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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 o

• adhamsalama 2y ago
  Try 3blue1brown. You'll love it.
• liammclennan 2y ago
  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.
  • aerhardt 2y ago
    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 2y ago
      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 2y ago
        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 2y ago
          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 2y ago
            This was in Spain, but I have no problem believing it's roughly the same everywhere in the world, with few exceptions.
    • deleted 2y ago
      [deleted]
    • ozim 2y ago
      I still would argue that weeder courses are fine. If you start as a major whatever the field is - you should be interested in the field enough to plow through. Otherwise at the end you get people dissatisfied they cannot get job but they have a degree.
  • vasco 2y ago
    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 2y ago
      Pretty common with lin alg and diffeq unfortunately. Many schools teach it as a toolbox instead of for understanding.
      • slowmovintarget 2y ago
        Math like words are tools. When you want application instead of tool operating instructions, go into physics or computer science (or both).
        • klysm 2y ago
          I disagree from a pedagogical standpoint. Math can be used as a tool, but that’s not the most effective way to teach math. I’m assuming the goal is to teach comprehension and understanding.
  • ozim 2y ago
    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.
    • jampekka 2y ago
      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.
    • z3phyr 2y ago
      > 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 2y ago
        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.
• jahnu 2y ago
  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 https://www.stevenstrogatz.com/books/infinite-powers
  • lo_zamoyski 2y ago
    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 https://www.newadvent.org/cathen/01760a.htm [1] https://a.co/d/hE5830i https://a.co/d/hE5830i
• threatofrain 2y ago
  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.
• abrookewood 2y ago
  So don't leave us hanging ...
• jampekka 2y ago
  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!
• Nevermark 2y ago
  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 2y ago
    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.
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• diffeomorphism 2y ago
  > 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).
• wodenokoto 2y ago
  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.
• nathan_compton 2y ago
  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 2y ago
    Is there someone now that can explain the intuition behind QFT?
    • nathan_compton 2y ago
      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.
• elric 2y ago
  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.
  • Moru 2y ago
    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 :-)
  • lo_zamoyski 2y ago
    > 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.
• analog31 2y ago
  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 2y ago
  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.
• mdavidn 2y ago
  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 2y ago
    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.
• 019341097 2y ago
  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.
• treflop 2y ago
  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.
• ndriscoll 2y ago
  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 2y ago
    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 2y ago
      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.
• deleted 2y ago
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• teleforce 2y ago
  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.