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 mabokin’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 on-the-spot)

Second Summer in Oxford

Euclidean algorithm and GL(2, Z)

Given a vector \(\mathbf{v} \in \mathbb{Z}^2\), what is its orbit under the action of the general linear group \(GL(2,\mathbb{Z})\) of order 2 over the integers?

Understanding the dual space

Networking to network - Phil Agre

Recently came across an article by Phil Agre on the significance, process and morality of networking in an academic setting. I’ve found it to be substantial, convincing and helpful. Here’s some quotes I found particularly inspiring.

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 could find the report here.

Calculator Programming

Years ago when I was still in secondary school, mobile phones were not that popular yet and as such I was obsessed with programming my fx-50FHII which was the only mobile computing platform I had. I ended up trying to write up an introductory course on calculator programming. Fun times. Here’s the [pdf]

Compilation of miscellaneous tools

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

Second Year at Oxford

Introduction to symmetries and its applications

Continuity and comparison of topologies

Through comparing topologies, one could deduce a more compact notation for continuity. It also leads to a nice description of continuous bijections and homemorphisms.

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.

Second Easter in Oxford

I Want to be a Mathematician - Paul Halmos

A book written by Paul Halmos, I’ve very much enjoyed it and have found the following quotes particularly inspiring.

Opportunities learning maths beyond the syllabus in HK

Hong Kong is filled with institutions that one can utilize to advance your learning in mathematics. I’m going to share a list of opportunities pursuing mathematics that I’m aware of.

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 which needs to be resolved. It turns out we can do so through calculating determinants of matrices.

Fifth Term at Oxford

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 continuous functions

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

The Art of Doing Science and Engineering - Richard Hamming

A book written by Richard W. Hamming, it has slowly became one of my favourite books of all time. I would mention a few snippets I’ve found particularly inspiring.

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.

On Stun-locks

Effort vs Quality

Across a wide range of technologies, there’s increasingly this tension between effort and quality. We want to minimize user effort and maximize quality of output, yet are we taking that too far?

Second Christmas in Oxford

DIY Pin Board of Tools and More

Recently I’ve been trying to make my room feel like home without resorting to consumerism and with function in mind. Here are some results.

LEON Discounts

Technical aspects of tobylam.xyz

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 \(e^{i \theta} = \cos \theta + i \sin \theta\).

Fourth Term at Oxford

Tackling Oxford Mathematics Interviews (Undergraduate)

Oxford Mathematics Interviews may feel daunting. Beyond the basic format of the interviews, here are some of my advice on tackling them.

First Summer in Oxford

Showing preservation of properties under multiplication via difference of squares

Making a mathematics personal statement personal

Audience and Tone of tobylam.xyz

Having maintained this website for over 2 years, I’d like to mention some of my thoughts on tobylam.xyz and where I would like to take it in the future.

Inspirations for tobylam.xyz

It has been two years since my first post and I’d like to share some of my thoughts on what made this blog a reality, and my thoughts on the direction of this blog. After all, the inspirations for this website heavily affect the direction I would like to take it.

First Year at Oxford

Second Term at Oxford

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.

First Term at Oxford

Applying to Oxford Mathematics (Undergraduate)

Using Zoom / Google Meet with OBS

Why start a website?