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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 273 0 "" }{TEXT 
274 23 "Calculus IV with Maple\n" }{TEXT 275 32 "Copyright 2002, Dr. J
ack Wagner\n" }{TEXT -1 30 "j.wagner@intelligentsearch.com" }}}{EXCHG 
{PARA 4 "" 0 "" {TEXT -1 28 "Lesson 10: Surface Integrals" }}}{EXCHG 
{PARA 5 "" 0 "" {TEXT -1 8 "Topics: " }{TEXT 276 64 "Integration over \+
a surface.  Integrating over the Moebius band. " }}}{EXCHG {PARA 5 "" 
0 "" {TEXT -1 27 "Maple commands introduced: " }{TEXT 256 1 " " }
{TEXT 257 20 "stackmatrix, augment" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 471 "In the same way that a vector fi
eld may be integrated over a curve, it may be integrated over a surfac
e.  To each parallelogram that forms an element of the area of the sur
face we assign the normal component of the vector field at some interi
or point.  As the partition of the surface is refined  the sum of the \+
products of the area of the parallelograms and the normal component of
 the vector field is the integral of the vector field over the surface
, usually written, " }{XPPEDIT 18 0 "Int(F,S);" "6#-%$IntG6$%\"FG%\"SG
" }{TEXT -1 31 "  , where S is the surface, or " }{XPPEDIT 18 0 "Int(F
[U].n,A);" "6#-%$IntG6$-%\".G6$&%\"FG6#%\"UG%\"nG%\"AG" }{TEXT -1 382 
", where dA is understood to represent the \"element of area\", n is t
he unit normal and U is the  parameterization.  This integral is the f
lux of F through the surface S. Because the normal to the surface play
s a key role it is important that we work only with surfaces over whic
h it is possible to consistently define a unit normal.  That is, there
 must be a vector valued function, " }{TEXT 264 1 "n" }{TEXT -1 88 "(p
), that assigns a unique unit normal to each point, p. Such a surface \+
is orientable. \n" }}{PARA 0 "" 0 "" {TEXT -1 112 "The archetypical no
n-orientable surface is the Moebius band, which may be parameterized a
nd plotted as follows.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 
"restart: with(plots):with(LinearAlgebra): with(VectorCalculus):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "F := <(2-v*sin(u/2))*sin(u),
 (2-v*sin(u/2))*cos(u), v*cos(u/2)>;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 60 "plot3d(F, u=-Pi..Pi, v=-1..1, axes=framed, style=wire
frame);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 150 "It is not orientable \+
because at each point the \"function\"  n(v, u),  creates two unit nor
mals  each pointing in the opposite direction. For a surface " }
{XPPEDIT 18 0 "S(u,v) = [x(u,v), y(u,v), z(u,v)];" "6#/-%\"SG6$%\"uG%
\"vG7%-%\"xG6$F'F(-%\"yG6$F'F(-%\"zG6$F'F(" }{TEXT -1 9 "  define " }
{XPPEDIT 18 0 "n(u,v) = S[u]*X*S[v]/abs(S[u]*X*S[v]);" "6#/-%\"nG6$%\"
uG%\"vG**&%\"SG6#F'\"\"\"%\"XGF-&F+6#F(F--%$absG6#*(&F+6#F'F-F.F-&F+6#
F(F-!\"\"" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "If \+
we plot " }{TEXT 265 1 "n" }{TEXT -1 117 "(u, v) over a surface, we ex
pect to get a copy of the original surface one unit away.  For the Moe
bius band, however:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dF[u
] := diff(F, u);\011" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dF[
v] := diff(F, v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "n := d
F[v] &x dF[u]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "N := Norm
alize(n, 2, inplace):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "P
1 := plot3d(F+N, u=-2*Pi..2*Pi, v=\n\011-1..1, axes=framed, style=wire
frame, color=blue, grid=[40,40]):\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "P2 := plot3d(F, u=-Pi..Pi, v=-1..1, axes=framed, colo
r=pink):\n\011" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "display(
\{P1, P2\}, scaling=constrained, orientation=[-135,75]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 378 "At each point on the surface u(v, u) has
 created two unit normals and therefore it is not well defined on this
 surface.  A bit further on we will look at how non-orientability inte
rferes with integration of a vector field over a surface.  We are also
 going to confine ourselves to simple surfaces, that is, surfaces whic
h do not contain singularities such as self intersections." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Let F be the vect
or field. The unit normal to the surface S, known by " }{XPPEDIT 18 0 
"S(u,v) = [x(u,v), y(u,v), z(u,v)];" "6#/-%\"SG6$%\"uG%\"vG7%-%\"xG6$F
'F(-%\"yG6$F'F(-%\"zG6$F'F(" }{TEXT -1 6 ", is  " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }{XPPEDIT 18 0 "S[u]*X*S[v]/abs(S[u]*X*S[v]);" "6#**&%\"
SG6#%\"uG\"\"\"%\"XGF(&F%6#%\"vGF(-%$absG6#*(&F%6#F'F(F)F(&F%6#F,F(!\"
\"" }{TEXT -1 24 " . We already know that " }{XPPEDIT 18 0 "dA = abs(S
[u]*X*S[v]);" "6#/%#dAG-%$absG6#*(&%\"SG6#%\"uG\"\"\"%\"XGF-&F*6#%\"vG
F-" }{TEXT -1 1 " " }{TEXT 266 4 "dudv" }{TEXT -1 15 " .  Therefore, \+
" }{XPPEDIT 18 0 "Int(F*n,A);" "6#-%$IntG6$*&%\"FG\"\"\"%\"nGF(%\"AG" 
}{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(Int(F*S[u]*X*S[v]*abs(S[u]*X*S[v
])/abs(S[u]*X*S[v]),u),v);" "6#-%$IntG6$-F$6$*.%\"FG\"\"\"&%\"SG6#%\"u
GF*%\"XGF*&F,6#%\"vGF*-%$absG6#*(&F,6#F.F*F/F*&F,6#F2F*F*-F46#*(&F,6#F
.F*F/F*&F,6#F2F*!\"\"F.F2" }{TEXT -1 2 " =" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "Int(Int(F*S[u]*X*S[v],u),v);" "6#-%$IntG6$-F$6$**%\"FG
\"\"\"&%\"SG6#%\"uGF*%\"XGF*&F,6#%\"vGF*F.F2" }{TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 12 "E
xample 10.1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 17 "Find the flux of " }{XPPEDIT 18 0 "F(x,y,z) = [1/(x^2),
 1/(y^2), 1/(z^2)];" "6#/-%\"FG6%%\"xG%\"yG%\"zG7%*&\"\"\"F,*$F'\"\"#!
\"\"*&F,F,*$F(F.F/*&F,F,*$F)F.F/" }{TEXT -1 40 "  through the surface \+
of the paraboloid " }{XPPEDIT 18 0 "z = x^2+y^2;" "6#/%\"zG,&*$%\"xG\"
\"#\"\"\"*$%\"yGF(F)" }{TEXT -1 2 "  " }{XPPEDIT 18 0 "1 <= x;" "6#1\"
\"\"%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y <= 3;" "6#1%\"yG\"\"$" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: wi
th(plots): with(LinearAlgebra): with(VectorCalculus):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "F := <1/x^2, 1/y^2, 1/z^2>;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "No
w select a parametrizarion for S. \{x=u, y=v\}" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "S := <u, v, u^2+v^2>;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "F := eval(F, \{x=S[1], y=S[2], z=S[3]\});" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "P1 := plot3d(S, u=1..3, v=1.
.3, color=aquamarine, style=wireframe, axes=framed):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 89 "P2 := fieldplot3d(F, u=1..3, v=1..3, z=1.
.18, arrows=THICK, grid=[5, 5, 5], color=black):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 16 "display(P1, P2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 15 "Now we compute." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 20 "dS[u] := diff(S, u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "dS[v] := diff(S, v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 
"n := dS[v] &x dS[u];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Note tha
t we have selected the \"outward\" pointing normal." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "FdA := F.n;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "The flux over the surface is" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "int(int(FdA, u=1..3), v=1..3);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 15 "Or numerically," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "In t
he standard basis, \{" }{TEXT 267 7 "i, j, k" }{TEXT -1 3 "\}, " }
{XPPEDIT 18 0 "S[u]*X*S[v] = Det(matrix([[i, j, k], [x[u], y[u], z[u]]
, [x[v], y[v], z[v]]]));" "6#/*(&%\"SG6#%\"uG\"\"\"%\"XGF)&F&6#%\"vGF)
-%$DetG6#-%'matrixG6#7%7%%\"iG%\"jG%\"kG7%&%\"xG6#F(&%\"yG6#F(&%\"zG6#
F(7%&F;6#F-&F>6#F-&FA6#F-" }{TEXT -1 74 " where the subscripts indicat
e the derivative.  This can be evaluated as  " }{TEXT 268 2 "i " }
{XPPEDIT 18 0 "Det(matrix([[y[u], z[u]], [y[v], z[v]]]));" "6#-%$DetG6
#-%'matrixG6#7$7$&%\"yG6#%\"uG&%\"zG6#F.7$&F,6#%\"vG&F06#F5" }{TEXT 
-1 3 " - " }{TEXT 269 2 "j " }{TEXT -1 0 "" }{XPPEDIT 18 0 "Det(matrix
([[x[u], z[u]], [x[v], z[v]]]));" "6#-%$DetG6#-%'matrixG6#7$7$&%\"xG6#
%\"uG&%\"zG6#F.7$&F,6#%\"vG&F06#F5" }{TEXT -1 3 " + " }{TEXT 270 1 "k
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Det(matrix([[x[u], y[u]], [x[v], x[v
]]]));" "6#-%$DetG6#-%'matrixG6#7$7$&%\"xG6#%\"uG&%\"yG6#F.7$&F,6#%\"v
G&F,6#F5" }{TEXT -1 15 " . Therefore,  " }{TEXT 271 6 "F dA =" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "Det(matrix([[F[1], F[2],
 F[3]], [x[u], y[u], z[u]], [x[v], y[v], z[v]]]));" "6#-%$DetG6#-%'mat
rixG6#7%7%&%\"FG6#\"\"\"&F,6#\"\"#&F,6#\"\"$7%&%\"xG6#%\"uG&%\"yG6#F9&
%\"zG6#F97%&F76#%\"vG&F;6#FC&F>6#FC" }{TEXT -1 13 " which yields" }
{XPPEDIT 18 0 "Int(F*n,A);" "6#-%$IntG6$*&%\"FG\"\"\"%\"nGF(%\"AG" }
{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(Int(Det(matrix([[F[1], F[2], F[3]
], [x[u], y[u], z[u]], [x[v], y[v], z[v]]])),u),v);" "6#-%$IntG6$-F$6$
-%$DetG6#-%'matrixG6#7%7%&%\"FG6#\"\"\"&F16#\"\"#&F16#\"\"$7%&%\"xG6#%
\"uG&%\"yG6#F>&%\"zG6#F>7%&F<6#%\"vG&F@6#FH&FC6#FHF>FH" }{TEXT -1 1 " \+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 
12 "Example 10.2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 11 "Integrate  " }{XPPEDIT 18 0 "F = [x^2, y^2, z^2];
" "6#/%\"FG7%*$%\"xG\"\"#*$%\"yGF(*$%\"zGF(" }{TEXT -1 30 " over the s
urface defined by  " }{XPPEDIT 18 0 "z = sin(x+y);" "6#/%\"zG-%$sinG6#
,&%\"xG\"\"\"%\"yGF*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "0 <= x;" "6#1\"
\"!%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y <= Pi;" "6#1%\"yG%#PiG" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "restart: wi
th(plots): with(LinearAlgebra): with(VectorCalculus):with(linalg):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "F := <x^2, y^2, z^2>;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "S := <u, v, sin(u+v)>;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "F := eval(F, \{x=S[1], y=S[2
], z=S[3]\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dS[u] := d
iff(S, u);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dS[v] := diff
(S, v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "FdA := <<F>|<dS[
u]>|<dS[v]>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Int(Int(De
terminant(FdA), u=0..Pi), v=0..Pi);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 158 "The
 determinant formulation leads to the following interpretation.  Consi
der the first term in the expansion of the determinant in minors of th
e first row.   " }{XPPEDIT 18 0 "F[1]*Det(matrix([[y[u], z[u]], [x[v],
 z[v]]]));" "6#*&&%\"FG6#\"\"\"F'-%$DetG6#-%'matrixG6#7$7$&%\"yG6#%\"u
G&%\"zG6#F37$&%\"xG6#%\"vG&F56#F;F'" }{TEXT -1 5 "  -  " }{XPPEDIT 18 
0 "F[2]*Det(matrix([[x[u], z[u]], [x[v], z[v]]]));" "6#*&&%\"FG6#\"\"#
\"\"\"-%$DetG6#-%'matrixG6#7$7$&%\"xG6#%\"uG&%\"zG6#F47$&F26#%\"vG&F66
#F;F(" }{TEXT -1 8 "   +    " }{XPPEDIT 18 0 "F[3]*Det(matrix([[x[u], \+
y[u]], [x[v], x[v]]]));" "6#*&&%\"FG6#\"\"$\"\"\"-%$DetG6#-%'matrixG6#
7$7$&%\"xG6#%\"uG&%\"yG6#F47$&F26#%\"vG&F26#F;F(" }{TEXT -1 5 " .   " 
}{XPPEDIT 18 0 "F[1];" "6#&%\"FG6#\"\"\"" }{TEXT -1 44 "  is the compo
nent of F in the x-y plane,   " }{XPPEDIT 18 0 "Det(matrix([[y[u], z[u
]], [y[v], z[v]]])) = y[u]*z[v]-z[u]*y[v];" "6#/-%$DetG6#-%'matrixG6#7
$7$&%\"yG6#%\"uG&%\"zG6#F/7$&F-6#%\"vG&F16#F6,&*&&F-6#F/\"\"\"&F16#F6F
=F=*&&F16#F/F=&F-6#F6F=!\"\"" }{TEXT -1 447 ". is the area of the proj
ection, onto the x-y plane,  of the element of area on the surface of \+
S.  Similarly for the other two components, except that the x-z plane \+
projection has a negative sign.  (Rotate the basis vector of the x axi
s into the z axis basis vector and, by the right hand rule, you will f
ind that the resulting vector has an opposite sense to that of the y a
xis.)  The sum of these projections is precisely the determinant given
. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 56 "From the basic definition of the surface integral, i.e. " }
{XPPEDIT 18 0 "Int(F.n,A);" "6#-%$IntG6$-%\".G6$%\"FG%\"nG%\"AG" }
{TEXT -1 17 " we also obtain \n" }{XPPEDIT 18 0 "Int(F.n,A) = Int(Int(
F.n*abs(S[u]*X*S[v]),u),v);" "6#/-%$IntG6$-%\".G6$%\"FG%\"nG%\"AG-F%6$
-F%6$*&-F(6$F*F+\"\"\"-%$absG6#*(&%\"SG6#%\"uGF4%\"XGF4&F:6#%\"vGF4F4F
<F@" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(Int(F.n*sqrt(E*G-F^2),u),v)
;" "6#-%$IntG6$-F$6$*&-%\".G6$%\"FG%\"nG\"\"\"-%%sqrtG6#,&*&%\"EGF.%\"
GGF.F.*$F,\"\"#!\"\"F.%\"uG%\"vG" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 12 "Example 10.3" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "C
ompute the integral of " }{XPPEDIT 18 0 "F(x,y) = [x, 2^y, 2^(-y)];" "
6#/-%\"FG6$%\"xG%\"yG7%F')\"\"#F()F+,$F(!\"\"" }{TEXT -1 19 " over the
 surface, " }{XPPEDIT 18 0 "S = [x, y, -2*x-3*y];" "6#/%\"SG7%%\"xG%\"
yG,&*&\"\"#\"\"\"F&F+!\"\"*&\"\"$F+F'F+F," }{TEXT -1 5 " ,   " }
{XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 
0 "y <= 1;" "6#1%\"yG\"\"\"" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 62 "restart:with(plots):with(LinearAlgebra): with(Vect
orCalculus):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f := <x, 2^
y, 2^(-y)>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "S := <x, y, \+
-2*x-3*y>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "P1 := plot3d(
S, x=0..1, y=0..1, color=blue, style=wireframe):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 86 "P2 := fieldplot3d(f, x=0..1, y=0..1, z=-6..1,
 color=red, grid=[5, 5, 5], thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "display(P1, P2, axes=normal );" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 123 "For a plane whose equation is 3x-2y-z=0, we know \+
that  [3, -2, -1] is the principal part of a normal vector at every po
int." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "N := <3, 2, 1>;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "n := Normalize(N, 2, inplac
e); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s[x] := diff(S, x);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "s[y] := diff(S, y);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "E := s[x].s[x];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "F := s[x].s[y];" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 15 "G := s[y].s[y];" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "sqrt(E*G-F^2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "FdA := f.n*sqrt(E*G-F^2);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "Int(Int(FdA, x=0..1), y=0..1);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Now
 let's look at what happens when we try to integrate a function over t
he surface of a Moebius band. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 261 12 "Example 10.4" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Calculate the int
egral of " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "S = [(2-v*sin(u/2)
)*sin(u), 2-v*sin(u/2)*cos(u), v*cos(u/2)];" "6#/%\"SG7%*&,&\"\"#\"\"
\"*&%\"vGF)-%$sinG6#*&%\"uGF)F(!\"\"F)F1F)-F-6#F0F),&F(F)*(F+F)-F-6#*&
F0F)F(F1F)-%$cosG6#F0F)F1*&F+F)-F:6#*&F0F)F(F1F)" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 64 "restart: with(plots): with(LinearAlgebra): wi
th(VectorCalculus):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "S :=
 <(2-v*sin(u/2))*sin(u), (2-v*sin(u/2))*cos(u), v*cos(u/2)>;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F := <u^2+v, v^2+u, sin(u+v)
>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dS[u] := diff(S, u);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dS[v] := diff(S, v);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "FdA := <<dS[u]>|<dS[v]>|<F
>>;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 156 "Now we integrate over the
 surface of the Moebius band.  First in a positive and then in a negat
ive direction.  The results should sum to zero.  They do not!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "int(int(Determinant(FdA), u=
0..2*Pi), v=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "int(
int(Determinant(FdA), u=0..-2*Pi), v=-1..1);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 5 "%+%%;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "Th
e surface integral of F computes the volume of a solid whose base is t
he surface and whose height is the normal projection of F.   If F happ
ens to be the unit normal, then  " }{XPPEDIT 18 0 "Int(n*n,A) = Int(1,
A);" "6#/-%$IntG6$*&%\"nG\"\"\"F(F)%\"AG-F%6$F)F*" }{TEXT -1 9 " = Are
a. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 262 0 "" }{TEXT 263 8 "Practice" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 65 "1.       Compute the indicated surface in
tegrals.\n           a)  " }{XPPEDIT 18 0 "F(x,y) = [x, sin(y), cos(y)
];" "6#/-%\"FG6$%\"xG%\"yG7%F'-%$sinG6#F(-%$cosG6#F(" }{TEXT -1 3 " , \+
" }{XPPEDIT 18 0 "S(x,y) = [x, y*cos(y), y*sin(y)];" "6#/-%\"SG6$%\"xG
%\"yG7%F'*&F(\"\"\"-%$cosG6#F(F+*&F(F+-%$sinG6#F(F+" }{TEXT -1 17 " , \+
x=0..1,  y=0.." }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "           b)  " }
{XPPEDIT 18 0 "F(x,y) = [x, x, exp(-.1*y)];" "6#/-%\"FG6$%\"xG%\"yG7%F
'F'-%$expG6#,$*&-%&FloatG6$\"\"\"!\"\"F2F(F2F3" }{TEXT -1 3 " , " }
{XPPEDIT 18 0 "S(x,y) = [y*cos(x), y*sin(x), y];" "6#/-%\"SG6$%\"xG%\"
yG7%*&F(\"\"\"-%$cosG6#F'F+*&F(F+-%$sinG6#F'F+F(" }{TEXT -1 18 " ,  x=
-6..6, y=0.." }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "           c)  " }{XPPEDIT 18 
0 "F(x,y) = [exp(.1e-1*x), exp(.1e-1*y), exp(.1e-1*(x+y))];" "6#/-%\"F
G6$%\"xG%\"yG7%-%$expG6#*&-%&FloatG6$\"\"\"!\"#F1F'F1-F+6#*&-F/6$F1F2F
1F(F1-F+6#*&-F/6$F1F2F1,&F'F1F(F1F1" }{TEXT -1 23 " , S(x, y) = \{x, y
, z| " }{XPPEDIT 18 0 "x+y+3*z = 1;" "6#/,(%\"xG\"\"\"%\"yGF&*&\"\"$F&
%\"zGF&F&F&" }{TEXT -1 6 " \},  \{" }{TEXT 272 4 "x, y" }{TEXT -1 3 "
\}| " }{XPPEDIT 18 0 "x^2+y^2 = 4;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F
(\"\"%" }{TEXT -1 3 "   " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "2.  \+
     Using Maple, show that for a vector field, F, symmetrical around \+
the axis of revolution, the surface integral over a surface of revolut
ion, between " }{XPPEDIT 18 0 "t[1];" "6#&%\"tG6#\"\"\"" }{TEXT -1 6 "
  and " }{XPPEDIT 18 0 "t[2];" "6#&%\"tG6#\"\"#" }{TEXT -1 6 ",  is " 
}{XPPEDIT 18 0 "F[1]*(A[t[1]]-A[t[2]]);" "6#*&&%\"FG6#\"\"\"F',&&%\"AG
6#&%\"tG6#F'F'&F*6#&F-6#\"\"#!\"\"F'" }{TEXT -1 9 " , where " }
{XPPEDIT 18 0 "F[1];" "6#&%\"FG6#\"\"\"" }{TEXT -1 66 "  is the vector
 field component along the axis of revolution and  " }{XPPEDIT 18 0 "A
[j];" "6#&%\"AG6#%\"jG" }{TEXT -1 36 " is the area of the surface disc
 at " }{XPPEDIT 18 0 "t[j];" "6#&%\"tG6#%\"jG" }{TEXT -1 2 " ." }
{MPLTEXT 1 0 3 " \011\n" }}}}{MARK "0 0 2" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }