• 0 Posts
  • 43 Comments
Joined 3 years ago
Cake day: June 12th, 2023
  • Kogasa@programming.devtoScience Memes@mander.xyz•cry harderEnglish
    4·
    1 month ago

    A Riemannian manifold isn’t necessarily non-Euclidean, it’s just a smooth manifold with a Riemannian metric, which is just sort of a way of defining local geometry in a coherent way. Namely it’s a smooth family of inner products on the tangent spaces at each point, where an inner product on the tangent space is sort of a way of comparing any two directions at a point and the smoothly varying part means that for sufficiently close points, the comparison function on their respective tangent spaces is similar.

    Anyway, like “manifold” is a formalism intended to capture the idea of a “shape or space,” a “Riemannian manifold” is just “a shape or space we can do geometry on.”

  • Kogasa@programming.devtoScience Memes@mander.xyz•number box o number boxEnglish
    3·
    9 months ago

    Which is really a roundabout way of saying a tensor is a multilinear relationship between arbitrary products of vectors and covectors. They’re inherently geometric objects that don’t depend on a choice of coordinate system. The box of numbers is just one way of looking at a tensor, like a matrix is to a linear transformation on a vector space

  • Kogasa@programming.devtoScience Memes@mander.xyz•Listen here, Little DickyEnglish
    2·
    11 months 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.

  • Kogasa@programming.devtoScience Memes@mander.xyz•Listen here, Little DickyEnglish
    1·
    11 months 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…

  • Kogasa@programming.devtoScience Memes@mander.xyz•ManifoldsEnglish
    12·
    1 year ago

    Manifolds and differential forms are foundational concepts of differential topology, and connections are a foundational concept of differential geometry. They are mathematical building blocks used in modern physics, essentially enabling the transfer of multivariable calculus to arbitrary curved surfaces (without relying on an explicit embedding into Euclidean space). I think the joke is that physics students don’t typically learn the details of these building blocks, rather just the relevant results, and get confused when they’re emphasized.

    For a tl;dr about the concepts mentioned:

    • A manifold is a curve, surface, or higher-dimensional object which locally resembles Euclidean space around each point (e.g. the surface of a sphere is a 2D manifold; tiny person standing on a big sphere perceives the area around them to resemble a flat 2D plane).

    • Differential forms are “things that can be integrated over a manifold of the corresponding dimension.” In ordinary calculus of 1 variable, that’s a suitably regular function (e.g. a continuous function), and we view such a function f(x) as a differential form by writing it as “f(x) dx.”

    • A connection is a way of translating local tangent vectors from one point on a manifold to another in a parallel manner, i.e. literally connecting the local geometries of different points on the manifold. The existence of a connection on a manifold enables one to reason consistently about geometric concepts on the whole manifold.