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.
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.
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.
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.
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.
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.
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.
> 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).
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?"
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.
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.
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.
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!
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.
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.
I think you hit the nailed on why most the textbooks are lousy at best on providing the 'why', apparently they are focusing more on the mechanical aspects and repeating exercises for scoring the exams, as a filler to the 1000 pages at US$100 textbook.
One of the best books on Electronics according to HN crowd is The Art of Electronics, and it is filled with pages over pages of how-to of designing circuits of more than 1000 pages. But if you want to know why a Colpitts oscillator is the best for your design, all the best for that.
Even the textbooks produced by professors from the best engineering schools (e.g MIT, Stanford, etc) are not spared of this issue. One of my former lecturers (not MIT) for linear algebra and numerical analysis courses claimed that he worked and consulted for NASA, but how I wished that he had cover some of the motivations of doing a dot product. For the ChatGPT responses of the reasons of doing dot product for two vectors see ChatGPT 4 prompt below. I think once ChatGPT/Gemini/etc can operate properly on math equations at least as good they can for NLP, I think we are going to have a wonderful treat even for the understanding the higly complex concepts for examples thermodynamics, electromagnetics and general relativity.
Another classic example is try to find inside your favorite DSP book on why analytic signal in complex number generated by Hilbert Transform is necessary for communication and there is a very high probability that that's not a single explanation for that but there are many equations and graphs related to analytic signal. If we understand why or the raison d'etre, perhaps we can further ask why stop at analytic signal in complex number form, can we use quaternion number and what is the benefit of having more elaborate quaternion number form rather than complex form. If you are trying to do that and check this fact with most of DSP textbooks on this particular important issue then you will probably end up dissapointed even if you have bookshelf that contains majority of DSP books that are published since the invention of FFT.
I have also included the ChatGPT prompt for the dot product, and the responses are delightful to say the least.
>User prompt: Why you want to perform dot product of two vectors
>ChatGPT 4:
The dot product of two vectors, also known as the scalar product, is a fundamental operation in mathematics and physics with various applications, including:
Computing angles between vectors: The dot product can be used to find the cosine of the angle between two vectors. This is particularly useful in geometry and 3D modeling to determine orientations and angles.
Projecting vectors: The dot product is used in the projection of one vector onto another. This is crucial in physics for resolving a vector into components parallel and perpendicular to another vector, and in computer graphics for rendering and shading calculations.
Checking orthogonality: Two vectors are orthogonal (i.e., the angle between them is 90 degrees) if their dot product is zero. This property is vital in many areas including computer graphics, signal processing, and linear algebra.
Work calculation in physics: In physics, the work done by a force (a vector) moving an object through a displacement (another vector) is calculated using the dot product. This is important for understanding how much energy is being used or transferred in a system.
Machine learning and data analysis: The dot product is used in algorithms such as those for finding similarities between data points, calculating vector norms, or in the training processes like those of neural networks where weights and input signals are combined.
Each of these applications demonstrates why the dot product is a key tool in both theoretical and applied disciplines, linking geometry, algebra, and practical computation.
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 :)
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.
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.
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.
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.)
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.
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.
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.
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.
I "knew" enough calculus to get good grades in math courses through high school and college as an engineer.
But I didn't really "know" calculus until I read a book not too dissimilar from this one, "A Course of Pure Mathematics" from 1908 (!), which constructs calculus up from number theory (I think the fundamental theorem of calculus comes halfway through the book?). From that point it's impossible to forget.
As for why it's not taught this way today, I'd blame our testing regime and large classrooms, which incentivizes temporary memorization of key formulae and knowing where to mindlessly apply them, over deep/lasting/semantic understanding. I'd also blame the fact that we have a different maths teacher every year, so students come in with heterogenous understandings of the pre-requisites for the next year's material, so the first part of a section is spent reviewing + consolidating.
It takes maybe 10-20% more time to get a rich understanding of the subject that lasts a lifetime. But we value compression and instantly-measurable results more than actual learning. :/
Encountering material like this makes you really happy, but it's kind of bittersweet because it makes you realize that the modal state of modern pedagogy is pretty abysmal.
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.
I appreciate the effort, but after a couple of pages I can already tell, that it wouldn't be something I wanted if it was the first time I'm learning calculus. And even for a refresher it isn't exactly perfect.
IMO, the author is right about what's wrong with the most "proper" books, but overcompensates, and thus makes it unnecessarily complicated to understand, maybe even more so, than some "proper" textbooks. It is too wordy, too informal and pretty hard to read and follow. I don't need any Dean Swift poetry and references to "Queen Elizabeth's days", I just want to know what is calculus, why do we need it, and how do I actually do it, goddammit… That's if we follow the seemingly easier "engineering" approach. And we need it to be a bit more formal still if we follow even slightly more mathematical approach, which I think is actually necessary if you want even a bit of real understanding, because math is this thing where it's very easy to fool yourself into thinking you understand something, when you actually don't, and then get all flabbergasted and helpless against paradoxical "fake proofs" or being asked to prove anything yourself. And to tell valid derivation from invalid, yes, you do need some formal definitions.
And in fact, the (formal) basics really aren't difficult to understand at all. Everyone can readily agree that x² grows faster than x, and being introduced into concept of limits to see why (dx)² can be considered negligible compared to dx. You don't need to be comparing weeks to minutes for that. If anything, the latter only throws you off. And I feel like it takes way less patience to read a handful of formal definitions than pages of these "old British"-stylized ramblings.
Edit: oh, it isn't "stylized". Still, doesn't change the main point, that there are far better more recent resources to learn calculus.
I got "Calculus Made Easy" back in high school, and it helped when I first took calc in 10th grade, and then BC Calc in 12th grade. It gave me some confidence to have seen the basic ideas presented in a gentle way. Actually, once you get past the old-fashioned language and layout many math books from the early 1900's are pretty good. They don't have a lot of the abstraction that came later, some of which makes you feel like you're trying to do surgery while wearing boxing gloves. Anyway, there are other calc books which have similar intents, though different styles. Here are two:
Daniel Kleppner and Norman Ramsey, "Quick Calculus" - We used this for math background when I was taking AP Physics in high school. The physics textbook was Resnick and Halliday, which uses calc throughout. It is not particularly intuitive, but the "programmed learning" approach makes it very easy to pick up enough basic computational ability to do applications. (The phrase "programmed learning" is so old that maybe I should clarify that it has nothing to do with computers.)
H. M. Schey, "Div, Grad, Curl, and All That" - This might be helpful for people taking vector calculus, or using it in an electromagnetism course in physics. In fact, a lot of his examples are from E-M. Very good at providing intuition without too many technicalities.
I think the best uses for books like these are: (a) You want to get the "big ideas" without worrying too much about technical facility (maybe because you won't ever need it - e.g. you're just learning for your own interest); (b) To provide a fair amount of the math you'll need in (say) a physics course; (c) As prep before taking a standard calc course, or maybe as a refresher. Books like Thompson, Kleppner and Ramsey, or Schey will help with the big picture and to boost confidence but it's best to read them before taking the "real" calc course - since once the "real" course is running you'll be too busy with the work in the course to read something else on the side. But do whatever works for you. Calculus really is wonderful - have fun!
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.
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?
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!
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.
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.
I skipped school and still trying to "catch up" to the all the stuff I was supposed to learn before gamedev, and I can't recommend these resources enough for actually grokking math stuff:
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.
> 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
[+] [-] _virtu|1 year ago|reply
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|1 year ago|reply
[+] [-] elric|1 year ago|reply
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.
[+] [-] Nevermark|1 year ago|reply
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.
[+] [-] jahnu|1 year ago|reply
https://www.stevenstrogatz.com/books/infinite-powers
[+] [-] ndriscoll|1 year ago|reply
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.
[+] [-] wodenokoto|1 year ago|reply
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.
[+] [-] diffeomorphism|1 year ago|reply
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).
[+] [-] nathan_compton|1 year ago|reply
[+] [-] threatofrain|1 year ago|reply
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.
[+] [-] ajkjk|1 year ago|reply
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.
[+] [-] 019341097|1 year ago|reply
[+] [-] jampekka|1 year ago|reply
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!
[+] [-] treflop|1 year ago|reply
I guess if you want to learn math, only a math textbook will actually care.
[+] [-] mdavidn|1 year ago|reply
[+] [-] analog31|1 year ago|reply
[+] [-] unknown|1 year ago|reply
[deleted]
[+] [-] adhamsalama|1 year ago|reply
[+] [-] abrookewood|1 year ago|reply
[+] [-] teleforce|1 year ago|reply
One of the best books on Electronics according to HN crowd is The Art of Electronics, and it is filled with pages over pages of how-to of designing circuits of more than 1000 pages. But if you want to know why a Colpitts oscillator is the best for your design, all the best for that.
Even the textbooks produced by professors from the best engineering schools (e.g MIT, Stanford, etc) are not spared of this issue. One of my former lecturers (not MIT) for linear algebra and numerical analysis courses claimed that he worked and consulted for NASA, but how I wished that he had cover some of the motivations of doing a dot product. For the ChatGPT responses of the reasons of doing dot product for two vectors see ChatGPT 4 prompt below. I think once ChatGPT/Gemini/etc can operate properly on math equations at least as good they can for NLP, I think we are going to have a wonderful treat even for the understanding the higly complex concepts for examples thermodynamics, electromagnetics and general relativity.
Another classic example is try to find inside your favorite DSP book on why analytic signal in complex number generated by Hilbert Transform is necessary for communication and there is a very high probability that that's not a single explanation for that but there are many equations and graphs related to analytic signal. If we understand why or the raison d'etre, perhaps we can further ask why stop at analytic signal in complex number form, can we use quaternion number and what is the benefit of having more elaborate quaternion number form rather than complex form. If you are trying to do that and check this fact with most of DSP textbooks on this particular important issue then you will probably end up dissapointed even if you have bookshelf that contains majority of DSP books that are published since the invention of FFT.
I have also included the ChatGPT prompt for the dot product, and the responses are delightful to say the least.
>User prompt: Why you want to perform dot product of two vectors
>ChatGPT 4:
The dot product of two vectors, also known as the scalar product, is a fundamental operation in mathematics and physics with various applications, including:
Computing angles between vectors: The dot product can be used to find the cosine of the angle between two vectors. This is particularly useful in geometry and 3D modeling to determine orientations and angles.
Projecting vectors: The dot product is used in the projection of one vector onto another. This is crucial in physics for resolving a vector into components parallel and perpendicular to another vector, and in computer graphics for rendering and shading calculations.
Checking orthogonality: Two vectors are orthogonal (i.e., the angle between them is 90 degrees) if their dot product is zero. This property is vital in many areas including computer graphics, signal processing, and linear algebra.
Work calculation in physics: In physics, the work done by a force (a vector) moving an object through a displacement (another vector) is calculated using the dot product. This is important for understanding how much energy is being used or transferred in a system.
Machine learning and data analysis: The dot product is used in algorithms such as those for finding similarities between data points, calculating vector norms, or in the training processes like those of neural networks where weights and input signals are combined.
Each of these applications demonstrates why the dot product is a key tool in both theoretical and applied disciplines, linking geometry, algebra, and practical computation.
[+] [-] melenaboija|1 year ago|reply
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|1 year ago|reply
[+] [-] jampekka|1 year ago|reply
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.
[+] [-] conwy|1 year ago|reply
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.
[+] [-] svat|1 year ago|reply
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.)
[+] [-] ecshafer|1 year ago|reply
[+] [-] getcrunk|1 year ago|reply
[+] [-] beltsazar|1 year ago|reply
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.
[+] [-] gofreddygo|1 year ago|reply
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.
[+] [-] unknown|1 year ago|reply
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[+] [-] unknown|1 year ago|reply
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[+] [-] willmeyers|1 year ago|reply
https://press.uchicago.edu/ucp/books/book/chicago/C/bo548572...
[+] [-] maroonblazer|1 year ago|reply
Genetic
[+] [-] gen220|1 year ago|reply
But I didn't really "know" calculus until I read a book not too dissimilar from this one, "A Course of Pure Mathematics" from 1908 (!), which constructs calculus up from number theory (I think the fundamental theorem of calculus comes halfway through the book?). From that point it's impossible to forget.
As for why it's not taught this way today, I'd blame our testing regime and large classrooms, which incentivizes temporary memorization of key formulae and knowing where to mindlessly apply them, over deep/lasting/semantic understanding. I'd also blame the fact that we have a different maths teacher every year, so students come in with heterogenous understandings of the pre-requisites for the next year's material, so the first part of a section is spent reviewing + consolidating.
It takes maybe 10-20% more time to get a rich understanding of the subject that lasts a lifetime. But we value compression and instantly-measurable results more than actual learning. :/
Encountering material like this makes you really happy, but it's kind of bittersweet because it makes you realize that the modal state of modern pedagogy is pretty abysmal.
[+] [-] spinlock_|1 year ago|reply
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...
[+] [-] lupire|1 year ago|reply
https://archive.org/details/calulusforthepra000526mbp
[+] [-] theophrastus|1 year ago|reply
[1] https://en.wikipedia.org/wiki/Calculus_Made_Easy
[+] [-] Jtsummers|1 year ago|reply
The link isn’t to the front page with his name.
[+] [-] barfbagginus|1 year ago|reply
"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...
[+] [-] krick|1 year ago|reply
IMO, the author is right about what's wrong with the most "proper" books, but overcompensates, and thus makes it unnecessarily complicated to understand, maybe even more so, than some "proper" textbooks. It is too wordy, too informal and pretty hard to read and follow. I don't need any Dean Swift poetry and references to "Queen Elizabeth's days", I just want to know what is calculus, why do we need it, and how do I actually do it, goddammit… That's if we follow the seemingly easier "engineering" approach. And we need it to be a bit more formal still if we follow even slightly more mathematical approach, which I think is actually necessary if you want even a bit of real understanding, because math is this thing where it's very easy to fool yourself into thinking you understand something, when you actually don't, and then get all flabbergasted and helpless against paradoxical "fake proofs" or being asked to prove anything yourself. And to tell valid derivation from invalid, yes, you do need some formal definitions.
And in fact, the (formal) basics really aren't difficult to understand at all. Everyone can readily agree that x² grows faster than x, and being introduced into concept of limits to see why (dx)² can be considered negligible compared to dx. You don't need to be comparing weeks to minutes for that. If anything, the latter only throws you off. And I feel like it takes way less patience to read a handful of formal definitions than pages of these "old British"-stylized ramblings.
Edit: oh, it isn't "stylized". Still, doesn't change the main point, that there are far better more recent resources to learn calculus.
[+] [-] bikenaga|1 year ago|reply
Daniel Kleppner and Norman Ramsey, "Quick Calculus" - We used this for math background when I was taking AP Physics in high school. The physics textbook was Resnick and Halliday, which uses calc throughout. It is not particularly intuitive, but the "programmed learning" approach makes it very easy to pick up enough basic computational ability to do applications. (The phrase "programmed learning" is so old that maybe I should clarify that it has nothing to do with computers.)
H. M. Schey, "Div, Grad, Curl, and All That" - This might be helpful for people taking vector calculus, or using it in an electromagnetism course in physics. In fact, a lot of his examples are from E-M. Very good at providing intuition without too many technicalities.
I think the best uses for books like these are: (a) You want to get the "big ideas" without worrying too much about technical facility (maybe because you won't ever need it - e.g. you're just learning for your own interest); (b) To provide a fair amount of the math you'll need in (say) a physics course; (c) As prep before taking a standard calc course, or maybe as a refresher. Books like Thompson, Kleppner and Ramsey, or Schey will help with the big picture and to boost confidence but it's best to read them before taking the "real" calc course - since once the "real" course is running you'll be too busy with the work in the course to read something else on the side. But do whatever works for you. Calculus really is wonderful - have fun!
[+] [-] a-dub|1 year ago|reply
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.
[+] [-] whatisthis9|1 year ago|reply
Just out of curiosity, does anyone know of similar sites for linear algebra, discrete mathematics, statistics, etc?
[+] [-] russfink|1 year ago|reply
[+] [-] threatofrain|1 year ago|reply
[+] [-] causality0|1 year ago|reply
[+] [-] Razengan|1 year ago|reply
https://www.youtube.com/c/3blue1brown
https://www.mathsisfun.com
[+] [-] cess11|1 year ago|reply
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.
[+] [-] raverbashing|1 year ago|reply
Yes this hits the nail on the head
The explanations overall look simple, maybe too long sometimes, but that's a killer prelude
[+] [-] jsmm|1 year ago|reply