**Exercise 1. **Show that the sphere

is a -dimensional smooth manifold.

**Exercise 2. **We consider the following two subsets of the plane

and

.

Are and smooth manifolds ? Of course, justify your answer with a proof.

**Exercise 3. **Let be a n-dimensional smooth manifold. Let . A smooth curve on is a smooth map . We denote by the set of smooth curves on such that . Show that the set

is a linear space isomorphic to .

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