A Natural Limit Definition

Often, the first exposure one gets to rigorous mathematics is the definition of a limit. Let’s consider what this is for a sequence. We say $\lim_{n\rightarrow \infty} a_n = A$ if

$\displaystyle \forall \epsilon \in \mathbb{R}^+\quad\exists N \text{ s.t.}\quad\quad n > N \implies |a_n-A| \leq \epsilon$

This, at first sight, is ugly. It takes a while to even understand what it’s saying, longer to see why it works, and much longer to apply it. It’s intimidating to say the least. I feel, however, there is another version that makes the idea of limits simple and very natural giving a deep insight into what a limit really is.

Quaternions Revisited

It has admittedly been quite a while since my last post over a year ago. I thought I would restart the posts by revisiting one of the first topics I discussed on the website: quaternions. My previous post, upon review, seems to be quite uninformative on what the nature and use of them are which I will attempt to show in this post.

Formalism

Quaternions are a generalization of complex numbers ($\mathbb{C}$) or hypercomplex numbers and they are denoted with $\mathbb{H}$. Below I write both in their general form.

$\displaystyle \mathbb{C}: a+bi$

$\displaystyle \mathbb{H}: a+bi+cj+dk$
where  $\displaystyle ijk=i^2=j^2=k^2=-1$

Now, there are 3 “imaginary” components and they are defined by that relation at the bottom. This is super interesting! What does this even mean though? A real number with some sort of 3-dimensional imaginary component?

Differential Forms Part 2: Differential Operators and Stokes Theorem

In the first post, we established a general intuition of how forms work and why they may provide a better geometric intuition of what is actually occurring. It was mentioned that these ideas extend the ideas of vector calculus so it seems natural to see how differential operators like gradient, curl, and divergence arise in the context of differential forms. It all comes out of the analysis of the exterior derivative $\textup{d}$. I will stick to 3 dimensions for now and explain the extension into higher dimensions at the end.

Cool Things

Math is cool. Here are some cool things in math that I don’t think are really extensive enough for their own post but I still want to share.

1. $\inline&space;0^0=1$

The following is a great reason why. Continue reading

Calculus of Variations Part 1: Establishing the Basis

Calculus of variations is an extremely useful and amazing tool in physics, math, computer science, and a variety of fields. Similar to how regular calculus is focused around functions and differentials, this field focuses on functionals and variations. A functional $\inline&space;F[y]$ takes in a function and spits out a number. The following are examples of functionals. Continue reading

Orders of ∞

The idea of infinity is easy to look over upon first glance. It can simply be defined as the idea that numbers go on forever and it is easy to end there. However, there are very developed and well-defined notions of infinity suggest that there are different orders and types of infinity which come with various properties. The consideration of all these are vital to our understanding of infinite quantities and especially in set theory. The exploration of these transfinite numbers Continue reading

Differential Forms Part 1: Dimensions and Notation

Differential forms is a topic that, in some sense, extends ideas presented in vector calculus with more suggestive notation and geometric intuition into higher dimensions. The distinction may seem small and insignificant especially in the third dimension that we live in but its results and implications are quite elegant and can lead to nice formalization of certain results such as Stokes’ Theorem. Continue reading

Some Resummation Theory

Perturbation theory, as mentioned in an earlier post, is a very important part of the study of many fields but a recurring problem is the issue of summing divergent sequences which sometimes arise in a solution. Even some convergent solutions are very hard to sum because we can only calculate the first two or three terms in a reasonable amount of time. As a result, the study of these infinite summation become very important to advance the field. Continue reading

Lebesgue Integration

Integrals are a great point of interest in many areas of mathematics and, when learned about, are often overlooked on the fundamental level. The ideas of Riemann integration, which is what many learn about, are very vast and complex and can provide powerful results but there exists, in some sense, a better and more general form of integration that can account for scenarios Riemann integration cannot. This is Lebesgue integration. Continue reading

Perturbation Theory

There exists a certain class of “hard” problems that can’t be solved with exact form. Examples include solutions to certain differential equation or higher order polynomials like quintics which can’t be solved with a simple cubic formula or quadratic formula. Perturbation theory is a tool commonly used in mathematical physics and can easily provide solutions to seemingly impossible problems. Continue reading