• Problem 1 shows that the manifold definition of a vector field concides with the Euclidean definition.
• Problem 2 shows that any tangent vector can be realised by a vector field.
• Problems 3 and 4 ask you to prove properties of the Lie bracket.
• Problem 5 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 6 generalises the pushfoward construction to smooth maps that are not diffeomorphisms.
• Problem 7 looks at how vector fields behave with respect to submanifolds.
• Problem 8 asks you to remove the compactness hypothesis from the version of the Whitney Embedding Theorem we proved.
• Problem 9 asks you to show that the (strong) Whitney Embedding Theorem is sharp.