Exercises for Monday February 11th: ¡­

Exercises for Monday February 11th: Chapter 2, problems 13, 21, 33, 34; MA 252 exam 2002, problem 1.

Exam problems from 1997 to 2002 can be found under "Oppgaver" in the left hand column on the course and semester pages. Translated to English, problem 1 from 2002 reads: Let N and P be smooth manifolds of dimension n and p, respectively, and let f : N --> P be a smooth map. Let psi = (psi^1, ..., psi^p) be a chart in P centered at a point f(a) (meaning that psi(f(a)) = 0), and define Q (a subset of P) by the equations psi^1 = 0, ..., psi^l = 0. (a) Explain why Q is a submanifold of P. What is its dimension? (b) Prove that if the map psi' o f with components psi^1 o f, ...., psi^l o f has rank l in the point a, then f^{-1}Q is a submanifold of N in a neighborhood of a. What is its dimension?