
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 preimage function. This will be useful as the definitions of topologies and continuity rely heavily on these functions.

The Gudermannian
Connecting Euclidean trigonometry and hyperbolic trigonometry. 
A physical interepertation of trigonometric substitution
Why does \(x = \tan \theta\) substitution work? 
Using matrices to understand polynomials
Coefficients of a product of polynomials can be generated by a multiplication between a matrix and a vector. 
Visualising bifurications
Bifurications made a lot more sense once I drew 3D plots that took account of the parameter and the phase space. 
Visualising linear dynamical systems directly
While eigendecomposition completely classifies the behaviour of linear dynamical systems, it's often too contrived for simple cases. 
Visualising invertibility and diagonalisability of matrices
also why the JNF is illconditioned and how the QR decomposition works by considering the induced action of matrices on the real projective line. 
Partial derivatives of the radius
A proof by picture of \(\partial_x (r) = \frac{x}{r}\) and by extension \(\nabla r = \frac{\mathbf{r}}{r} \) 
Understanding the hyperbolic plane with refraction
We could think of the Poincaré’s halfplane 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.

Complex function plotter
You can check out the plotter at tobylam.xyz/plotter! For the more technically savvy, here’s the GitHub repo. A fork of mabotkin’s complex function plotter.

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 onthespot)

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
The dual space of \(\mathbb{R}\) is \(\{f\colon \mathbb{R} \to \mathbb{R} \text{ s.t. } f(x) \vcentcolon= cx \;\vert\; c \in \mathbb{R}\}\). Could we find the dual space of \(\mathbb{R}^2\) just by using that? 
Understanding linear fractional transformations
How could \(f(x) = 1/x\) be linear? We'd try to show that to some extent, it is linear. The main idea is that we embed the real line into the cartesian plane and cheat a bit by using projections. 
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.

Compilation of miscellaneous tools
Useful web tools I have found for maths. In descending order of how well known they are (approximately)

Introduction to symmetries and its applications
Speaking of symmetries, you may think of the rotational symmetries of a circle or the reflection symmetry of a rectangle. It turns out that symmetries like these greatly help us in simplifying, solving and understanding physical problems in the real world. 
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.

LeviCivita 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 LeviCevita symbols. Fortunately, we can achieve this by calculating the determinant of a product of two matrices.

Structures closed under unions
Mathematical structures that are closed under countable or uncountable unions often give rise to a maximal structure generated by some set. 
LeviCivita Symbol (Part I)
Increasingly I’ve found the LeviCivita 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
Recall the definition of the subgroup generated by a set and the closure of a set in a topological space. They look curiously similar, involving intersections of subgroups and closed sets respectively. 
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
How good are 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)
A lot of the formulae given to you in HKDSE Physics, as it turns out, can be derived from the calculus taught in M2. In this series of posts we're going to go through deriving some of them. 
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
If you want to show a property is preserved under multiplication, try showing that the property is preserved under linear combinations and squaring. 
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.
On maths I find interesting