Sam Auyeung

Sam Auyeung

Blog Posts - 2023

2023

The Divergence Theorem and Gravity

8 minute read

Published:

In this post, we’ll consider an example related to the Divergence Theorem and gravity. I think it’s a nice example to show students taking multivariable/vector calculus (which I’m teaching this semester). Let’s first recall the statement of the Divergence Theorem. Let $W \subset \mathbb{R}^3$ be a 3-dim region with boundary being a piecewise smooth, closed surface $S$. It’s a somewhat nontrivial fact that all closed hypersurfaces of $\mathbb{R}^n$ are orientable but in this case, one can use the cross product on $\mathbb{R}^3$ to produce an orientation on a closed surface. Note the closedness is essential; the Möbius band is a nonorientable surface in $\mathbb{R}^3$ but it is not closed. Moreover, in $\mathbb{R}^3$, having a unit normal vector field on $S$ induces an orientation on $S$. So let’s suppose that $W$ is oriented and $S$ has the induced orientation from $W$. If $F$ is a vector field defined on a neighborhood of $W$, we may find the divergence of $F$ which is a scalar function that intuitively, measures the net flow of $F$ at a given point.

Integrability and the Three Body Problem

7 minute read

Published:

The standard definition for integrable system, which, for example, can be found on the Wikipedia article, is due to Liouville. Given a Poisson manifold $(P,\{ \})$ parametrising the states of a mechanical system, a Hamiltonian function $H$ defines a vector field $X_H := \{H,-\}$, whose flows are the classical trajectories of the system. A function $f$ which Poisson-commutes with $H$ is constant along the classical trajectories and hence is called a conserved quantity. For example, if $(P,\omega)$ is a symplectic manifold, then there is a Poisson bracket given by ${f,g} = \omega(X_f,X_g)$ where $X_f,X_g$ are Hamiltonian vector fields; i.e. $\iota_{X_f} \omega = df$ and similarly for $g$. Note that $\{f,H\} = \omega(X_f,X_H) = df(X_H)$ and so if this is zero, then $f$ is constant in the direction of $X_H$ which are the classical trajectories.

On the Stages of Learning and Doing Mathematics

18 minute read

Published:

I’m teaching Multivariable and Vector Calculus at Trinity College this fall semester and have students with all kinds of backgrounds. For some of them, this is their first semester of college and are used to year-long math classes rather than a whole course in one semester. The calculus they had was high-school AP Calculus which is at a different level than college-level calculus and can vary a lot depending on, say, whether it’s a public high school or a private science and technology magnet school. Some of them finished high school on Zoom because of COVID and for many, distance learning is not as effective as in-person learning. And then they took a break from math for a full year. Some of them don’t speak English as their first language. For others, they’ve taken the first two college-level calculus courses and have also taken Physics: Mechanics and Physics: Electromagnetism where vector calculus already appears. So this class is a breeze for them. One of my students went the first three months of his first year before losing a single point in any of his classes. English is not his first language and so he seems to appreciate that in a math course, the most important language to understand is mathematics (though of course that happens by means of English in my classroom).

The Generalized Stokes Theorem

3 minute read

Published:

In a class on vector calculus, the students are presented with a few theorems such as the Fundamental Theorem of Line Integrals, Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. The first one relates differentiation and integration in the following way. Take the gradient of a multivariable function $f:\mathbb{R}^n \to \mathbb{R}$ and consider a smooth path parametrized by $\vec{r}(t)$ with $t\in [a,b]$. Then the definite integral of the dot product $\nabla f \cdot d\vec{r} \, dt$, a quantity arising from a 1-dimensional object has a 0-dim interpretation: $\int^a_b \nabla f \cdot d \vec{r}\,dt = f(\vec{r}(b)) - f(\vec{r}(a))$. The proof of this really is just to use the usual Fundamental Theorem of Calculus.

Center of Mass of $N$ Points in Space

7 minute read

Published:

Recently, a friend introduced me to a problem posed by Presh Talwalkar in a video here. The problem is as follows. Take an equilateral triangle and inscribe a circle into it. Pick any point $P$ on the circle and connect line segments from $P$ to each of the vertices. The challenge is to show that the sum of the squares of the lengths is actually a constant, independent of the placement of $P$. The video gives a nice proof of the fact but when one studies it more closely, we see that we can actually generalize the statement quite a bit.

Fun Fact About Partitions

2 minute read

Published:

Let $p(n)$ denote the number of unordered partitions of $n$. For example, if $n=5$, we can partition this as $1+1+1+1+1,1+1+1+2,1+1+3,1+4,5,1+2+2,2+3$. So $p(5)=7$. We can write down a generating function $P(x) = \sum^\infty_{n=0} p(n)x^n$. But also, observe that if we wrote out $(1+x+x^2+x^3+…)(1+x^2+x^4+x^6…)…(1+x^k+x^{2k}+x^{3k})…$ and start multiplying things out, it amounts to picking an element from each “bubble.” If we want a finite product, we eventually only choose 1’s. So pick something from the first bubble, say $x^2$, $x^6$ from the second, and $x^6$ from the third, then all 1’s. The product is $x^{14}$ and these choices gave us a partition $14=1+1+2+2+2+3+3$. How did we produce this? Our choice of $x^2$ from the first bubble says we want two contributions of 1. Our choice of $x^6 = (x^2)^3$ from the second bubble says we want three contributions of 2. And our choice of $x^6=(x^3)^2$ from the third bubble says we want two contributions of 3.

A Simple But Powerful Thought Experiment for Special Relativity

4 minute read

Published:

Imagine two parallel straight copper rods set a centimeter apart that extend infinitely out in both directions and suppose there are positive unit charges spaced out evenly on the two rods. We’ll also suppose that the charges on one rod are lined up with charges on the other. You’re an observer some distance away and you take note that the rods want to repel each other because everything is positively charged; call this amount $\alpha$. We’ll suppose the rods are magical and can never be more than one centimeter apart. So they aren’t actually repelled.

On “Anxiety is the Dizziness of Freedom”

13 minute read

Published:

This post is about a short story by Ted Chiang entitled Anxiety is the Dizziness of Freedom. It is found in his collection Exhalation which has several other short stories, all worth reading. In case you don’t know, Ted Chiang is a Chinese-American writer and his short story Story of Your Life is possibly his most well-known work since it was adapted into the movie Arrival by the visionary Denis Villeneuve. Though Chiang doesn’t write science fiction exclusively, many of his stories can be considered science fiction though I’d argue that his focus is often not so much on the science side of things; I’ll try to convey what I mean in this post. A fun fact (for me anyways) is that he was born in Port Jefferson, NY where I lived for a year. His father is a professor at Stony Brook University in mechanical engineering though we never crossed paths while I was at SBU.

Two Elegant Proofs of the Pythagorean Theorem

9 minute read

Published:

In this post, I’ll like to provide two proofs of the Pythagorean theorem which I find elegant and also, don’t really require anything other than pictures though, of course, I’ll discuss them with words. Now, recall that if we have a right triangle in the Euclidean plane with side lengths $a,b,c$ where $c$ is the length of the hypotenuse (the longest side), the Pythagoreans understood that the three numbers are related to each other by the equation $a^2 + b^2 = c^2$.

A Tiny Slice of Quantum Chromodynamics

21 minute read

Published:

Yesterday, I attended the last summer Sunday at Brookhaven National Labs and in particular, did a brief tour of the engineering department that worked on building the RHIC (relativistic hadron ion collider) as well as hear a lecture by Gene van Buren on a solenoidal tracker attached to RHIC called the STAR detector. The RHIC is able to accelerate gold nuclei to 99.995% the speed of light; the track has about a 2.5 mile circumference and the particles travel around it about 80,000 per second. At speeds like this, the particles are very massive and experience very little time. This statement is not something we usually say in our everyday experiences but if you have some experience with special relativity, you know that in order to accelerate something, you need to apply force. At speeds near the speed of light, there is a diminishing return. In order to gain a fraction of speed, one needs to apply more and more force and asymptotically, one needs an infinite amount of force to get to the speed of light. If we view mass as a property which resists being moved (faster), then one way to view relativistic particles is that they’re very massive; cf. moment of inertia.

A “Basic” Probability Problem

5 minute read

Published:

This is yet another post in the series of “I am not at all an expert in this but enjoyed thinking about it.” Also, I wanted to write something shorter, that doesn’t require a big time commitment to read. This seems to fit the bill. The problem that will be posed is basic in the sense that it is easy to understand the question but I did not solve it right away.

Lattices and Modular Forms

12 minute read

Published:

I am not a number theorist but this topic has fascinated me and I’ve decided to write about it as one of the first blog posts. The classification of definite forms is difficult but also appears in the study of 4-manifolds since Freedman proved that closed, oriented, simply-connected, topological 4-manifolds are classified by their intersection form. This post isn’t about that so much. Much of what I write here can be found in A First Course in Modular Forms by Fred Diamond and Jerry Shurman.

Theorem of Rokhlin

22 minute read

Published:

The goal of this post is to discuss Rokhlin’s Theorem and also the Rokhlin invariant which is for 3-manifolds. Theorem (Rokhlin): Let $M^4$ be a closed, smooth, spin 4-manifold. Then its signature $\sigma(M)$ is divisible by 16.

Morava K-Theory

20 minute read

Published:

I’m not a homotopy theorist but ever since the paper by Abouzaid-McLean-Smith was released which showed the power of homotopy theory towards answering questions in symplectic geometry, I’ve been interested in stable homotopy theory. In particular, what the role of the Morava K-theories play. Of course, the Abouzaid-Blumberg paper is an earlier application of Morava K-theory to the Arnold conjecture. In that paper, one would like to define Hamiltonian Floer theory over $\mathbb{F}_p$ but because our counts of curves ought to divide by the order of finite stabilizer groups, it is too naive to do mod $p$ counts since $p$ would not be invertible. Instead, one uses the hierarchy of generalized cohomology theories $K_p(n)$ which approximates $\mathbb{F}_p$ as $n$ gets larger. The notation here with the subscript $p$ is unconventional but I wanted to emphasize that we’ve fixed a prime $p$. I’ll continue to do this for a while but will eventually suppress that part of the notation.

Most Recent
2024 →