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

a. Describe [math]\displaystyle{ M }[/math].

b. Show that [math]\displaystyle{ M }[/math] is a manifold.

c. Find the boundary [math]\displaystyle{ \partial M }[/math] of [math]\displaystyle{ M }[/math].

d. Find the volume [math]\displaystyle{ V(M) }[/math] of [math]\displaystyle{ M }[/math].

e. Find [math]\displaystyle{ \int_MzdV }[/math] (where [math]\displaystyle{ z }[/math] denotes the third coordinate of [math]\displaystyle{ {\mathbb R}^3 }[/math]).

Student Solutions

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