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{SECT 0 {PARA 258 "" 0 "" {TEXT 256 13 "Flux and flow" }{TEXT 257 31 "
 of vector fields across curves" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 21 "Let C be the ellipse\n" }{XPPEDIT 18 0 "x
=3*cos(t) " "6#/%\"xG*&\"\"$\"\"\"-%$cosG6#%\"tGF'" }{TEXT -1 2 ", " }
{XPPEDIT 18 0 "y=2*sin(t)" "6#/%\"yG*&\"\"#\"\"\"-%$sinG6#%\"tGF'" }
{TEXT -1 2 ", " }{TEXT -1 8 "   t=0.." }{XPPEDIT 18 0 "2*Pi" "6#*&\"\"
#\"\"\"%#PiGF%" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 24 "and F th
e vector field  " }}{PARA 0 "" 0 "" {TEXT -1 77 "F = (x+y)i + (y-x)j.
\nFind the flux of F across C  - and the flow of F along C" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "We name th
e components of the vector field and the curve" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 76 "restart: with(plots):\nM := x + y;  N := y-x;\nf := 3
*cos(t);   g := 2*sin(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Now \+
we plot the vector field and the curve" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 149 "ellipse := plot( [f,g,t=0..2*Pi] , color=blue):\nfield := fie
ldplot( [M,N], x=-3..3, y=-3..3, color=green, arrows=THICK ):\ndisplay
( \{ellipse,field\} );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Will th
e flux of F across C be positive or negative?" }}{PARA 0 "" 0 "" 
{TEXT -1 51 "Will the flow of F along C be positive or negative?" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "T
he flux of F across C" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "dx := diff
(f,t):\ndy := diff(g,t):\ntemp := M*dy - N*dx;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 73 "Since the integration is on the curve C,  on C   x=3
cos(t), y=2sin(t), so" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "integrand \+
:= eval( temp, \{x=f, y=g\} );\nflux := int( integrand, t=0..2*Pi );" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "We now compute the flow of f al
ong C" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "temp := M*dx + N*dy;\nint
egrand := eval( temp, \{x=f, y=g\} );\nflowalong := int( integrand, t=
0..2*Pi );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "1
3" 0 }{VIEWOPTS 1 1 0 1 1 1803 }