17/3, 24/3 and 31/3 I ¡­

17/3, 24/3 and 31/3 I covered the material in 4.1 (I think that 4.1 contains few details soo I lectured the multilinear algebra from Spivak instead), 4.2, 4.3 (here (as examples) I proved that the spheres were orientable and that projective spaces are orientable if and only if the dimension is odd) and 5.1 and 5.2. Today I talked about manifolds with boundaries (5.3), integration of forms on compact orientable manifolds (4.4). Then I formulated and (tried to) proved Stokes theorem (5.4) (in the end of the proof I unfortunately ran out of time so the calculations became rather sketchy, but since the proof in the textbook is rather readble I will not repeat the proof.) There will (of course) be no lecture next friday, but 21/4 I will do some applications of Stokes Theorem and I will then defer 5.5 and start on chapter 6