17 VECTOR CALCULUS

CHAPTER 17
VECTOR CALCULUS
GROUP WORK 3: Self-Intersection of Surfaces
This one is a challenge for the students. For Problem 2, it will probably be necessary to give them a hint. If
f (u 1 ,v 1 ) = f (u 2 ,v 2 ), then we know that u 1 =±u 2 and v 1 =±v 2 in order that the first two coordinates be
the same. Given this fact, we have to find the relationship between the pairs to make u 1 + 2v 1 = u 2 + 2v 2 .
There are three nontrivial cases to check:
u 1 = u 2 ,v 1 =−v 2 u 1 =−u 2 ,v 1 = v 2 u 1 = u 2 ,v 1 =−v 2
When all is said and done, the only nontrivial result occurs when u 1 =−u 2 and v 1 =−v 2 , and that result is
u =−2v. So the self-intersecting set occurs when u =−2v.
Answers:
1. u 1 =±u 2 , v 1 =±v 2 , u 1 + 2v 1 = u 2 + 2v 2 2. v =− 1 2 u
)
3.
(u, ± 1 2 u, 0 , a pair of intersecting lines.
GROUP WORK 4: Setting Up Surface Integrals
All methods give the area as 24π.
LABORATORY PROJECT: A Difficult Plot
[
1. Have the students plot the surface described by x 2 + y 2 = z 1 +
(
tan −1 x ) ] 2
. Allow them to notice
y
that this is very hard to do, even with a sophisticated graphing package like Maple or Mathematica. They
may fail to get a picture of this surface that makes sense. This is okay; let them fail this time.
〈
s 2 〉
2. Now have them draw the parametric surface r (s, t) = s cos t, s sin t,
1 + t 2 , − π 2 ≤ t ≤ π 2
. This one
should be straightforward. They should get a picture like this:
3. Now have them discover, using algebra, that the two surfaces are the same. When they do this, discuss
how important it is to be able to parametrize surfaces.
4. Let them investigate the surface for t ∈ R. Ask what the relationship is between the two intersecting
surfaces which result.
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