Math 3B03 – Test 3
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Math 3B03 – Test 3
Problems
1. Consider the ellipsoid E = ((z. y. 3) { R3 | 
+ 
+ 32 = 1}.
(a) (3 points) Let Uz = ((u. w) | 
+ 
< 1} and let Uy = ((
. 
) | 
+ 
2 < 1}. Consider the two surface patches
7z(+) : Uz e R3 . 7z(+)(u. w) = ╱u. w. ′ 1 → 
→ 
\
and
7一y : Uy e R3 . 7一y(
. 
) = ╱
. → ′9 → 
→ 9
2 . 
\．
Compute the transition map between these surface patches.
(b) (3 points) Compute the standard unit normal for the surface patch 7z(+) .
2. Let s be the unit sphere s = ((z. y. 3) | z2 + y2 + 32 = 1}. Consider the map f : s e s given by rotation about the 3-axis counterclockwise by 
. Let p = ( 
. 
. 
). (It may help to recall that f '(┌)zy3'(┐) = '(┌)
1 
'(┐) '(┌)zy3'(┐).)
(a) (4 points) Find the tangent plane T6s and the tangent plane Tf(6)s. (b) (4 points) Let -w = (w1 . w2 . w3 ) { T6s. Find D6f(-w).
3. Give an example of each of the following. Justify your answers.
(a) (3 points) A surface in R3 that is not compact.
(b) (3 points) A smooth map between smooth surfaces that is not a local diffeomor- phism.
4. Consider the surface patch 7 : R2 e R3 given by 7 (u. w) = (u + w2 . u2 → w2 + uw. 3w).
(a) (4 points) Prove that 7 is an allowable surface patch. Be sure to carefully verify all
parts of the definition of allowable surface patch.
(b) (4 points) Compute the first fundamental form of 7.
5. (4 points) Consider the map f from the zy-plane to the unit cylinder C = ((z. y. 3) | z2 + y2 = 1} defined by f (u. w. 0) = (cos(2u). sin(2u). w). Is f a local isometry? Justify your answer.
2022-02-21