The Poetry of Mathematics

Linear algebra: this branch of mathematics developed from trying to solve numerous linear equations all at once! The new objects used in this pursuit and examining the underlying structures at play led to this powerful and highly applicable field of mathematics. It is where most students first encounter the concept of a matrix, an object made of numbers but which does not behave like "normal" numbers. For example, the order in which we multiply two matrices together matters, as you may get different results depending on which way round you do it! (Compare this with multiplying two "normal" numbers where the order doesn't matter: $2\times 3$ is the same as $3\times 2$).

"The Matrix"
Numbers in arrays.
Rows and columns in brackets.
The matrix appears.

Real analysis: a fundamental branch of mathematics where students take a peek behind the curtain to understand the theory behind calculus. The topic is introduced in the first year of the majority of mathematics degrees and contains the first "new" material encountered by undergraduate students (real analysis is not part of the standard A-level syllabus). Consequently, it is often considered a "difficult" subject and provides a challenge for most students.

"A Real Challenge"

There is doubt
But then at each stage, removed
It contains difficult questions, all rational
We see there is strong evidence and logic
Defining the information we collect
Reason, strong arguments
With a convincing way to create form
For all, an established point in time.

Differential geometry: this topic studies the geometry of curves and surfaces, and is not covered in all undergraduate degree schemes in the UK. Over the years, after various generalisations, it has grown into a large field with applications to many areas of mathematics, as well as to physics, such as the language used to describe Einstein's general theory of relativity. The basic questions asked in an undergraduate course is how do we describe the "curve" (more precisely, curvature) of a curve? What about the curvature of a surface? Perhaps surprisingly, there is more than one way to describe how a surface is curved.

"Curvature is Normal"

Curvature is a choice, no?
Shape influences shape
Curves in a curve
Always restricted, different
No interpretation is the one
The face shows only the shape
All curves will have the same name
This is the standard
Reducing it to a point called normal.

Number theory: A fundamental topic in mathematics, number theory is the study of whole numbers and the relationships between them. They are familiar objects and questions can often be easy to state, yet often be suprisingly difficult to answer. A basic concept is that of a prime number, studied in detail still in research-level mathematics despite being an idea introduced to school children.

"Primes"
A prime follows prime
Infinitely many, yet
There are questions still

Numerical analysis: It is not uncommon in mathematics and the physical sciences to encounter a problem that offers no analytical solution. In such cases, one resorts to numerical techniques to understand the situation. The field of numerical analysis is devoted to understanding and applying these methods. A simple example, encountered in most UK undergraduate mathematics degrees, is that of interpolation: estimating unknown data values based upon a given set of known data points.

"Interpolation"
We like as a tool
An essential numerical rule
For the approximate solution of an equation
Polynomial interpolation
With continuous functions on the interval [a, b]
The basic algebraic polynomials we use frequently

Fourier analysis: an important branch of mathematics first examined in the 18th century which has wide applicability within physics and engineering. Named after the French mathematician Joseph Fourier, it has been studied and developed for over 200 years and is covered in the majority of UK undergraduate STEM subjects. Historically, it was explored how to represent periodic functions using sums of trigonometric functions and has now been generalised to more abstract situations, leading to the development of new branches of mathematics.

"Fourier Knew"
Fourier knew
That this was true
The infinite sum of f(r)exp(irt) [f of r, e to the i r t]
Could be used to solve equations of the 19th century.

Perturbation theory: When one is faced with a complex problem in mathematics, one can often build approximate solutions if one can notice "small" terms that relate the complex problem to a much simpler, solvable problem. These solutions, built as a power series in the "small" terms, are called perturbative solutions and this technique - perturbation theory - is used to analyse a wide class of physical problems, from quantum mechanics to fluid dynamics.

"Perturbations"
When a model is represented by equations
One must resort to numerical approximations
Terms that are small
May be low order, that's all
When scaled the solution is a perturbation.

Stokes' Theorem: Stokes' Theorem is a classical result in vector calculus, an important branch of mathematics covering integration and differentiation of scalar and tensor fields (generally in a Euclidean three-dimensional space). It is often considered a difficult topic and is taught to many undergraduate students in wide range of STEM subjects. Stokes' Theorem can be considered a generalisation of the Fundamental Theorem of Calculus, a result in single variable calculus, to the case of three dimensions. It can, as with the whole topic of vector calculus, be generalised to an arbitrary number of dimensions.

"Stokes"
Here we contemplate
Stokes's Theorem, which relates
The surface integral of the curl
With the boundary curve(s) line integral.