• Problem 1 shows that local operators can be reconstructed from their local behaviour. (In sightly fancier language: local operators are sheaf morphisms).
• Problem 2 explains the relationship between $\varphi$-related vector fields and the pullback operator $\varphi^*$ on tensors of type $(0,k)$.
• Problem 3 is linear algebra.
• Problem 4 shows that the interior product is a graded derivation of degree $-1$.
• Problem 5 proves that the induced orientation is well defined.
• Problem 6 gives examples of manifolds that are (not) orientable.
• Problem 7 explains why our preferred half-space is “better” than the standard upper half-space.
• Problem 8 is an introduction to the best area of mathematics.
• Problem 9 shows that the product of two smooth manifolds with boundary can fail to be a smooth manifold with boundary.
• Problem 10 is a joke. Seriously. Don’t do it.