Part 1

How to study

University Study
Pattern of work
Lectures
Tutorials
Cooperation
Books and libraries
Vacation work

University Maths
Introduction
Studying the theory
Problem solving
Writing mathematics

Applied Maths
Pure vs applied
Applied problems
Writing solutions

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Writing mathematics discusses in general terms the task of writing out the solution accurately and completely. When you read Part 2: Proof, you will find some examples taken mainly from pure mathematics, where the task is to write out a proof at the appropriate level of `rigour'. In applied mathematics, very little (but still some) of your work will involve writing out formal proofs of precisely formulated general propositions. You will spend much more time developing techniques to solve particular problems, without trying to push up against the limits of their applicability.

Here we deal with the following topics:

• Inappropriate rigour
• Clear writing
• Accurate manipulation
• Concise writing
• Diagrams

Inappropriate rigour

Aiming for rigour in your applied work can lead you into inappropriate and distracting arguments.

In your analysis courses you may well work through proofs of very simple and familiar propositions, such as that the derivative of the function x® x² is the function x® 2x. It takes a while to see why it is important to do this: it is that you need to put the foundations of calculus on a sound basis in order to build on them later. If you rely on intuitive and vague definitions of concepts like limit and derivative, you will eventually come up against the same barriers that stalled the development of mathematics in the eighteenth century. In pure mathematics, the danger is that, in exploring the foundations, you accidently allow in intuitive arguments based on your informal understanding of the concepts you are playing with. Hence the emphasis on the discipline of rigour, of taking great care to check that your argument follows logically, step-by-step from the definitions. There is a corresponding danger in your applied work that you will be so unnerved by concern for rigour that you will be unable to write down ``if f(x) = x², then f¢(x) = 2x'' without proving it. Taken to its extreme, this attitude would imply that you could not use integrals at all in your first two terms because you do not begin the rigorous theory of integration until your third term. This would be to misunderstand the purpose of rigour, which is precisely to allow you to use the familiar rules of calculus (and the less familiar ones that you will meet in the coming year) with confidence. The purpose is to reinforce that confidence, not to undermine it.

It is very easy to misinterpret these remarks as saying that ``rigour has no place in applied mathematics; any argument, however sloppy, will do''. That is wrong. All that is being said is that to sum up the instructions for writing good applied mathematics under the heading of `rigour' would be misleading. The important point is that your written work should be clear, accurate, and concise (I shall expand on this below). Rigour is, in any case, a relative term. One of the disconcerting lessons of twentieth century mathematics is that, if you insist on proving everything by strict rules of inference from a finite set of axioms, then your mathematical horizons will be very limited. A celebrated theorem of Gödel's implies that you will not even be able to prove rigorously all the propositions in arithmetic that you know to be true by informal argument. In almost all your work, pure and applied, you will construct arguments by building on other results that you take for granted. Read what is said about this in Part 2: What can you assume? and look carefully at some of the examples in Part 2: Proof when you work through them. In your applied work, you will build on the results you prove in your pure work.

Clear writing

A solution to a mathematical problem takes the form of an argument. It should be written in good grammatical English with correct punctuation. It should make sense when you read it out loud. This applies to the mathematical expressions as well as to the explanatory parts of the solution. A solution should never take the form of a disconnected sequence of equations. When you are writing up a calculation, you must make clear the logical connection between successive lines (does the first equation imply the second, or the other way around?), and you must make the equations fit in a coherent way into your sentences and paragraphs (see the remarks in Part 2: Making the proof precise). One minor point: many mathematicians are poor spellers. If you are not confident about your spelling, keep a dictionary on your desk, and use it. A good way to improve your style is to browse occasionally in Fowler's Modern English usage.

You will never write a clear argument unless you are clear in your own mind what it is it that the argument is proving, and you make this clear to your reader. Sometimes it is simplest to do this by using the `proposition-proof' style of pure mathematics, but more often this is inappropriate in applied mathematics (it looks very artificial to formulate as a `proposition' a result that is special to a particular problem). It is very helpful, both for yourself and for your reader, to say what you are going to do before you do it, for example, by writing something like ``We shall now show that every function given by an expression of the form ... satisfies the differential equation''. It is also helpful to break up a long argument into short steps by using sub-headings. For example, if you are asked to show that ``X is true if and only if Y is true'', then you might head the first part the argument with ``Proof that X implies Y'' and the second with ``Proof that Y implies X'' (see Part 2: Implications).

Of course it is in the nature of much of applied mathematics that you have to do quite a lot of thinking to extract from the original formulation of the problem the precise statements that you have to establish. Even something as apparently simple as

Solve the equation y¢¢ = y

contains potential pitfalls. The correct answer is: y = Ae^x+Be^-x, where A and B are arbitrary constants. Your argument in this case will establish two things: first, that every function of the form y = Ae^x+Be^-x satisfies the equation, and, second, that every function that satisfies the equation is of the form y = Ae^x+Be^-x. In a simple example like this, you can establish both statements by the same short calculation, and it would be rather fussy to break the argument up into two parts. But you do not have to go much further to find an example in which you can go badly wrong if you simply manipulate without thinking. Consider the following problem.

Solve the simultaneous differential equations y¢ = z, z¢ = y.

If you ignore what has been said above, you might be tempted to write the following.

Solution.

y¢ = z,        z¢ = y y¢¢ = z¢,        z¢¢ = y¢ y¢ ¢ = y,        z¢¢ = z Answer       y = Aex+Be-x,       z = Cex+De-x.

However, if you take particular values for the `arbitrary constants' A, B, C, D, say, A = 1, B = 0, C = 0, D = 0, then you rapidly find that your `answer' does not work. The problem is that it is not clear what the `argument' in the solution is establishing. If you look carefully, you will see that the first line implies the second line, that the first two lines together imply the third, and that the third implies the fourth. However, when you try to go backwards, you cannot get from the second line to the first. To solve this problem without confusing yourself and your reader, you should break it up into two parts, along the following lines.

Solution. Suppose that y and z are functions such that y¢ = z and z¢ = y. Then

y¢¢ = z¢ = y,        z¢¢ = y¢ = z.

But the general solution to the differential equation y¢¢ = y is y = Ae^x+Be^-x, where A and B are constant. Therefore y and z must be of the form

y = Aex+Be-x,       z = Cex+De-x,

for some constants A, B, C and D. [This does not say that all y and z of this form are solutions.]

Conversely, suppose that y and z are of this form. Then

y¢ = Aex-Be-x,       z¢ = Cex-De-x.

Therefore y¢ = z and z¢ = y if A = C and B = -D.

We conclude that the general solution is y = Ae^x+Be ^-x, z = Ae^x-Be ^-x, where A and B are arbitrary constants.

Even in such a very simple problem, you must think carefully about the logical structure of your argument if you are going to avoid mistakes. Two final remarks: (1) you should have spotted in the initial stage of thinking about this problem that the solution would contain two arbitrary constants, and not four; (2) if you think about rigour, then there are a lot of other questions that come up here, such as ``What are the domains of the functions? Are they twice differentiable?'', and so on. A skill that you must learn by practice is that of finding the appropriate level of rigour. There are no absolute rules. What is appropriate depends on the context. Too little rigour, and you will end up saying things that are untrue; too much will be distracting.

Accurate manipulation

For the most part, this is a matter of taking trouble, and giving yourself time to check your work carefully. However, the following tips may help.

(i)

If you have sketched out a solution in rough, do not just make a fair copy of your notes. Having seen how to solve the problem in rough, put you notes aside, and write out the argument from scratch, checking each step carefully as you go. It is very hard to see mistakes when you are simply copying, rather than reworking.

(ii)

If you are prone to making careless slips, treat your work with suspicion, and look for ways of seeing that bits of it must be wrong. In mechanics problems, you can often pick up sign errors by thinking about whether your answer makes good physical sense. If your trajectory for a projectile implies that it falls upwards rather than downwards, then you have made a mistake. If you have found a formula involving parameters for the probability of some event, check that it always gives an answer between 0 and 1. If a particular choice of parameter values gives a negative probability, then you have made a mistake. Test your conclusions against common sense. If you calculation gives 2.31 as the expected number of tosses of a coin needed to get three heads in a row, then you have made a mistake.

(iii)

Wherever possible, check your answer. For example, always substitute your solution back into a differential equation (this will also pick up the sort of problem we met in the example above). If this is too messy, try doing it with special values of the constants. It may be too much of a labour to check that

y = cos-1 æ è Ö   klog(sec2x + xx)+x2   ö ø

satisfies some complicated differential equation, but it may be much easier when the constant k vanishes.

(iv)

Never bring a rough and sloppy piece of work to a tutorial in the hope that your tutor will not spot the mistakes. The chances are that your tutor will not spot the mistakes, in which case nobody learns anything; if your tutor does spot them, then your tutor learns something about you, but you learn little about mathematics. If you are uncertain about something you have written, even after spending a lot of time on it, draw it to your tutor's attention, and ask to go through it carefully together.

Concise writing

Pascal wrote in a letter to a friend ``I have made this letter longer than usual, only because I have not had time to make it shorter''. Long-winded arguments in mathematics are hard to follow. It is well worth spending time and effort to extract the essential steps from your original rough notes and to present your argument in as concise as form as is consistent with clarity. But do not be surprised if you find it hard work, sometimes much harder than spotting the solution in the first place.

There are two points to keep in mind here: first, tutors very rarely criticise undergraduate work for being over-concise. Second, a good test to apply is the `text-book test': ``if my solution were printed as a worked example in a text-book, would I find it helpful and easy to follow''

Diagrams

One other topic needs special mention: the use of diagrams. It is a classic howler, and one that is almost too easy for a tutor to spot, to argue ``from the diagram it is obvious that ... ''. If you try it, you will be left in no doubt that such arguments cannot be rigorous (incidentally, you should in any case avoid phrases like ``it is obvious that'': either it is obvious, in which case you do not have to say that it is obvious, or it is not, in which case you should be more honest). Undergraduates who have been chastened by the scornful reaction to a `diagramatic' proof sometimes draw the false conclusion that diagrams have no place in mathematics. This is quite wrong. First, you should use diagrams freely in the experimentation stage. Second, it is quite legitimate to use diagrams to help your reader to follow an argument, even in the most abstract parts of pure mathematics. The mistake is to make deductions from a diagram. It is similar to the mistake of arguing a general proposition from a single example. In fact, it is bad practice to suppress diagrams which have played an important part in the construction of your solution, particularly if you give away the fact that you have a diagram in mind by using phrases like ``above the x-axis'' or ``in the upper half-plane''. Finally, it is quite legitimate to use a diagram to set up notation. For example, there is nothing wrong with using a diagram in a geometric proof to specify the labelling of points (``where A, B, C are as shown ... ''); or in analytical proof to give the definition of a function that takes different constant values in different ranges (a picture here may be much easier to take in at a glance than a long sequence of expressions with inequalities defining the various parts of the domain). The only test you must satisfy is: is the use of the diagram clear and unambiguous?

Dr N. Woodhouse