Some golden rules for new mathematicians

We start this semester’s posts with a contribution from one of our fourth-year students.

The beginning of a new academic year always holds promises of “I will try harder this year”; none more so than those, like myself, heading into their honours year. I felt that I could reflect on past experience and hopefully pass it onto the newest mathematicians at Strathclyde.

Over my four years I have developed two golden rules:

  1. Understand proofs and what is going on.
  2. Understand and be careful with notation.

These may seem glaringly obvious, but I and others will testify that these are often overlooked, even if unintentionally.

Allow me to elaborate on my first point, this being the key one, of course. Lecturers will tell you, time and again, to read over and understand your notes until you know what is going on; but what do they know, right? You just earned a place in university, maybe even earned an Advanced Higher in maths — you are master of the universe. Wrong. Incorrect. Generic game show dud alarm. They say this because they are, in fact, right. I fell into the trap “I can do the problems; the exam questions will be similar; practise; practise; practise…” This is the attitude that was, and will be, many an undoing. The truth is that schools teach you how to pass the exam, not to understand what is actually happening.  The reason I stress this is that it is paramount for learning mathematics: if you understand the proofs and what is actually happening, then this can be applied to any question. Trying this process the opposite way around, practising lots of similar questions but not fully understanding the material, will prove erroneous. What’s to say the nasty lecturer won’t change the question slightly, in such a way that you would only notice the difference if you had understood what was going on? “I will still get some credit”, I fear you answer, for this is what makes a bad mathematician, and that is ultimately what you wish to qualify as. Don’t do it!!

The second point, notation, is another pitfall that catches many early — and some not so early — mathematicians. One that immediately springs to mind is the following: compare

f(x(t)) = (x(t))^2 + x(t)-1

and f(x) = x^2 + x - 1.

In some cases the two expressions are equivalent — so long as they are defined correctly. Often the t is omitted because it is assumed to be implied, but the reader may not know that. If it isn’t properly defined then certain operations will become troublesome: what, for example, do we mean by the function f'? Another example is the dummy variable; this is exactly what it says it says on the tin. I may have 4x, but it may as well be 4y, or 4 batman symbols! It doesn’t matter so long as they are suitably defined beforehand. There are many traditional symbols used, which you will probably be acquainted with, but what of those traditional symbols in one field that is different from another? A classic example of this is the imaginary number, i, as you may have seen: in engineering this is often j. In mechanics there is the vector notation in \mathbb{R}^3 of \mathbf{i}, \mathbf{j} and \mathbf{k}, but also \mathbf{e}_1\mathbf{e}_2 and \mathbf{e}_3. They are equivalent in this context, but that does not mean mixing and matching until your heart is content. This is a big no-no! Consistency is key here, so that you do not confuse a marker, and ultimately yourself.

As a further example, this time from fourth year fluid mechanics: the Navier–Stokes equations are often written, for example, as

\displaystyle\frac{\partial u_i}{\partial x_i} = 0 and \displaystyle\rho\left(\frac{\partial u_i}{\partial t} + u_j\frac{\partial u_i}{\partial x_j}\right) = \rho f_i + \frac{\partial\sigma_{ij}}{\partial x_j}, where \sigma_{ij} = -p\delta_{ij} + \displaystyle\mu\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right),

but also appear frequently in the form

\nabla\cdot \mathbf{u} = 0 and \displaystyle\frac{\mathrm{D}\mathbf{u}}{\mathrm{D}t} = \mathbf{f} -\displaystyle\frac{1}{\rho}\nabla p + \nu\nabla^2\mathbf{u}

(and indeed in several other ways!) Hopefully the need to distinguish between vectors and scalars, between variables and subscripts, between D and d and so on, is painfully clear… not to mention the need to keep track of what each of these symbols represents. The main point, in any case, is to make sure you define everything you use, do not get lost in confusing notation, and keep track of all of the symbols you are using.

I learned these lessons the difficult way in my 3rd year in the USA, but rectified it as quickly as I could, being in another country where the notation and the methods of learning differ somewhat from Strathclyde. Don’t get stuck in one method of learning: not all lecturers teach the same, some grade more strictly, and some set easier exams; the only way to learn is to adapt. Books are great to see things from a different perspective; sometimes the one you are viewing isn’t best suited.

The important point here is that you must understand the proofs, the questions will follow, and to be careful with notation and defining things. If you master these rules early then you will do well so long as you apply yourself — student nights are tempting!

(Vincent Keenan)

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