Kogasa

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

In the context of differential forms, an integral expression isn’t complete without an integral symbol and a differential form to be integrated.

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.

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…

No 👍

The other thing is that it’s legit not a fraction.

Not really, you need to have a basic understanding at least

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.

Yes, OP only gets you to Q[i]

Yeah. Normal whoppers are crunchy. 1 in 4 whoppers is soggy and chewy and hard to eat

Whoppers are good but the risk of getting a bad one is not worth it. Ech

It can be, usually for college credit though

At the universities I went to, Calc 2 was integration, sequences and series, then Calc 3 was multivariable. They really pack all the harder parts into 2.

Yes. A matrix is unitary if the conjugate transpose (conjugate as in “complex conjugate”) is equal to its inverse.

Quoting a relatively famous mathematician, linear algebra is one of the few branches of math we’ve really truly understood. It’s very, very well behaved

Stokes’ theorem. Almost the same thing as the high school one. It generalizes the fundamental theorem of calculus to arbitrary smooth manifolds. In the case that M is the interval [a, x] and ω is the differential 1-form f(t)dt on M, one has dω = f’(t)dt and ∂M is the oriented tuple {+x, -a}. Integrating f(t)dt over a finite set of oriented points is the same as evaluating at each point and summing, with negatively-oriented points getting a negative sign. Then Stokes’ theorem as written says that f(x) - f(a) = integral from a to x of f’(t) dt.

Only if you’re trying to get a numerical point evaluation. For example, one can use Fourier series to represent complex signals in terms of sine waves, and then reproduce the sine waves with hardware to reproduce the original signal. This is how a simple synthesizer produces different kinds of tones.

Nullable reference types are (a completely mandatory) bandaid fix in my opinion as a .net dev. You will encounter lots of edge cases where the compiler is unable to determine the nullability of an object, e.g. when using dependency injection to populate a field, or when using other unusual control flows like MediatR. You can suppress the warnings manually at the slight risk of lying to the analyzer. Objects supplied by external library code may or may not be annotated, and they may or may not be annotated correctly. The lack of compile-time null checking is occasionally an issue. But that said, NRT makes nullability a significantly smaller issue in C# than it used to be

Reminds me of 2048 making a slightly worse clone of Threes and then releasing it for free.

But not all mathematicians are logicians.