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{SECT 0 {PARA 257 "" 0 "" {TEXT -1 9 "Math 4530" }}{PARA 256 "" 0 "" 
{TEXT -1 39 "Solutions to hw set 4 - due February 11" }}{PARA 0 "" 0 "
" {TEXT -1 40 "I and II from page 7 of February 4 notes" }}{PARA 0 "" 
0 "" {TEXT -1 59 "Chapter 2: 1.12, 1.20, 1.21, 1.22, 2.7, 2.9, 5.6, 5.
7, 5.8 " }}{PARA 0 "" 0 "" {TEXT -1 129 "This file contains some of th
e solutions to the chapter 2 problems.  Problems 1.22, I, II, 2.7, 2.9
 are done separately, by hand." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 12 "with(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 123 "1.12:  The catenoid is the surface of revolution obtai
ned by revolving y=cosh(x) about the x-axis.  So a patch is given by \+
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "X:=(u,v)->[u,cosh(u)*cos
(v),cosh(u)*sin(v)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XGf*6$%\"u
G%\"vG6\"6$%)operatorG%&arrowGF)7%9$*&-%%coshG6#F.\"\"\"-%$cosG6#9%F3*
&F0F3-%$sinGF6F3F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 33 "the patch will
 be 1-1 as long as " }{XPPEDIT 18 0 "0 < v;" "6#2\"\"!%\"vG" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "` ` < 2*Pi;" "6#2%\"~G*&\"\"#\"\"\"%#PiGF'" }
{TEXT -1 161 ".  The general discussion of what makes a surface of rev
olution patch in general is contained on the final page of the hand-wr
itten solutions.  Here's a picture:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 93 "plot3d(X(u,v),u=-1..1,v=0..2*Pi,axes=boxed,color=black,\nstyle=w
ireframe,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT3D 276 276 
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{PARA 0 "" 0 "" {TEXT -1 56 "1.20) The Monge parameterization yields a
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>[u,v,u*v];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"ZGf*6$%\"uG%\"vG6\"
6$%)operatorG%&arrowGF)7%9$9%*&F/\"\"\"F.F1F)F)F)" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 330 "We can write Z(u,v) as u * <1,0,v> + <0,v,0> or a
s v*<0,1,u > + <u,0,0>.  These are different rulings because the direc
tion vectors are linearly independent (look at the first two entries.)
 Since Maple shows the coordinate lines when we plot a parametric surf
ace, you can see the double ruling by using the parameterization above
!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "plot3d(Z(u,v),u=-1..1,
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0000000002-%&COLORG6&%$RGBG$\"\"!F/F.F.-%*AXESSTYLEG6#%$BOXG-%(SCALING
G6#%,CONSTRAINEDG-%&STYLEG6#%%LINEG" 1 6 0 1 10 0 2 1 1 2 1 1.000000 
76.000000 69.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 72 "1.2
1)  The helicoid parameterization already shows it is a ruled surface
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "H:=(u,v)->[a*u*cos(v),a*
u*sin(v),b*v];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HGf*6$%\"uG%\"vG
6\"6$%)operatorG%&arrowGF)7%*(%\"aG\"\"\"9$F0-%$cosG6#9%F0*(F/F0F1F0-%
$sinGF4F0*&%\"bGF0F5F0F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 89 "which we
 can write as H(u,v) = u* < a*cos(v),a*sin(v),0> + <0,0,v*b>.  Here is
 a picture:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "a:=1:b:=1:\n
plot3d(H(u,v),u=-2..2,v=0..2*Pi,scaling=constrained,color=black,\naxes
=boxed,style=wireframe);" }}{PARA 13 "" 1 "" {GLPLOT3D 299 264 264 
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1 10 0 2 1 1 2 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 
0 "" 0 "" {TEXT -1 102 "1.22, 2.1.6, 2.1.7: These are done by hand, se
e the hand work file.  Related pictures are shown below." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 352 "5.6) For fixed u \+
the ruling below traces out a horizontal line with angle u (and u+Pi) \+
from the x-direction.  As u varies the line moves up and down, with ve
rtical coordinate sin(ku).  Thus, the larger the k value, the more qui
ckly, or often, the ruling line oscillates.  As long as k is an intege
r the surface will close up as u varies from 0 to 2*Pi." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Rul:=[v*cos(u),v*sin(u),sin(k*u)]:
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