1617-257/homework 13 assignment solutions

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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_MzdV (where z denotes the third coordinate of {\mathbb R}^3).

Student Solutions

Student 1, LaTeX PDF

Student 2