15h0m

Those trig identities were tremendously useful throughout Calculus. Having them readily accessible in memory was very helpful, from what I recall. Why do you think this pre-calc teacher was “stupid”?

thoaw342
1 replies
14h12m

I never took "Pre-Calc" and got easy A's in the calculus series.

In my opinion, college calculus is poorly named and instead should be named "Pre-Calc" and the real calculus course is then physics.

oaktowner
0 replies
14h8m

This gibes with my experience in college -- I did more calculus in my physics courses than in my calculus courses.

qball
0 replies
12h21m

Oh, don't worry; university professors do this too.

Most of those courses are not calculus, it's just advanced algebra, and we do those trying for success in them a disservice with the dishonesty.

BenFranklin100
2 replies
15h34m

This is actually not a bad way to go. Understanding which is then followed by rote memorization of basic identities and formulas frees up your brain to focus on higher level concepts. Like the tennis amateur who didn’t make pro because they never bothered to drill on the fundamentals, many people get bogged down in higher level math and physics courses because they’ve long lost or never developed a fluency with the basics needed to derive and understand the advanced concepts.

raincole
1 replies
7h20m

I would have downvoted this comment one year ago.

But my perspective changed. "Memorizing to free up brain space" is very real.

pictureofabear
0 replies
49m

This is a good example of why upvotes and downvotes don't always work well when you have class imbalances (more young people than old people or vice versa). We upvote or downvote mostly based on intuition, but wisdom is not always intuitive--we don't always recognize it when we see it.

extraduder_ire
1 replies
5h56m

I've always found needing to rote-memorize formulae and identities kind of contrived in an academic setting, unless you're being quizzed on how to derive them.

This is probably due to the education I got. Every maths or science exam in my country hands copies of this booklet out for reference: https://www.examinations.ie/misc-doc/BI-EX-7266997.pdf

pictureofabear
0 replies
46m

So you get this booklet to reference on tests? If so, that sounds fantastic. It's the concepts that matter.

Either way, thanks! I'm saving this. It's a wonderful reference.

mettamage
9 replies
16h45m

After years of wanting to but never really being disciplined or motivated enough, I have finally been able to tackle math. My math skills stagnated in high school and then I went on studying computer science (avoiding almost all of the math) and came out pretty good in programming but still bad in math. I tried to read books like calculus made easy but it was still too dry and too hard.

How I fixed it:

1. I learned to become bored and be okay with it. I sometimes lie on the ground for 15 minutes to an hour and do nothing (and that's different than meditation!). I feel incredibly bored, bored to tears actually. I have noticed when I'm in this state, I'm in a much better position to do math or anything else that I slightly find boring but also really interesting.

2. I went to the root of my problem. I'm Dutch. I failed a course in high school called wiskunde B (math B). It teaches calculus, vectors and trig. I'm currently doing that [1]. It's been a few months, but it's going well.

Results and observations:

I think I'm almost at the level to do actual college level math. It helps that I studied CS, because I know a ton of math channels on YouTube and they are now immensely helpful. The thing is the course I'm following isn't explaining the theory well. It is amazing in making me practice, so fortunately I just need to go to good YouTube channels such as 3Blue1Brown or Khan Academy for an adequate explanation on theory.

Another observation is that it has made me a better programmer. My debugging style has always been "turn on the debugger". Now my debugging style is: think deeply about code execution first. Math makes me comfortable with "code execution" because with math you have no choice!

__Looking for a math tutor (email in profile)__

I'm looking for a math tutor. I'd love to ask questions during the week that I don't need an immediate answer to. I think it'll take about 1 hour per week for you. I have noticed I'll need one after the wiskunde B course because I have asked about 100 questions to them in total over the duration of 4 months.

Edit:

To add. I think for me what made calculus so hard was not having a strong understanding of the fundamentals. The gist of calculus of slope and area is easy. Manipulating the equations perfectly 100% of the time. That used to be really hard and it is getting easier the more I practice.

I just happen to watch a YouTube video where a math professor mentions the same thing [2].

[1] The math course I've followed (400 euro - includes exam): https://kdvi.uva.nl/nl/onderwijs/wiskundecursussen/e-winterc...

It's in Dutch, but they have an English version in the language settings.

[2] https://www.youtube.com/watch?v=M7febmLhS6E&ab_channel=BigTh...

syndicatedjelly
1 replies
14h55m

I'm looking for a math tutor. I'd love to ask questions during the week that I don't need an immediate answer to. I think it'll take about 1 hour per week for you. I have noticed I'll need one after the wiskunde B course because I have asked about 100 questions to them in total over the duration of 4 months.

I used to tutor math (10 years ago), but would be happy to provide some asynchronous help if you like. Time permitting of course. Let me know how I can help.

Math background: Have completed and done well in all math courses I’ve taken, thru graduate level Chaos Theory (last math class I took).

mettamage
0 replies
13h30m

Fun! Asynchronous help is all I'd need. I'm the type of person who sometimes want a deeper explanation than theory provides, so I'd basically be asking for those type of things.

I can't seem to contact you. My email is in my profile.

gmays
1 replies
16h32m

You might want to take a look at Math Academy. I'm similar to you and have been using it since last year.

My tool to get me through slogs is streaks, so I commited to doing a lesson (or at least part of a lesson) every day and I'm at 198 days so far.

I wrote this at 100 days in case it's helpful: http://gmays.com/math I'm not sure if I'll have time to write an update for 200 days, but maybe at the 1 year mark.

mettamage
0 replies
15h43m

Wanna see if we could be an accountability partner to each other? Let me know! You can email me or I can contact you via LinkedIn.

fuzztester
1 replies
13h27m

I know a ton of math channels on YouTube

Can you share links to some of those?

dboreham
1 replies
15h26m

I have noticed when I'm in this state...

There are psych experiments to back this up. Also Seinfeld's central thesis is that humans are essentially always just frantically trying to avoid becoming bored.

mettamage
0 replies
13h29m

I wonder what psych experiments those are. Do you have some sources? I did a bachelor in psych but I don't remember getting any (partial) lecture on boredom.

Seinfeld seems to be right :P

melenaboija
7 replies
10h7m

After 20 years dealing with calculus in school, professional career and spare time there is always joy and inevitably a smile when I see writings like this. The feeling is that the intuition that took me years to build it is a matter of minutes when properly developed.

Things like:

Then (dx)^2 will mean a little bit of a little bit of x;

is one of the pillars for one of my last struggles that has taken tens of my hours just to have a basic understanding of stochastic calculus and why this actually matters in this specific case.

When I see things like this makes me think that humanity is progressing as new generations having access to this information will make them learn faster. Thanks :)

smatija
5 replies
9h59m

The problem isn't that resources like this didn't exist in the past (Calculus Made Easy was written in 1910 after all), problem is that they aren't wellknown - and that is unlikely to change even today.

melenaboija
2 replies
9h53m

My opinion is that the problem (difference) is that digital format and internet is not the same as paper and libraries.

smatija
1 replies
9h21m

I would argue that internet makes good resources even harder to find - there is so much of everything, with no curation, that any resource's chance of gaining long term recognition is practically zilch.

melenaboija
0 replies
7h18m

For myself (not a mathematician) I would find hard to argue that in 1920 would have been easier to have access to this specific information. I was sitting in my sofa when I found this.

Also, it should be easy to ask people attending school now if they would prefer books to internet. And to me if the justification is that youth doesn’t know what they are talking about is like denying progress.

cal85
1 replies
4h34m

"that is unlikely to change even today" - huh? Doesn't the internet make it much easier to discover and access resources?

Liquix
0 replies
2h59m

While the internet does make it easier to access educational and useful materials, it also makes it easier to access X,000 hours of $videoGame or X0,000 dopamine hits from endless TikTok videos. Perhaps parent comment meant that there does not appear to be a significant trend towards more people reading or making use of these types of resources.

jampekka
0 replies
9h10m

I think the notation and conceptualization is needlessly confusing. Some of it relates to old, often philosophical and even theological, debates on ontological status of infinitesimals.

The difference quantinent ended up as the Official Blessed Formulation of differential calculus, but it's very rarely used in practice, even though that's how calculus is used. And in practice calculus is still done using ad-hoc infinitesimal notations, but they are some weird thing with rules of their own which very few actually know (at least I don't).

Nonstandard calculus allows using infinitesimals in algebra with more or less the usual rules. Not sure if it's not more popular due to some fundamental technical or philosophical problems, or if it's just conservatism.

Stochastic calculus is quite bizarre indeed. Never understood e.g. the "proper" formulation of continuous time Kalman filters. Just limiting the timestep to zero seems to make sense and produces the right result with some massaging, but I've understood it's not really formally correct.

whatisthis9
4 replies
16h31m

man this is awesome. Thank you! I took my last calculus course in undergrad over 10 years ago and honestly forgot a lot of math including calculus. I am currently back in school doing my masters and struggling at the moment after forgetting this and a lot of other undergrad math topics.

Just out of curiosity, does anyone know of similar sites for linear algebra, discrete mathematics, statistics, etc?

threatofrain
0 replies
11h11m

In general, proof-oriented undergrad classes for Linear Algebra would be either the second class they take, or it's a math major going a pathway specifically for math people. Similar to Sheldon Axler's famous book.

nerdponx
0 replies
11h16m

I learned right from FIS without any "warmup" from Strang-level material and it was rough to say the least. I came away with a good final grade and a very poor understanding of most of what I was doing, I was just grinding through proofs with little practical intuition. Good reference book, but very difficult to learn from, especially with a professor who seemed to love abstract math and didn't place high value on geometric intuition.

j_bum
0 replies
12h12m

+1 for Gilbert Strang’s linear algebra textbookcourse. Im working through his textbook now as I’m diving deeper into ML/DL methods.

Here is an additional link to the Spring 2023 course materials that follow along the 6th ed. Of his textbook [0].

[0] https://github.com/mitmath/1806

noduerme
3 replies
12h11m

Hand up, I failed precalc twice and never made it past high school. But give me a blank SQL query where I can use window functions on a large dataset, I will tease out the differentiating variables that matter to a business. I think I missed the point of precalc. I never had to solve any real world problems, and the jargon was too impenetrable. Grouping and summarizing sets within sets is like, piece of cake. Just as is formal logic. The way it's taught is completely upside down.

PennRobotics
1 replies
9h41m

My brother had a strong dislike for math that wasn't immediately practical. He ended up understanding integrals (at least numerical approximation) after building a small boat and needing to calculate how much air it would hold: obviously less than the containing rectangle, more than the contained rectangle; and then he figured out you can split the boat into sections and get a good guess of each section's volume, move from rectangles to trapezoids, etc.

noduerme
0 replies
8h33m

I'm a bit like that. I scored surprisingly high on the math portion of the SAT, but couldn't remember which thing was sin and which was cosine. In coding for 2D game platforms I think I've needed to reinvent parts of trigonometry and spatial geometry every time I encounter a new version of the problem. What your brother did makes perfect sense to me; not that I dislike math per se, but his way is probably how I would approach that problem too. I need to be able to visualize it in space. By breaking it up into discreet buckets if necessary. Once I do, I can figure out an iterative logic function that approximately describes it... although it may take me much longer if I'm trying to figure out an actual equation.

woopsn
0 replies
11h37m

The lead up to calculus is brutal. I did poorly in high school math, got my requirements finished by the grace of BYU online. Never took precalc. Then in college I crammed for the placement exam and got directly into the upper level calculus track - and from then on I loved math. Unfortunately in the traditional track, calculus is the first and only class that "makes sense", as in someone would want to take it. But it takes years of preparation (supposedly). In the meantime like you I took some interest in programming.

aaronbrethorst
0 replies
12h56m

Ohh, this really is from 1910. And here I just thought the author was being obnoxiously cutesy with their language.

coolThingsFirst
2 replies
12h3m

it's already easy

-eastern european

emeril
0 replies
6h24m

some americans find it easy too!

coolThingsFirst
0 replies
11h51m

mericans jealous

beckthompson
2 replies
12h43m

Whenever I see integrals I think "area under the curve" and when I see derivatives I think "slope of a function"!

beckthompson
0 replies
12h32m

Haha that's a funny video!

WhitneyLand
2 replies
5h29m

How much UX design / usability tests do the authors of math textbooks do?

dagw
1 replies
5h26m

When I was at university I had probably 2 or 3 courses where we were beta testing textbooks that the lecturer was in the process of writing. On the downside the book had some rough edges, on the plus side the books where free

WhitneyLand
0 replies
3h59m

Interesting, did they ask you to look for mistakes, or also validate if things were easy to understand and learning efficient?

maroonblazer
0 replies
5h14m

Generic

Genetic

xNeil
0 replies
15h4m

Feynman started with Calculus Made Easy, and only did Calculus for the Practical Man after he finished it.

HarHarVeryFunny
1 replies
3h38m

Easy derivatives are indeed easy - easier than that site makes them seem!

We can approximate the slope (derivative) of any function y = f(x), at (x,y) as the slope of a straight line from (x,y) to another nearby point (x',y') where y' = f(x').

The slope of this line is (y'-y)/(x'-x).

Since we need x' close to x to make this accurate, we use x' = x + dx where dx is small, and we can also represent y' as y' = y + dy.

So, then our slope is (y'-y)/(x'-x) = dy/dx = (f(x+dx)-f(x))/dx

And that's all there is to it.

e.g.

For f(x) = x^2

dy/dx = ((x+dx)^2-x^2)/dx = 2x + dx

We get the true value of the slope at (x,y) in the limit of dx -> 0, giving us the derivative of f(x) = x^2 as f'(x) = 2x.

HDThoreaun
0 replies
2h43m

Probably why derivatives are the first semester of calculus,. they are indeed the easiest part and a good way to introduce limits.

yazzku
0 replies
17h28m

Looks great, thank you.

threatofrain
0 replies
14h26m

If people here are looking to catchup or refresh their old calculus knowledge, I'd recommend Terry Tao's Analysis 1. It's pedagogically friendly, conversational, but also rigorous.

spinlock_
0 replies
8h46m

The last couple of months, I have been studying the fundamentals of algebra using Professor Leonard's YT Channel[0]. My goal is to fill in the gaps in my knowledge before I refresh my Calculus. It takes a while to go through all this stuff, if you do it right. But man, I have so much more confidence in my skills now than I had before, which to me is in itself rewarding and motivating. I had no idea how big my knowledge gaps in algebra were before I started going through his playlists.

My end goal is to be able to follow Andrej Karpathy's "Neural Networks: Zero to Hero"[1] without any big problems So starting basically from "zero" in order to learn the prerequisites before learning what you actually want to learn on your own can feel daunting at times. But I think taking shortcuts will result in frustration. So, here I am taking algebra courses on YT with 38 years.

[0] https://youtube.com/@ProfessorLeonard?si=0kiGvmbZv4b9Sgf9

[1] https://youtube.com/playlist?list=PLAqhIrjkxbuWI23v9cThsA9Gv...

russfink
0 replies
5h6m

WRITTEN IN 1910! Why oh why has every text written after this failed to include such a clear understanding of the concepts. There is no excuse for this, academia! Shame, shame! I should engage thee in a bout of fisticuffs were thou a being!

raverbashing
0 replies
6h53m

The fools who write the textbooks of advanced mathematics — and they are mostly clever fools — seldom take the trouble to show you how easy the easy calculations are.

Yes this hits the nail on the head

The explanations overall look simple, maybe too long sometimes, but that's a killer prelude

math_dandy
0 replies
15h25m

If this book has exercises at the end of each chapter, I’d assign it to my calculus classes.

jsmm
0 replies
16h44m

Wow! this taught me more than 2 semesters of calculus. Thanks for sharing.

ineedaj0b
0 replies
17h37m

This taught me Calculus! It’s fantastic and if you need to learn calculus this was better than any teacher I ever had.

graycat
0 replies
14h19m

I gave my 6th grade nephew a 90 second "Calculus Made Easy":

First, credibility: Ah, I never took the first year of college calculus -- to make faster progress in college math, got a good book and taught myself. After an oral exam at a black board, was admitted to the second year. Majored in math. Took (so called) advanced calculus from Rudin's Principles .... Took applied advanced calculus from a famous MIT book from an MIT Ph.D. Taught calculus at Indiana University. Studied Fleming's Functions of Several Variables, right, through the inverse and implicit function theorems, Stokes formula, exterior algebra, etc. Published some advanced math, essentially advanced calculus. Once, with some calculus, at FedEx pleased the most serious investors on the Board, had them return their airline tickets back to Texas, stay after all, and saved the company.

Okay, the 90 seconds:

Consider a car, its speedometer and odometer.

Calculus has two parts.

For the first part, you read the data from the odometer and reconstruct the speedometer readings. Doing this, you take changes in (increments of, differences in) the odometer readings, say, every second. This is called differentiation.

For the second part, you read the data from the speedometer and reconstruct the odometer readings. Doing this, you add the speedometer readings, say, every second. This is called integration.

Starting with the odometer readings and differentiating to get the speedometer readings and then integrating the speedometer readings will give back the odometer readings -- this is the "fundamental theorem of calculus".

~90 seconds.

For more, instead of the 1 second steps could use 0.1 seconds, .... 0.0001 seconds, etc. With really small steps, making them smaller will make no or nearly no difference. So, the reconstructions will have converged, reached a limit.

Reaching this limit is mostly what was novel when Newton, Leibnitz, etc. invented calculus.

It is fair to say that the first big application was to Newton's law F = ma where have some object -- baseball, airplane, rocket -- with mass m and are applying to the object force F. Then a is the acceleration of the object. Integrate the acceleration and get the velocity v. Integrate v and get distance d. So, can find where the rocket is after, say, 10 seconds. Other early applications were to planetary motion.

There are applications to areas, volumes, classical mechanics. fluid flow, mechanical engineering, electricity and magnetism, quantum mechanics, relativity, electrical and electronic engineering, e.g., Fourier theory.

Physics and engineering are big users. And there are applications in economics, e.g., work of Arrow, Hurwicz, and Uzawa on the Kuhn-Tucker conditions.

By the early 20th century, calculus was refined, e.g., presented with careful assumptions, definitions, theorems, and proofs, e.g., B. Riemann and, soon, H. Lebesgue. By then there was the idea of the highly irregular Brownian motion and the observation that differentiation wouldn't work there -- Brownian motion was differentiable nowhere!

Calculus? A pillar of science, technology, and, thus, civilization.

cess11
0 replies
11h58m

When I was in school calculus was the first time we encountered sleight-of-hand tricks as a technique in mathematical reasoning, along with math basically being a symbolic form of fiction it made it quite hard.

While I enjoy Silvanus book, I don't think calculus can be made easy without preparing the students to accept the handwaving and trickery involved in the jump from an equation to its 'area' or 'velocity'. Compared to the solemn majesty of euclidic trigonometry or relatively straightforward step-by-step solving of quadratic equations the techniques foundational to calculus are rather devious (as are those that bring in complex numbers).

In high school the combination with physics made it harder for some students, they had the impression that learning math amounted to learning about nature, rather than a language for expressing fictions about a view of nature. In turn the approximative nature of the problem solving and calculus didn't fit very well.

canjobear
0 replies
15h35m

(1910)

barfbagginus
0 replies
2h47m

The title claims it makes calculus easy, and yet there is no category theory.

"How can this be possible?!" I hear you ask.

Well, the book was written in 1910, 50 years before category theory appeared.

But do not worry! There is a book that uses categories to develop ordinary differential and integral calculus!

What could be easier than that? I don't know! But if I find it, I'll let you know!

Enjoy!

https://books.google.com/books?id=gaE5EAAAQBAJ&newbks=1&newb...

anthk
0 replies
9h38m

Re-edit that with metrical units and I'd recommend that.

a-dub
0 replies
12h44m

the hardest part of doing calculus is fluency in arithmetic, algebra and trigonometry.

the most important idea in calculus is often glossed over, weakly presented or omitted entirely: the continuity of the reals. i feel that once this is fully understood, most of the ideas in calculus become intuitive.

DoreenMichele
0 replies
14h58m

I will recommend the book A tour of the calculus as another potential resource for those interested.