• someacnt@sh.itjust.worksEnglish
    1·
    1 day ago

    But df/dx is a fraction, is a ratio between differential of f and standard differential of x. They both live in the tangent space TR, which is isomorphic to R.

    What’s not fraction is \partial f / \partial x, but likely you already know that. This is akin to how you cannot divide two vectors.

  • Caveman@lemmy.worldEnglish
    91·
    2 days ago

    The thing is that it’s legit a fraction and d/dx actually explains what’s going on under the hood. People interact with it as an operator because it’s mostly looking up common derivatives and using the properties.

    Take for example ∫f(x) dx to mean "the sum (∫) of supersmall sections of x (dx) multiplied by the value of x at that point ( f(x) ). This is why there’s dx at the end of all integrals.

    The same way you can say that the slope at x is tiny f(x) divided by tiny x or d*f(x) / dx or more traditionally (d/dx) * f(x).

      • jsomae@lemmy.mlEnglish
        42·
        2 days ago

        it’s legit a fraction, just the numerator and denominator aren’t numbers.

          • jsomae@lemmy.mlEnglish
            82·
            2 days ago

            try this on – Yes 👎

            It’s a fraction of two infinitesimals. Infinitesimals aren’t numbers, however, they have their own algebra and can be manipulated algebraically. It so happens that a fraction of two infinitesimals behaves as a derivative.

            • Kogasa@programming.devEnglish
              1·
              2 days ago

              Ok, but no. Infinitesimal-based foundations for calculus aren’t standard and if you try to make this work with differential forms you’ll get a convoluted mess that is far less elegant than the actual definitions. It’s just not founded on actual math. It’s hard for me to argue this with you because it comes down to simply not knowing the definition of a basic concept or having the necessary context to understand why that definition is used instead of others…

              • jsomae@lemmy.mlEnglish
                3·
                2 days ago

                Why would you assume I don’t have the context? I have a degree in math. I could be wrong about this, I’m open-minded. By all means, please explain how infinitesimals don’t have a consistent algebra.

                • Kogasa@programming.devEnglish
                  21·
                  2 days ago

                  1. I also have a masters in math and completed all coursework for a PhD. Infinitesimals never came up because they’re not part of standard foundations for analysis. I’d be shocked if they were addressed in any formal capacity in your curriculum, because why would they be? It can be useful to think in terms of infinitesimals for intuition but you should know the difference between intuition and formalism.

                  2. I didn’t say “infinitesimals don’t have a consistent algebra.” I’m familiar with NSA and other systems admitting infinitesimal-like objects. I said they’re not standard. They aren’t.

                  3. If you want to use differential forms to define 1D calculus, rather than a NSA/infinitesimal approach, you’ll eventually realize some of your definitions are circular, since differential forms themselves are defined with an implicit understanding of basic calculus. You can get around this circular dependence but only by introducing new definitions that are ultimately less elegant than the standard limit-based ones.

  • socsa@piefed.socialEnglish
    3·
    2 days ago

    The world has finite precision. dx isn’t a limit towards zero, it is a limit towards the smallest numerical non-zero. For physics, that’s Planck, for engineers it’s the least significant bit/figure. All of calculus can be generalized to arbitrary precision, and it’s called discrete math. So not even mathematicians agree on this topic.

  • chortle_tortle@mander.xyzEnglish
    87·
    3 days ago

    Mathematicians will in one breath tell you they aren’t fractions, then in the next tell you dz/dx = dz/dy * dy/dx

  • benignintervention@lemmy.worldEnglish
    81·
    4 days ago

    I found math in physics to have this really fun duality of “these are rigorous rules that must be followed” and “if we make a set of edge case assumptions, we can fit the square peg in the round hole”

    Also I will always treat the derivative operator as a fraction

  • rudyharrelson@lemmy.radioEnglish
    67·
    4 days ago

    Derivatives started making more sense to me after I started learning their practical applications in physics class. d/dx was too abstract when learning it in precalc, but once physics introduced d/dt (change with respect to time t), it made derivative formulas feel more intuitive, like “velocity is the change in position with respect to time, which the derivative of position” and “acceleration is the change in velocity with respect to time, which is the derivative of velocity”

    • Prunebutt@slrpnk.netEnglish
      361·
      4 days ago

      Possibly you just had to hear it more than once.

      I learned it the other way around since my physics teacher was speedrunning the math sections to get to the fun physics stuff and I really got it after hearing it the second time in math class.

      But yeah: it often helps to have practical examples and it doesn’t get any more applicable to real life than d/dt.

      • exasperation@lemmy.dbzer0.comEnglish
        4·
        3 days ago

        I always needed practical examples, which is why it was helpful to learn physics alongside calculus my senior year in high school. Knowing where the physics equations came from was easier than just blindly memorizing the formulas.

        The specific example of things clicking for me was understanding where the “1/2” came from in distance = 1/2 (acceleration)(time)^2 (the simpler case of initial velocity being 0).

        And then later on, complex numbers didn’t make any sense to me until phase angles in AC circuits showed me a practical application, and vector calculus didn’t make sense to me until I had to actually work out practical applications of Maxwell’s equations.

  • Mubelotix@jlai.luEnglish
    2·
    2 days ago

    We teach kids the derive operator being ' or ·. Then we switch to that writing which makes sense when you can use it properly enough it behaves like a fraction

    • Kogasa@programming.devEnglish
      12·
      2 days ago

      It doesn’t. Only sometimes it does, because it can be seen as an operator involving a limit of a fraction and sometimes you can commute the limit when the expression is sufficiently regular

  • vaionko@sopuli.xyzEnglish
    42·
    4 days ago

    Except you can kinda treat it as a fraction when dealing with differential equations