• someacnt@sh.itjust.worksEnglish
    1·
    2 hours 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.

  • socsa@piefed.socialEnglish
    2·
    15 hours 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.

  • Caveman@lemmy.worldEnglish
    91·
    22 hours 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·
        20 hours ago

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

          • jsomae@lemmy.mlEnglish
            72·
            19 hours 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·
              17 hours 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
                2·
                17 hours 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·
                  16 hours 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.

  • Mubelotix@jlai.luEnglish
    2·
    22 hours 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

  • chortle_tortle@mander.xyzEnglish
    85·
    2 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

  • Gladaed@feddit.orgEnglish
    13·
    21 hours ago

    Why does using it as a fraction work just fine then? Checkmate, Maths!

    • Kogasa@programming.devEnglish
      12·
      21 hours 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

  • shapis@lemmy.mlEnglish
    11·
    2 days ago

    This very nice Romanian lady that taught me complex plane calculus made sure to emphasize that e^j*theta was just a notation.

    Then proceeded to just use it as if it was actually eulers number to the j arg. And I still don’t understand why and under what cases I can’t just assume it’s the actual thing.

    • jsomae@lemmy.mlEnglish
      2·
      19 hours ago

      e𝘪θ is not just notation. You can graph the entire function ex+𝘪θ across the whole complex domain and find that it matches up smoothly with both the version restricted to the real axis (ex) and the imaginary axis (e𝘪θ). The complete version is:

      ex+𝘪θ := ex(cos(θ) + 𝘪sin(θ))

      Various proofs of this can be found on wikipeda. Since these proofs just use basic calculus, this means we didn’t need to invent any new notation along the way.

      • shapis@lemmy.mlEnglish
        1·
        16 hours ago

        I’m aware of that identity. There’s a good chance I misunderstood what she said about it being just a notation.

        • jsomae@lemmy.mlEnglish
          1·
          11 hours ago

          It’s not simply notation, since you can prove the identity from base principles. An alien species would be able to discover this independently.

    • carmo55@lemmy.zipEnglish
      1·
      22 hours ago

      It is just a definition, but it’s the only definition of the complex exponential function which is well behaved and is equal to the real variable function on the real line.

      Also, every identity about analytical functions on the real line also holds for the respective complex function (excluding things that require ordering). They should have probably explained it.

      • shapis@lemmy.mlEnglish
        1·
        21 hours ago

        She did. She spent a whole class on about the fundamental theorem of algebra I believe? I was distracted though.

    • frezik@lemmy.blahaj.zoneEnglish
      101·
      2 days ago

      Let’s face it: Calculus notation is a mess. We have three different ways to notate a derivative, and they all suck.

      • JackbyDev@programming.devEnglish
        4·
        20 hours ago

        Calculus was the only class I failed in college. It was one of those massive 200 student classes. The teacher had a thick accent and hand writing that was difficult to read. Also, I remember her using phrases like “iff” that at the time I thought was her misspelling something only to later realize it was short hand for “if and only if”, so I can’t imagine how many other things just blew over my head.

        I retook it in a much smaller class and had a much better time.

  • benignintervention@lemmy.worldEnglish
    79·
    3 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