Department of Mathematics & Computer Science | ||||
Credits: 10 | Convenor: Dr Christiane Tretter | Semester: 2 |
Prerequisites: | essential: MC144,MC145,MC146,MC147,MC248,MC241 | |
Assessment: | Continual assessment: 20% | hour exam in January: 80% |
Lectures: | 18 | Problem Classes: | 5 |
Tutorials: | none | Private Study: | 52 |
Labs: | none | Seminars: | none |
Project: | none | Other: | none |
Surgeries: | none | Total: | 75 |
Let's start with continuity. Suppose we have a function 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 ? Well, roughly speaking, if is close to , then must be close to . 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 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 , (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 is close to if 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 would be a `point'. Now consider the mapping called `differentiation'. Applying this mapping to the `point' gives the new function defined by the rule . 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 and and the rule for getting from to , often called . Nobody encouraged you to believe that either or were made up of numbers. Instead, we tried to convince you that and 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 and 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 between and can be
evaluated if only you know a function such that . One version of the
fundamental theorem is then that
At the end of the course, students should
The ability to present and understand arguments in an abstract setting.
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.
Author: S. J. Ambler, tel: +44 (0)116 252 3884
Last updated: 2001-09-20
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