# Problem Sheet D

- 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.

Comments and questions?