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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 31 "Line Integrals of Vector Fields
" }}{PARA 18 "" 0 "" {TEXT -1 20 "Using Maple and the " }{TEXT 256 8 "
vec_calc" }{TEXT -1 8 " Package" }}{PARA 0 "" 0 "" {TEXT -1 74 "This w
orksheet shows how to compute line integrals of vector fields using " 
}{TEXT 257 5 "Maple" }{TEXT -1 9 " and the " }{TEXT 258 8 "vec_calc" }
{TEXT -1 33 " package.  As examples we compute" }}{PARA 0 "" 0 "" 
{TEXT 260 36 "* The Work Done by an Electric Field" }{TEXT 263 0 "" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 261 28 "* The Circulation of a \+
Fluid" }{TEXT 262 0 "" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 13 "To start the " }{TEXT 259 8 "vec_calc
" }{TEXT -1 41 " package, execute the following commands:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "libname:=libname,\"C:/mylib/vector\";" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%(libnameG6$Q=C:\\Program~Files\\Maple~6/lib6\"
Q0C:/mylib/vectorF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with
(vec_calc): vc_aliases:" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the p
rotected names norm and trace have been redefined and unprotected\n" }
}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been \+
redefined\n" }}{PARA 6 "" 1 "" {TEXT -1 33 "Package:   vec_calc   Vers
ion 4.3" }}{PARA 6 "" 1 "" {TEXT -1 32 "For all HELP, execute: ?vec_ca
lc" }}{PARA 6 "" 1 "" {TEXT -1 38 "To use aliases, execute:   vc_alias
es;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 4 
"" 0 "" {TEXT 264 34 "The Work Done by an Electric Field" }{TEXT 265 
0 "" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The electric
 field of a line of charge with charge density " }{XPPEDIT 18 0 "lambd
a;" "6#%'lambdaG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "E = [2*k*lambda*x
/sqrt(x^2+y^2), 2*k*lambda*y/sqrt(x^2+y^2), 0];" "6#/%\"EG7%*,\"\"#\"
\"\"%\"kGF(%'lambdaGF(%\"xGF(-%%sqrtG6#,&*$F+F'F(*$%\"yGF'F(!\"\"*,F'F
(F)F(F*F(F2F(-F-6#,&*$F+F'F(*$F2F'F(F3\"\"!" }{TEXT -1 7 " where " }
{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 38 " is a constant.  A particle
 of charge " }{XPPEDIT 18 0 "q;" "6#%\"qG" }{TEXT -1 54 " moves throug
h this field along the line segment from " }{XPPEDIT 18 0 "[1, 2, 3];
" "6#7%\"\"\"\"\"#\"\"$" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "[3, 2, 1];
" "6#7%\"\"$\"\"#\"\"\"" }{TEXT -1 80 ".  We want to find the work don
e by this electric field on the charged particle." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We input the field as the
 function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "E:=MF([x,y,z]
,[2*k*lambda*x/sqrt(x^2+y^2),2*k*lambda*y/sqrt(x^2+y^2),0]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"EG7%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&
arrowGF+,$*&*(%\"kG\"\"\"%'lambdaGF39$F3F3*$-%%sqrtG6#,&*$)F5\"\"#F3F3
*$)9%F=F3F3F3!\"\"F=F+F+F+RF'F+F,F+,$*&*(F2F3F4F3F@F3F3*$-F86#F:F3FAF=
F+F+F+\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "We input the line \+
segment as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "r:=MF(t, eval
l([1,2,3]+t*([3,2,1]-[1,2,3])));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"rG7%R6#%\"tG6\"6$%)operatorG%&arrowGF),&\"\"\"F.*&\"\"#F.9$F.F.F)F)F
)F0RF'F)F*F),&\"\"$F.*&F0F.F1F.!\"\"F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{XPPEDIT 18 0 "0 <= t;" "6#1\"\"!%\"tG" }
{XPPEDIT 18 0 "` ` <= 1;" "6#1%\"~G\"\"\"" }{TEXT -1 21 ".  So the vel
ocity is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "v:=D(r);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%\"\"#\"\"!!\"#" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 42 "On the line segment, the electric field i
s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Er:=E(op(r(t)));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ErG7%,$*&*(%\"kG\"\"\"%'lambdaGF*,&
F*F**&\"\"#F*%\"tGF*F*F*F**$-%%sqrtG6#,&*$)F,F.F*F*\"\"%F*F*!\"\"F.,$*
&*&F)F*F+F*F**$-F26#F4F*F8F7\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
23 "The work is defined as " }{XPPEDIT 18 0 "Work = Int(q*E*`.`,r);" "
6#/%%WorkG-%$IntG6$*(%\"qG\"\"\"%\"EGF*%\".GF*%\"rG" }{TEXT -1 3 " = \+
" }{XPPEDIT 18 0 "Int(q*E(r(t))*`.`*v(t),t = 0 .. 1);" "6#-%$IntG6$**%
\"qG\"\"\"-%\"EG6#-%\"rG6#%\"tGF(%\".GF(-%\"vG6#F/F(/F/;\"\"!F(" }
{TEXT -1 16 ".  Consequently," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "qEv:=q*Er &. v(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$qEvG,$*
&**%\"qG\"\"\"%\"kGF)%'lambdaGF),&F)F)*&\"\"#F)%\"tGF)F)F)F)*$-%%sqrtG
6#,(\"\"&F)*&\"\"%F)F/F)F)*&F7F))F/F.F)F)F)!\"\"F7" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 38 "Work:=Int(qEv,t=0..1); Work:=value(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG-%$IntG6$,$*&**%\"qG\"\"\"%\"k
GF,%'lambdaGF,,&F,F,*&\"\"#F,%\"tGF,F,F,F,*$-%%sqrtG6#,(\"\"&F,*&\"\"%
F,F2F,F,*&F:F,)F2F1F,F,F,!\"\"F:/F2;\"\"!F," }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%WorkG,&**-%%sqrtG6#\"#8\"\"\"%\"qGF+%\"kGF+%'lambdaG
F+\"\"#*,F/F+-F(6#\"\"&F+F,F+F-F+F.F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 56 "Another way to compute this line integral is to use the \+
" }{TEXT 267 15 "Line_int_vector" }{TEXT -1 23 " command (or its alias
 " }{TEXT 268 3 "Liv" }{TEXT -1 11 ") from the " }{TEXT 269 8 "vec_cal
c" }{TEXT -1 79 " package which works directly with the parametrized c
urve and the vector field:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "Liv(E,r,t=0..1); Work:=q*value(%);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%$IntG6$,$*&*(%\"kG\"\"\"%'lambdaGF*,&F*F**&\"\"#F*%\"tGF*F*F*F
**$-%%sqrtG6#,(\"\"&F**&\"\"%F*F/F*F**&F7F*)F/F.F*F*F*!\"\"F7/F/;\"\"!
F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG*&%\"qG\"\"\",&*(-%%sqrt
G6#\"#8F'%\"kGF'%'lambdaGF'\"\"#**F0F'-F+6#\"\"&F'F.F'F/F'!\"\"F'" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "There is a second way to compute \+
this work.  The electric field is conservative because it is curl-free
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "CURL(E);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
46 "So the electric field has a scalar potential, " }{XPPEDIT 18 0 "ph
i;" "6#%$phiG" }{TEXT -1 3 ".  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "POT(E,'phi');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "phi(x,y,z);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*(%\"kG\"\"\"%'lambdaGF&-%%sqrtG6#,&*$)%\"xG\"\"
#F&F&*$)%\"yGF/F&F&F&F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "By the
 Fundamental Theorem of Calculus, the work is the change in the potent
ial energy:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Work:=q*phi(
3,2,1)-q*phi(1,2,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG,&**-%
%sqrtG6#\"#8\"\"\"%\"qGF+%\"kGF+%'lambdaGF+\"\"#*,F/F+-F(6#\"\"&F+F,F+
F-F+F.F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 266 26 "The Circulation of a Fluid" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The velocity of the water in a sin
k going down the drain is " }{XPPEDIT 18 0 "V = [(-y-z)/(x^2+y^2), (x-
z)/(x^2+y^2), z^2*(-x-y)/((x^2+y^2)^2)];" "6#/%\"VG7%*&,&%\"yG!\"\"%\"
zGF)\"\"\",&*$%\"xG\"\"#F+*$F(F/F+F)*&,&F.F+F*F)F+,&*$F.F/F+*$F(F/F+F)
*(F*F/,&F.F)F(F)F+*$,&*$F.F/F+*$F(F/F+F/F)" }{TEXT -1 84 " .  We want \+
to find the circulation of the fluid counterclockwise around the circl
e " }{XPPEDIT 18 0 "x^2+y^2 = 4;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(
\"\"%" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "z = 2;" "6#/%\"zG\"\"#" }
{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 42 "We enter the fluid velocity as a function:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "V:=MF([x,y,z],[(-y-z)/(x^2+y^2), (x
-z)/(x^2+y^2), z^2*(-x-y)/((x^2+y^2)^2)]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"VG7%R6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF+*&,
&9%!\"\"9&F2\"\"\",&*$)9$\"\"#F4F4*$)F1F9F4F4F2F+F+F+RF'F+F,F+*&,&F8F4
F3F2F4F5F2F+F+F+RF'F+F,F+*&*&)F3F9F4,&F8F2F1F2F4F4*$)F5F9F4F2F+F+F+" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "The circle may be parametrized a
s" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "r:=MF(t,[2*cos(t),2*si
n(t),3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG7%R6#%\"tG6\"6$%)op
eratorG%&arrowGF),$-%$cosG6#9$\"\"#F)F)F)RF'F)F*F),$-%$sinGF0F2F)F)F)
\"\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Its tangent vector (velo
city) is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "v:=D(r);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG7%R6#%\"tG6\"6$%)operatorG%&arro
wGF),$-%$sinG6#9$!\"#F)F)F)R6#%\"tGF)F*F),$-%$cosGF0\"\"#F)F)F)\"\"!" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Caution:  Don't confuse the vel
ocity (tangent vector) of the curve " }{XPPEDIT 18 0 "v;" "6#%\"vG" }
{TEXT -1 32 " with the velocity of the fluid " }{XPPEDIT 18 0 "V;" "6#
%\"VG" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 41 "On the curve, the fluid velocity becomes:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "V(op(r(t)));  Vr:=simplify(%
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%*&,&-%$sinG6#%\"tG!\"#\"\"$!\"
\"\"\"\",&*$)-%$cosGF(\"\"#F-\"\"%*&F4F-)F&F3F-F-F,*&,&F1F3F+F,F-F.F,,
$*&,&F1F**&F3F-F&F-F,F-*$)F.F3F-F,\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#VrG7%,&-%$sinG6#%\"tG#!\"\"\"\"##\"\"$\"\"%F,,&-%$cosGF)#\"\"
\"F-#F/F0F,,&F2#!\"*\"\")*&#\"\"*F:F5F'F5F," }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 30 "The circulation is defined as " }{XPPEDIT 18 0 "Circ = \+
Int(V*`.`,r);" "6#/%%CircG-%$IntG6$*&%\"VG\"\"\"%\".GF*%\"rG" }{TEXT 
-1 3 " = " }{XPPEDIT 18 0 "Int(V(r(t))*`.`*v(t),t = 0 .. 2*Pi);" "6#-%
$IntG6$*(-%\"VG6#-%\"rG6#%\"tG\"\"\"%\".GF.-%\"vG6#F-F./F-;\"\"!*&\"\"
#F.%#PiGF." }{TEXT -1 16 ".  Consequently," }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 17 "Vr_v:=Vr &. v(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%%Vr_vG,(-%$sinG6#%\"tG#\"\"$\"\"#*&#F+F,\"\"\"-%$cosGF(F/!\"\"F/F/
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Circ:=Int(Vr_v,t=0..2*P
i); Circ:=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%CircG-%$IntG
6$,(-%$sinG6#%\"tG#\"\"$\"\"#*&#F.F/\"\"\"-%$cosGF+F2!\"\"F2F2/F,;\"\"
!,$%#PiGF/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%CircG,$%#PiG\"\"#" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Another way to compute this line \+
integral is to use the " }{TEXT 270 15 "Line_int_vector" }{TEXT -1 23 
" command (or its alias " }{TEXT 271 3 "Liv" }{TEXT -1 11 ") from the \+
" }{TEXT 272 8 "vec_calc" }{TEXT -1 79 " package which works directly \+
with the parametrized curve and the vector field:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 35 "Liv(V,r,t=0..2*Pi); Circ:=value(%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(-%$sinG6#%\"tG#\"\"$\"\"#*&
#F,F-\"\"\"-%$cosGF)F0!\"\"F0F0/F*;\"\"!,$%#PiGF-" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%CircG,$%#PiG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}{MARK "12 13 2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }