Main References. Some general remarks: Location. Mathematics Analysis and Topology Technology and Science 3. block

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1 Location Kroghstræde 3, room Main References A T. Apostol, Mathematical Analysis, Addison-Wesley. BV M. Bökstedt and H. Vosegaard, Notes on point-set topology, electronically available at Some general remarks: We have now completed half of the course, and it seems that many of you do not take full advantage of the resources available to you during the course, namely the possibility of assistance with writing mathematics. The purpose of the exercises is twofold. On the one hand they are meant to illustrating and illuminating the subjects of the lectures, but just as important, they are there to give you an opportunity to formulate and write down the solutions and ask us how to do this. The latter is the reason why we have exercise sessions and do not just give you the solutions, or tell you to do it all at home. Some of you may now find that you are far behind with the exercises, and hence we have made a selection for you of exercises from the previous sessions, which we find particularly important for the two above purposes. Moreover on Tuesday 30.11, we will present solutions to some of the exercises. Please send an if you have suggestions for which problems we should present solutions to. It does not have to be some of the exercises singled out below. From block 1: Metric Spaces: BV and Topological spaces: BV and Convergence in metric and topological spaces: BV 2.5.5, A 4.7 p.96 Uniform Convergence: There is one exercise in this session about Newton iteration (see the plan for 1. block). This one is a bit technical, so first make sure you understand the statement of the fixed-point theorem. Then try to follow the steps outlined in the problem and convince yourself that the theorem can be applied in the last step. From block 2: Connected Spaces and Compact Spaces: BV 6.1.6, A 4.38, BV 3.2.8

2 Functions on compact spaces: There is one exercise in this session about Dini s theorem (see the plan for 2. block). Try to follow the steps outlined in the problem. Stone Weierstrass: There is one exercise in this session, namely A 4.67 plus understanding the solution of This is tough, if you are a bit lost at this point; in that case, you should perhaps have a closer look at the sup-norm, for instance by doing exercise 4.66a. The hard part is the triangle inequality. Differentiability. Mon, 29.11, 9 12 Startup Discussion of topics related to the 2. block: your questions related to the lectures problems with exercises Lecture Aims and Content After having emphasized continuity, we now get to differentiability, a concept that cannot be defined on general topological spaces, but only on open subsets of Euclidean space (or spaces built out of those): From the basic year and also from math in high school, you already know about many differentiable functions and you can certainly calculate the derivative or the partial derivatives of many functions. You have seen a definition of differentiability of a function f : R R, and you have perhaps been working with the Jacobi matrix for functions f : R n R m. For a function like f(x) = sin(x), everything is clear, but what about { x sin(1/x) for x > 0 f(x) = 0 for x 0 For this function your definition of differentiability from high school will tell you what to do, but what about functions of more than one variable? The function { xy 2 for (x, y) (0, 0) f(x, y) = x 2 +y 4 0 for (x, y) = (0, 0)

3 has derivatives in all directions in particular it has all its partial derivatives, but it is not even continuous at (0, 0)! Differentiable functions ought to be continuous, at least, and thus the right definition requires more than the existence of partial and directional derivatives. The aim of this part of the course is to give the right definition of differentiability. We will emphasize an interpretation via approximations by linear maps or linear tangent spaces to the graph of the function. References A Chapter and Exercises 1. In Apostol p. 345, the function f(x, y) = { xy 2 x 2 +y 4 for (x, y) (0, 0) 0 for (x, y) = (0, 0) is studied. Prove, or convince yourself, that you understand Apostols proof, that this function has all directional derivatives, and that it is not continuous at (0, 0). A 12.4, 12.5, Mathematical problems/subjects from the thesis work of the participants. Presentations by the participants Group 1 The members of this group are Jan Jakob Jessen, Jacob Illum, Steffen Præstholm, Jens Dalsgaard, Jens A. Hansen, Morten Holm Larsen, Brian Solberg and Lars Alminde. Group 2 The members of this group are Jesper Michael Kristensen, Anders Brødløs Olsen, Christian Rom, Michael Nielsen, Malek Boussif, Mangesh A. Ingale and Søren Skovgaard Christensen

4 Group 3 Nicola Marchetti, Daniel V.P. Fiqueiredo, Huan C. Nguyen, Thomas Nielsen, Menghua Zhao, AlFredo Chavez Plascencia, Zhuang Wu, Muhammad Imadur Rahman and Suvra Sehkar Das. After the presentations, comments and discussion, there will be time to work on the exercises that you did not get to during the last two sessions. See the suggestions above, if you feel too much behind with the exercises. Complex differentiability. Higher order derivatives. Taylor expansion. Tue, , 9 12 Startup Questions? Problems? concerning Monday s lectures and/or exercise session. Lecture Aims and Content Complex differentiable functions enjoy especially nice properties. In particular, the real and imaginary parts have to satisfy the Cauchy-Riemann differential equations, and this condition is also sufficient! Can one ensure differentiability of a function of several variables by inspections of its partial derivatives? Yes, there is a sufficient (but not necessary) condition, using the continuity of the partial derivatives. An important tool in the proof is the mean value theorem in one and in several variables, which is interesting for other purposes, as well. The partial derivatives of a function may be differentiable and thus give rise to higher order derivatives. Under which conditions is the result independent of the order of differentiation? And how can one use the 1st and higher order derivatives for approximation purposes? The answer is given by Taylor 1 s formula (best approximation by a multivariate polynomial of a given degree) and an estimation of the remainder term. Real functions on an open set U R n that are k-times differentiable with continuous derivatives form a specific subspace C k (U,R) of the set of continuous functions on U. The sup-norm is replaced by a finer norm, that also takes care of the higher order derivatives and thus it requires more for a sequence to converge. 1 history/mathematicians/taylor.html

5 References A Sections 5.16, , 12.6, 9.10 Exercises 1. Show: There is no C 2 -function f : R 3 R such that f(x, y, z) = (y 2 z, 2xyz, xy 2 + y). 2. A (interpret the result by a drawing); A 5.36: a)-d). Presentations by the participants plus solutions to some exercises Tue, , 12:30 15 Group 4 The members of this group are Jens Peter Hedelund Larsen, Anders Risom Korsgaard, Daniel K. Jensen, Lars Chr. Terndrup Overgaard, Hans Laurberg, Jesper Højvang, and Anders Jørgensen. Group 5 Course participants not in Groups 1 4. We did not get a list of group members, but you still have to give the presentation! Exercises from earlier We give solutions to some of the exercises. Please tell us in advance which ones you would like us to present. Plan for the 4. block Date: , 2004 Existence and uniqueness of solutions to ordinary differential equations. The inverse function theorem. The implicit function theorem.

6 Implicitely defined functions and their derivatives. The Lagrange and Kuhn Tucker methods for optimization. Course Evaluation. Discussion of demands for other math courses at the Ph.D.-level.