#Differentiation in Math

[spedresources4all]

Mastering Multiplication: A Teacher-Tested System for Success

Multiplication fluency is one of the biggest hurdles in elementary math, but it doesn’t have to be a struggle. As a veteran special education teacher and labor activist, I know firsthand how hard it is to juggle differentiation, engagement, and skill mastery. That’s why I created a color-coded, chunking-based multiplication system designed for real classrooms, where time is short and every…

[thesomebodywho]

The biggest step of a mathematicians journey to maturity is no longer having to resist giggling at words like homogenous and homomorphic.

[algebraic-dumbass]

this neighborhood is totally tubular

[greatbeholderbrooke]

Just started differential geometry, I'm starting to understand the appeal

[yoneda-emma]

:D

[randomuniversityquotes]

Differential Equations Professor: "...so this is like, super easy."

[brandinotbroke]

at this point i am feeling secondhand embarrassment seeing how EA has been treating the sims franchise for years, and now the dragon age franchise too apparently

like i'm not into dragon age at all but wtf is this shit

[positively-knotted]

Dissertationposting 1 - Motivation & Introduction

A favourite result from undergraduate differential geometry is Gauss-Bonnet, which says that given a smooth surface M and scalar curvature R,

In particular, if M has positive curvature everywhere it must be a sphere, and if it is flat it must be a torus.

This is a crazy strong result, given that (in 2 dimensions) scalar curvature completely determines the geometry of the surface, while topology generally tells you very little. So it makes sense to try to generalise this to higher dimensions. Unfortunately, this doesn't work very well. Lohkamp (1992) showed that having R < 0 everywhere tells us nothing about the topology, and we can even construct spheres with negative curvature everywhere. However, there are some interesting things that happen if you force R > 0. In fact, it has some interesting restrictions on the algebraic topology of the manifold!

We'll explore a couple of these consequences, including the wacky proof techniques needed to get there. Some bits will require more technical prereqs (never more than Hatcher + do Carmo), but I'll try to keep them separate from the more intuitive and cute bits of the discussion.

Some highlights to wet the appetite:

Well start by induction on the dimension of submanifolds until we get down to surfaces. But in a few instances "submanifolds" won't be general enough and we'll have to come up with a more abstract idea called "μ-bubbles"

"Infinite cyclic covers"

These friendly guys:

Some small original results by yours truly

[secondwheel]

+c always forgotten :/

[mythicalcoqui]

The chain rule deserves to burn in the depths of hell

[amateurphysicist]

controversial opinion but..

maths is my favorite subject <3

[algebraic-dualist]

I don't know much about analysis, so perhaps this is easy... but nevertheless here is a question:

Let f : \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^{n} be Lipschitz continuous.

Let g : \mathbb{R} \to \mathbb{R}^n decay exponentially.

Let x, y : \mathbb{R} \to \mathbb{R}^n be such that x'(t) = f(t, x(t)) and y'(t) = f(t, y(t)) + g(t)

Does it follow that x(t) = y(t) as t -> infinity? Where does one find stuff about this?

[augmentedpolls]

If you are currently in a math class you can choose that option

[gay-mathematician-portmanteau]

@calculus students, i just need you to know that a differential in the form dy/dx or similar is absolutely a fraction and you may treat it as such.

[randomuniversityquotes]

Differential Equations Professor: "I have a matrix here, and a number here. That's a crime."