Credits: 10 | Convenor: Will Light | Semester: 2 |
Prerequisites: | essential: MC144,MC145,MC146,MC147,MC248,MC241 | |
Assessment: | Continual assessment: 20% | 1.5 hour exam in January: 80% |
Lectures: | 18 | Classes: | none |
Tutorials: | 6 | Private Study: | 51 |
Labs: | none | Seminars: | none |
Project: | none | Other: | none |
Total: | 75 |
Let's start with continuity. Suppose we have a function f mapping some sort
of
objects (which we will call `points') to numbers. What does it mean for the
function to be continuous at the point p? Well, roughly speaking, if q is
close to p, then f(q) must be close to f(p). OK so far? Well, not really.
We've already been much too loose in the way we have formulated the problem.
First off, we described the things on which f acts as `objects', and then
used
the more emotive word `points'. Then we went on to talk about these `objects'
or
`points' being close together. This places some restriction on the sort of
objects allowed. For example, our set of objects could consist of all cheeses
currently lying on the shelves of Sainsbury's supermarket. For a particular
cheese p, f(p) (which remember has to be a real number) could be the weight
of the cheese. Now I think you're going to have a hard job coming up with an
appropriate definition of two cheeses being close together! Note here that the
problem in determining a sensible meaning for `closeness' only occurs in the
input space, or the domain of the function - the set of all cheeses in
Sainsbury's. Closeness in the output space is easy to define isn't it? We would
say that f(p) is close to f(q) if |f(p)-f(q)| is small. But hang on a
minute. Even this idea is not precise enough for a mathematician (or even for
an applied scientist, or person working is some sort of technological
industry).
If I was to discover that the north wall of my house was 2mm higher than the
south wall, I wouldn't exactly be rushing to sue the builder for negligence. On
the other hand, if the garage sets the gap between my spark plug electrodes
with
an error of 2mm, then I would be bleating in their ears that something is wrong
with my car. So the notion of closeness needs to be expressed a lot more
carefully. (The formalisation of the concept of closeness in leads
inexorably via MC146 to the formal definition of continuity using
and
technology.) But around the turn of the century,
mathematicians wanted to get away from the idea of `closeness'. This desire was
bound up with another very sophisticated development in thinking, which has
today become second nature to professional mathematicians. Traditionally,
functions used to be things which mapped numbers to numbers, or more generally,
`points' in
to `points' in
. But it became increasingly common
to
think of functions in a much more abstract way. In the vanguard of this
development was the idea that the so-called `points' could themselves be
functions. For example, the `points' could consist of all real-valued functions
differentiable on the whole real line. Each `point' is now a function from
to
. Thus the function
defined by the rule
f(x)=x2 would be a `point'. Now consider the mapping called
`differentiation'. Applying this mapping to the `point' f gives the new
function
defined by the rule g(x)=2x. Mathematicians
wanted to study such types of mappings or functions acting on
`points' which are themselves very sophisticated objects. A major force in this
movement was the French mathematician Maurice Fréchet (1878-1973). The work
of
Fréchet and his contemporaries has had a profound effect on the way we teach
you! In MC144, you first met the idea of a function. It probably seemed then
pretty weird, because all you were given were two abstract sets A and B and
the rule for getting from A to B, often called f. Nobody encouraged you
to
believe that either A or B were made up of numbers. Instead, we tried to
convince you that A and B could be pretty much anything you liked: cheeses
from Sainsbury's, functions, points in
. However, because we are
sensitive to the difficulty of this abstraction, we often used examples in
which
A and B were both (subsets of)
. OK, let's hope you see that working
with pretty abstract objects is part of modern mathematics, and let's return to
our idea of continuity. My previous arguments have been designed to show that
we
need some idea of `closeness' before we can begin to make sense of continuity.
In the example I have already introduced, where the `points' were themselves
functions from
to
, it is not too difficult to come up with a
sensible notion of when two `points' (functions) are close. MC248 takes this
route, but we are bound for higher things! The fundamental question we shall
answer in the first lecture of the course is
Of course, we cannot in this course be anarchists, and develop our own notion
which fails to coincide with the MC146 notion of continuity. So there are
constraints.
Let's go on now to talk about the other great concept of integration. At the outset, the `A'-level student knows that integration has two meanings - definite integration (which is the area under the curve), and indefinite integration (which is the reverse process to differentiation). Of course, the two are linked through the fundamental theorem of calculus. On a personal note, it's one of the great surprises to me, and something that I find absolutely fascinating, that the area under a curve f between a and b can be evaluated if only you know a function g such that g'=f. One version of the fundamental theorem is then that
The ability to apply taught concepts to new situations.
The ability to write mathematics concisely.
The ability to construct logical arguments.
The ability to carry out elementary manipulations with unions and intersections of sets.
W. Rudin, Real and Complex Analysis, McGraw Hill, 1970.
W. A. Sutherland, Introduction to metric and topological spaces, Oxford : Clarendon Press, 1975.
W. Light, Introduction to Abstract Analysis, Chapman and Hall, 1990.