1617-257/homework 13 assignment solutions

Doing

Solve all the problems in sections 25-26, but submit only your solutions of problems 4 and 8 in section 25 and problems 5 and 6 in section 26. In addition, solve the following problem, though submit only your solutions of parts d and e:

Problem A. Let $\alpha \colon \{(u,v)\in {\mathbb {R} }^{2}\colon u^{2}+v^{2}\leq 1\}\to {\mathbb {R} }^{3}$ be given by $\alpha (u,v)=\left(u-v,\,u+v,\,2(u^{2}+v^{2})\right)$. Let $M$ be the image of $\alpha$.

a. Describe $M$.

b. Show that $M$ is a manifold.

c. Find the boundary $\partial M$ of $M$.

d. Find the volume $V(M)$ of $M$.

e. Find $\int _{M}zdV$ (where $z$ denotes the third coordinate of ${\mathbb {R} }^{3}$).