This Problem Sheet is based on Lectures 6 and 7.

• Problem 1 shows that any tangent vector can be realised by a vector field.
• Problems 2 and 3 ask you to prove properties of the Lie bracket that I skipped in class today.
• Problem 4 asks you to prove another (somewhat mysterious looking) identity about the Lie bracket. This identity will make more sense by the end of next week.
• Problem 5 generalises the pushfoward construction to smooth maps that are not diffeomorphisms.
• Problem 6 looks at how vector fields behave with respect to submanifolds.