[!]. This exercise is used in Section 15.3. Translation on Rn by an a ERn is defined by O'(x) = x +a for x ERn. Dilation on Rn by a J > 0 is defined by O'(x) = Jx for x ERn.

(a) Prove that if A is an oriented atlas of a manifold M and 0' is a translation or a dilation, then 13 = {(V, 0' 0 h) : (V, h) E A} is an atlas of M that is orientation compatible with A.

(b) Let A be an orientation of a manifold M and let x E M be an interior point.

Prove that there is a chart (V, h) at x such that h(V) = BI (0) and h(x) = O.