A circle and a hyperbola living in one plot

We will see that the 3D plot of \(x^2 + (y+zi)^2 = 1\) contains a circle and a hyperbola, where \(i\) is the imaginary number. Beyond the visuals, this helps us understand the (complex) eigenvalues of real matrices.

Taylor series raised to a power

Given some \(f(x)\), what’s the taylor series of \(f(x)^p\) for some real \(p\)?

Alternate notation for topology

We introduce some notation for the image and pre-image function. This will be useful as the definitions of topologies and continuity rely heavily on these functions.

The Gudermannian

A physical interepertation of trigonometric substitution

Using matrices to understand polynomials

Visualising bifurications

Visualising linear dynamical systems directly

Visualising invertibility and diagonalisability of matrices

Partial derivatives of the radius

Understanding the hyperbolic plane with refraction

We could think of the Poincaré’s half-plane model as a plane of transparent material with a specified refraction index. The light rays would correspond to geodesics.

Understanding potential wells

To think about potential wells, we often imagine little beads rolling on the plot of the potential well (as a wire) under gravity without friction. How precise is this analogy?

Understanding Green's theorem

The statement of Green’s theorem is not particularly illustrative of the ideas driving it. We would try to gain some intuition by considering a simple example: finding the area bounded by a curve in the plane.

Another visualization of the second fundamental theorem of calculus

Calculus of inverse functions: Inverse function theorem and the Legendre transformation (Part I)

Given a function such as \(\tan x\), could you write \(\frac{d}{dx} \arctan x\) and \(\int \arctan x \; dx\), just from \(\tan x\), \(\frac{d}{dx} \tan x\) and \(\int \tan x \; dx\)? With some caveats, the inverse function theorem answers the former while the Legendre transformation answers the later.

Graph sketching techniques

We’d cover techniques of sketching graphs of compositions of functions (i.e. \(f(g(x))\)) and when one could approximate graphs with simpler ones.

Generalising function plots to topological spaces

How could we plot a function \(f: X \to Y\) for two general topological spaces \(X,Y\)? Firstly, let’s look at some examples of plots of functions that we are used to.

Pedagogically weaker notions of continuity

We introduce a weaker notion of continuity, similar to continuity at a point, that might be useful pedalogically.

Constructing homotopies using gravity

To show that the open disc without the origin is homotopy equivalent to the circle, one could imagine having a circle with radius \(1/2\) centred at the origin with uniform mass inside the disc.

Geometry of curves and projectile motion

To describe a differentiable curve, one could use a parameterisation. Imaginatively, one could imagine driving a car with the objective of aliging the car’s path with the curve. One has unlimited control over the car’s throttle (the velocity) and “ability to turn” (the centripetal acceleration, one could imagine a car that can turn on-the-spot)

Euclidean algorithm and GL(2, Z)

A question from stack exchange, given a vector \(\mathbf{v} \in \mathbb{Z}^2\), what is its orbit under the action of the general linear group \(\text{GL}(2,\mathbb{Z})\) of order 2 over the integers? Through answering this question, we relate the Euclidean algoirthm from number theory to the study of the general linear groups.

Understanding the dual space

Understanding linear fractional transformations

Applications of Lie Groups to Differential Equations

I recently completed a summer project on applications of lie groups to differential equations under the supervision of Jason D. Lotay. You can find the report here. I have given a talk on the topic using these slides.

Introduction to symmetries and its applications

Understanding the Jordan Normal Form

We aim to understand the strategy of the proof of Jordan Normal Form theorem through analyzing a very simple example of a nilpotent matrix.

Playing with the real projective plane

The desmos graph below is one of the many interpretations of the projective plane. In particular, we aim to demonstrate duality between points and lines.

Visualising composition series

I’ve found visualising composition series using curved exact sequences to be very helpful. The notation could be used to condese elementary theorems about solvable groups into a single picture.

Visualising the group isomorphism theorems

I’ve found the following diagrams to be very helpful at understanding and remembering the isomoprhism theorems.

Three ways of thinking about matrix multiplication

I’ve found thinking about matrix multiplication in the following three ways to be exceptionally helpful, especially at understanding why elementary row operations are on the left and why elementary column operations are on the right.

Levi-Civita Symbol (Part II)

Continuing from part I, we’d like to go further and handle objects of form \(\mathbf{u} \times (\mathbf{v} \times \mathbf{w})\). However, this would involve multiplication of two Levi-Cevita symbols. Fortunately, we can achieve this by calculating the determinant of a product of two matrices.

Structures closed under unions

Levi-Civita Symbol (Part I)

Increasingly I’ve found the Levi-Civita symbol to be incredibly useful at deducing equalities involving cross products. I’d go through some basic applications of the symbol in linear algebra and multivariable calculus.

Structures closed under intersections

Extending sequences into smooth functions using trignometric functions

Sequences sometimes extend to good examples of smooth functions with special properties e.g. interesting limsups and liminfs.

Interiors, closures and boundaries of topological spaces

I’d show alternate definitions of interiors, closures and boundaries for topological spaces which I’ve found immensely helpful.

LEON Discounts

Compilation of miscellaneous ideas

HKDSE Physics and M2 (Part III)

In the final post of this series, we would discuss waves. How do we formulate waves mathematically? Why are waves often depicted by sine curves?

HKDSE Physics and M2 (Part II)

Continuing from part I, we would now look into projectile motion and uniform circular motion.

HKDSE Physics and M2 (Part I)

Euler's formula and compound angle formulae

Compound angle formulae are one of the most useful tools you have in high school trigonometry, yet reciting them could be quite difficult. It turns out one could deduce them from the more straightforward Euler’s formula \(e^{i \theta} = \cos \theta + i \sin \theta\).

Showing preservation of properties under multiplication via difference of squares

Another example of pointwise but not uniform convergence

There’s a lot of examples of pointwise but not uniform convergence. But I suppose it wouldn’t harm to have another one. The following is certainly not original.