Below you will find solutions ¡­

Below you will find solutions of the problem sets from the exam of MA 252

solutions from the exam 2000 are

here

solutions from the exam 2001 are

here

and solutions from the exam 2002 are

here

The syllabus is the chapters 1-5 and 7-8 in Spivak with the following modifications:

Given an oriented manifold with boundary, you need to know the definition of the induced orientation of the boundary. This definition is given in problem 16 of chapter 3 which thus is included in the syllabus (the definition of induced orientation is repeated in chapter 8 on p. 260).

In chapter 5 I have not lectured proposition 15 and this is not part of the syllabus.

Chapter 6, integral manifolds is not part of the syllabus.

Proposition 14, chapter 7 (the Frobenius integrability theorem)and the following corollary (Corollary 15) is not part of the syllabus.

I have not said anything about volume elements and therefore the stuff about this (page 258 in chapter 8) is not part of the syllabus.

Finally all candidates for the exam in MAT 4520 and MAT 9520 have to meet at room B62 (N.H.A-building) at 9.00 AM June 12.. Then the time for the examination for each individual candidate will be announced. Also, if we need more than one day in total for the examination, some of the candidates will be examined June 13. These (possible) candidates will be informed on June 12. 9.00 AM (so everybody MUST meet at 9.00 June 12).