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0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 
2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 
259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 
0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 3 260 1 {CSTYLE "" -1 -1 "" 1 12 
0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE 
"" 3 261 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 
0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 
1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }
{PSTYLE "" 3 263 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 
1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 418 33 "Math  233  Spring 2004 F
inal Exam" }}{PARA 0 "" 0 "" {TEXT 256 2 "  " }}}{SECT 0 {PARA 3 "" 0 
"" {TEXT 258 1 "1" }{TEXT 267 2 ". " }{TEXT 450 43 "A line that is per
pendicular to the plane  " }{XPPEDIT 455 1 "4*x+3*y+2*z=13" "6#/,(*&\"
\"%\"\"\"%\"xGF'F'*&\"\"$F'%\"yGF'F'*&\"\"#F'%\"zGF'F'\"#8" }{TEXT 
452 33 "   has symmetric equations \n\n    " }{TEXT 460 2 "  " }{TEXT 
657 5 "     " }{XPPEDIT 461 1 "(4*x+1)/a = (3*y-2)/b" "6#/*&,&*&\"\"%
\"\"\"%\"xGF(F(F(F(F(%\"aG!\"\"*&,&*&\"\"$F(%\"yGF(F(\"\"#F+F(%\"bGF+
" }{TEXT 458 2 "  " }{XPPEDIT 462 1 "` `=4*z" "6#/%\"~G*&\"\"%\"\"\"%
\"zGF'" }{TEXT 459 1 "\n" }{TEXT 463 14 "\n     What is " }{TEXT 451 
6 "  a?\n\n" }{TEXT 457 142 "a)  1            b)  2           c)  3   \+
      d)  4          e)   5  \nf)   6           g)   7          h)  8 \+
         i)   9          j)   10" }{TEXT 456 1 "\n" }}{PARA 3 "" 0 "" 
{TEXT 453 8 "Solution" }{TEXT 454 6 ": (b)\n" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 0 {PARA 260 "" 0 "" {TEXT -1 8 "2. If   " }{XPPEDIT 
19 1 "`<`*14 , -2*`>`=`<`*a,b*`>`+`<`*c,d*`>`" "6&*&%\"<G\"\"\"\"#9F%/
,$*&\"\"#F%%\">GF%!\"\"*&F$F%%\"aGF%,&*&%\"bGF%F+F%F%*&F$F%%\"cGF%F%*&
%\"dGF%F+F%" }{TEXT -1 9 "   where " }{XPPEDIT 19 1 "`<`*a,b*`>`" "6$*
&%\"<G\"\"\"%\"aGF%*&%\"bGF%%\">GF%" }{TEXT -1 18 "   is parallel to \+
" }{XPPEDIT 19 1 "`<`*3,1*`>`" "6$*&%\"<G\"\"\"\"\"$F%*&F%F%%\">GF%" }
{TEXT -1 5 " and " }{XPPEDIT 19 1 "`<`*c,d*`>`" "6$*&%\"<G\"\"\"%\"cGF
%*&%\"dGF%%\">GF%" }{TEXT -1 28 "  is perpendicular \n    to  " }
{XPPEDIT 19 1 "`<`*3,1*`>`" "6$*&%\"<G\"\"\"\"\"$F%*&F%F%%\">GF%" }
{TEXT -1 16 ",  then what is " }{TEXT 471 1 "b" }{TEXT -1 1 "?" }
{TEXT 265 2 "\n\n" }{TEXT 470 142 "a)  1            b)  2           c)
  3         d)  4          e)   5  \nf)   6           g)   7          \+
h)  8          i)   9          j)   10" }{TEXT 469 3 "\n \n" }}{PARA 
3 "" 0 "" {TEXT 419 8 "Solution" }{TEXT 420 6 ": (d)\n" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 
"" 0 "" {TEXT 257 32 "3. A curve is parameterized by  " }{XPPEDIT 464 
1 "r(t)=`<`*t,t^3/3,t^2*`>`" "6%/-%\"rG6#%\"tG*&%\"<G\"\"\"F'F**&F'\"
\"$F,!\"\"*&F'\"\"#%\">GF*" }{TEXT 465 51 " .  Calculate its principal
 unit normal vector \n   " }{XPPEDIT 468 1 "N=`<`*a,b,c*`>`" "6%/%\"NG
*&%\"<G\"\"\"%\"aGF'%\"bG*&%\"cGF'%\">GF'" }{TEXT 472 40 "   at the po
int (1, 1/3, 1).  What is   " }{XPPEDIT 467 1 "a" "6#%\"aG" }{TEXT 
466 3 "?\n " }{TEXT 424 1 "\n" }{TEXT 425 7 "a)     " }{XPPEDIT 540 1 
"1/3" "6#*&\"\"\"F$\"\"$!\"\"" }{TEXT 526 20 "                b)  " }
{XPPEDIT 541 1 "-1/3" "6#,$*&\"\"\"F%\"\"$!\"\"F'" }{TEXT 527 17 "    \+
        c)   " }{XPPEDIT 542 1 "1/sqrt(2)" "6#*&\"\"\"F$-%%sqrtG6#\"\"
#!\"\"" }{TEXT 528 17 "            d)   " }{XPPEDIT 543 1 "-1/sqrt(2)
" "6#,$*&\"\"\"F%-%%sqrtG6#\"\"#!\"\"F*" }{TEXT 531 1 " " }{TEXT 529 
2 "  " }{TEXT 530 10 "       e) " }{TEXT 532 3 "   " }{TEXT 533 1 " " 
}{XPPEDIT 544 1 "1/sqrt(3)" "6#*&\"\"\"F$-%%sqrtG6#\"\"$!\"\"" }{TEXT 
534 8 "      \n " }{TEXT 422 5 "f)   " }{TEXT 521 1 " " }{XPPEDIT 545 
1 "-1/sqrt(3)" "6#,$*&\"\"\"F%-%%sqrtG6#\"\"$!\"\"F*" }{TEXT 522 7 "  \+
     " }{TEXT 523 7 "   g)  " }{XPPEDIT 546 1 "1/sqrt(6)" "6#*&\"\"\"F
$-%%sqrtG6#\"\"'!\"\"" }{TEXT 524 8 "        " }{TEXT 525 3 "   " }
{TEXT 520 1 " " }{TEXT 423 3 "h) " }{XPPEDIT 547 1 "-1/sqrt(6)" "6#,$*
&\"\"\"F%-%%sqrtG6#\"\"'!\"\"F*" }{TEXT 535 7 "       " }{TEXT 536 12 
"     i)     " }{XPPEDIT 548 1 "2/sqrt(6)" "6#*&\"\"#\"\"\"-%%sqrtG6#
\"\"'!\"\"" }{TEXT 537 6 "      " }{TEXT 538 11 "      j)   " }
{XPPEDIT 549 1 "-2/sqrt(6)" "6#,$*&\"\"#\"\"\"-%%sqrtG6#\"\"'!\"\"F+" 
}{TEXT 539 6 "      " }}{PARA 3 "" 0 "" {TEXT 421 1 " " }}{PARA 3 "" 
0 "" {TEXT 426 8 "Solution" }{TEXT 427 6 ": (d)\n" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 473 17 
"4. Suppose that  " }{XPPEDIT 474 1 "x^2 +y*z-z^3=2*x*z-9" "6#/,(*$%\"
xG\"\"#\"\"\"*&%\"yGF(%\"zGF(F(*$F+\"\"$!\"\",&*(F'F(F&F(F+F(F(\"\"*F.
" }{TEXT 475 16 " .  Calculate   " }{XPPEDIT 476 1 "diff( z ,x)" "6#-%
%diffG6$%\"zG%\"xG" }{TEXT 477 8 "   at   " }{XPPEDIT 478 1 "``(3,1,2)
" "6#-%!G6%\"\"$\"\"\"\"\"#" }{TEXT 479 202 ".\n\na)  3/8             \+
  b)  4/15             c)  2/9             d)  3/7             e)  5/1
3               \nf)   2/17             g)  2/11             h) 3/19  \+
          i)  4/17             j)  8/21 " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 3 "" 0 "" {TEXT 480 8 "Solution" }{TEXT 481 6 ": (f)\n" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 28 "eqn := x^2 +y*z-z^3=2*x*z-9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%$eqnG/,(*$%\"xG\"\"#\"\"\"*&%\"yGF*%\"zGF*F**$F-\"\"$!\"\",&*&F(F*F-
F*F)!\"*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "F := (x,y,z) \+
-> lhs(eqn)-rhs(eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6%%\"
xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,&-%$lhsG6#%$eqnG\"\"\"-%$rhsGF1
!\"\"F*F*6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "expr := - d
iff(F(x,y,z),x)/diff(F(x,y,z),z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%%exprG,$*&,&%\"xG\"\"#%\"zG!\"#\"\"\",(%\"yGF,*$F*F)!\"$F(F+!\"\"F1" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(\{x=3,y=1,z=2\},expr
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"#\"#<" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT 260 1 " " }{TEXT 482 2 "5." }{TEXT -1 1 " " }{TEXT 283 15 "The f
unction   " }{XPPEDIT 282 1 "f(x,y)=3*x^2*y- x*y^2- 9*x*y" "6#/-%\"fG6
$%\"xG%\"yG,(*(\"\"$\"\"\"*$F'\"\"#F,F(F,F,*&F'F,*$F(F.F,!\"\"*(\"\"*F
,F'F,F(F,F1" }{TEXT 281 133 "  has exactly one critical point P = (a,b
) that does not lie on a coordinate axis.  Choose the ordered list [b,
what] where b is the  " }{TEXT 655 1 "y" }{TEXT 656 82 "-coordinate of
 the critical point P = (a,b) and \"what\" describes the behavior of  \+
" }{XPPEDIT 285 1 "f" "6#%\"fG" }{TEXT 284 12 "  at P. \n\na)" }{TEXT 
286 145 " [-1,local minimum]         b) [-2,local minimum]          c)
 [-3,local minimum]          \nd)  [-1,local maximum]       e) [-2,loc
al maximum]    " }{TEXT 289 4 "    " }{TEXT 292 3 " f)" }{TEXT 293 1 "
 " }{TEXT 290 18 "[-3,local maximum]" }{TEXT 291 1 " " }{TEXT 287 95 "
       \ng)  [-1,saddle point]            h) [-2,saddle point]        \+
      i) [-3,saddle point]" }{TEXT 288 5 "    \n" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 439 8 "Solution" }{TEXT 440 6 ":
 (f)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f := (x,y) -> 3*x^
2*y - x*y^2- 9*x*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG
%\"yG6\"6$%)operatorG%&arrowGF),(*&9$\"\"#9%\"\"\"\"\"$*&F/F2F1F0!\"\"
*&F/F2F1F2!\"*F)F)6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "so
lve(\{diff(f(x,y),x) = 0 , diff(f(x,y),y) = 0\}, \{x,y\} );" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6&<$/%\"yG\"\"!/%\"xGF&<$/F(\"\"$F$<$/F%!\"*F'
<$/F(\"\"\"/F%!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "diff(
f(x,y),x$2)*diff(f(x,y),y$2)-(diff(f(x,y),x,y))^2;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,&*&%\"xG\"\"\"%\"yGF&!#7*$,(F%\"\"'F'!\"#!\"*F&\"\"#
!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(\{x=1,y=-3\},
\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#F" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "subs(\{x=1,y=-3\},diff(f(x,y),x$2));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#!#=" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 35 "f(1,-3);  f(.99,-3.1); f(.99,-2.9);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'<#**)!\"&
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'B')*)!\"&" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 1 " " }{TEXT 444 8 "6. Let  " }{XPPEDIT 511 1 "u" "6#%\"uG
" }{TEXT 508 43 "  be the upward unit normal of the plane   " }
{XPPEDIT 510 1 "x-2*y+2*z=7" "6#/,(%\"xG\"\"\"*&\"\"#F&%\"yGF&!\"\"*&F
(F&%\"zGF&F&\"\"(" }{TEXT 509 46 ".  Calculate the directional derivat
ive of    " }{XPPEDIT 517 1 "F(x,y,z) = x^3+3*y*z+z^2" "6#/-%\"FG6%%\"
xG%\"yG%\"zG,(*$F'\"\"$\"\"\"*(F,F-F(F-F)F-F-*$F)\"\"#F-" }{TEXT 512 
25 "   in the direction of   " }{XPPEDIT 519 1 "u" "6#%\"uG" }{TEXT 
518 28 "  at the point  ( 2, 2, 3). " }{TEXT 513 2 "\n\n" }{TEXT 515 
142 "a)  1            b)  2           c)  3         d)  4          e) \+
  5  \nf)   6           g)   7          h)  8          i)   9         \+
 j)   10" }{TEXT 514 1 "\n" }{TEXT 516 7 "      \n" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 445 8 "Solution" }{TEXT 446 6 ":
 (f)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}
{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}
{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F := (x,y,z) -> x^3+3*y*z+z^
2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6%%\"xG%\"yG%\"zG6\"6$%)
operatorG%&arrowGF*,(*$9$\"\"$\"\"\"*&9%F29&F2F1*$F5\"\"#F2F*F*6\"" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "u := [1/3,-2/3,2/3];  v := \+
grad(F(x,y,z),[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"uG7%#\"
\"\"\"\"$#!\"#F(#\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"vG-%'V
ECTORG6#7%,$*$%\"xG\"\"#\"\"$,$%\"zGF-,&%\"yGF-F/F," }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "dp := dotprod(u,v);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#dpG,(*$%\"xG\"\"#\"\"\"%\"zG#!\"#\"\"$%\"yGF(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(\{x=2,y=2,z=3\}, dp );
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " " }{TEXT 259 8 "7.  Le
t " }{TEXT 488 1 " " }{TEXT 485 1 "(" }{TEXT 483 7 " a , b " }{TEXT 
484 48 ")   be the point in the first quadrant at which " }{TEXT 272 
1 " " }{XPPEDIT 268 1 "f(x,y)=x*y+1" "6#/-%\"fG6$%\"xG%\"yG,&*&F'\"\"
\"F(F+F+F+F+" }{TEXT 269 1 " " }{TEXT 270 46 " has a maximum value sub
ject to the constraint" }{TEXT 271 3 "   " }{XPPEDIT 486 1 "x^2+2*y^2=
1" "6#/,&*$%\"xG\"\"#\"\"\"*&F'F(*$%\"yGF'F(F(F(" }{TEXT 487 3 ".  " }
{TEXT 428 10 "Calculate " }{TEXT 489 1 "a" }{TEXT 490 6 "  and " }
{TEXT 491 1 "b" }{TEXT 492 25 ".  What is the value of  " }{XPPEDIT 
494 1 "a^2+b^2" "6#,&*$%\"aG\"\"#\"\"\"*$%\"bGF&F'" }{TEXT 493 4 " ?\n
\n" }{TEXT 273 4 "a)  " }{XPPEDIT 497 1 "1/8" "6#*&\"\"\"F$\"\")!\"\"
" }{TEXT 429 15 "            b) " }{XPPEDIT 498 1 "1/4" "6#*&\"\"\"F$
\"\"%!\"\"" }{TEXT 430 22 "                 c)   " }{XPPEDIT 499 1 "3/
8" "6#*&\"\"$\"\"\"\"\")!\"\"" }{TEXT 431 26 "                      d)
  " }{XPPEDIT 500 1 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT 432 19 "    \+
           e)  " }{XPPEDIT 501 1 "5/8" "6#*&\"\"&\"\"\"\"\")!\"\"" }
{TEXT 433 11 "     \n f)  " }{XPPEDIT 502 1 "3/4" "6#*&\"\"$\"\"\"\"\"
%!\"\"" }{TEXT 434 15 "            g) " }{XPPEDIT 503 1 "7/8" "6#*&\"
\"(\"\"\"\"\")!\"\"" }{TEXT 435 20 "                 h) " }{XPPEDIT 
504 1 "3/16" "6#*&\"\"$\"\"\"\"#;!\"\"" }{TEXT 436 26 "               \+
       i)  " }{XPPEDIT 505 1 "5/16" "6#*&\"\"&\"\"\"\"#;!\"\"" }{TEXT 
495 18 "              j)  " }{XPPEDIT 506 1 "7/16" "6#*&\"\"(\"\"\"\"#
;!\"\"" }{TEXT 496 3 "   " }{TEXT 507 3 "   " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 3 "" 0 "" {TEXT 437 8 "Solution" }{TEXT 438 6 ": (f)\n
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f := (x,y) -> x*y+1:  ph
i := (x,y) -> x^2 + 2*y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "eqn1 := diff(f(x,y),x) = lambda*diff(phi(x,y),x);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%eqn1G/%\"yG,$*&%'lambdaG\"\"\"%\"xGF*\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "eqn2 := diff(f(x,y),y) = lam
bda*diff(phi(x,y),y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"x
G,$*&%'lambdaG\"\"\"%\"yGF*\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "eqn3 := phi(x,y) = 1;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%%eqn3G/,&*$%\"xG\"\"#\"\"\"*$%\"yGF)F)F*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 39 "solve( \{eqn1,eqn2,eqn3\}, \{x,y,lambda\});" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%/%\"xG-%'RootOfG6#,&!\"\"\"\"\"*$%
#_ZG\"\"#F./%\"yG#F+F./%'lambdaG,$F&F1" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "c := 1/sqrt(2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 98 "levelCurve[1] := implicitplot( f(x,y)=f(c,1/2), x = -3/2..3/2,y=
-3/2..3/2,color=pink,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 94 "levelCurve[2] := implicitplot( x^2+2*y^2=1, x = -3/2.
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" {MPLTEXT 1 0 110 "levelCurve[3] := contourplot(f(x,y), x = -3/2.. 3/
2, y= -3/2..3/2, contours=10,color=COLOR(RGB,0.75,0.7,0.7)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display(seq(levelCurve[j],j=
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve
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{TEXT 447 33 "8.  Calculate  the integral of   " }{XPPEDIT 554 1 "x+2*
y" "6#,&%\"xG\"\"\"*&\"\"#F%%\"yGF%F%" }{TEXT 550 96 "  over the regio
n in the first quadrant that is bounded by the coordinate axes and the
 circle   " }{XPPEDIT 555 1 "x^2+y^2=4" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"y
GF'F(\"\"%" }{TEXT 551 3 "  ." }{TEXT 552 1 "\n" }{TEXT 553 145 "\na) \+
 1            b)  2           c)  3         d)  4          e)   5  \nf
)   6            g)   7          h)  8          i)   9          j)   1
0\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 448 8 "So
lution" }{TEXT 449 6 ": (h)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "int(int( (r*cos(t)+2*r*sin(t
))*r,r=0..2),t=0..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 15 "9. The vector
  " }{XPPEDIT 274 1 "`<`*a,-3,-1*`>` " "6%*&%\"<G\"\"\"%\"aGF%,$\"\"$!
\"\",$*&F%F%%\">GF%F)" }{TEXT 275 30 "  is tangent to the surface   " 
}{XPPEDIT 276 1 "x^2+2*y^3-3*z^2=3" "6#/,(*$%\"xG\"\"#\"\"\"*&F'F(*$%
\"yG\"\"$F(F(*&F,F(*$%\"zGF'F(!\"\"F," }{TEXT 277 36 " at the point (2
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      b)  2           c)  3         d)  4          e)   5  \nf)   6   \+
        g)   7          h)  8          i)   9          j)   10\n" }}
{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 298 1 " " }
{TEXT 441 8 "Solution" }{TEXT 442 5 ": (c)" }{TEXT 443 1 "\n" }}
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> " 0 "" {MPLTEXT 1 0 83 "solve(dotprod(v,[a,-3,-1])=0,a); \n#Tangent \+
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"" {TEXT 621 27 "10. Calculate\n     \n       " }{BITMAP 91 45 45 1 "?
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kyyI::ZykyryyA::jyrYvyy=B:xY::ryxI<:;:Zyuy::y]:J:::y^yyY:rY:ryxCy_YL::
y]yyY:y_:jyjy;:::::Z:B:xiy=::::jyrYvyy=::rYvIxyy;::vIxAyyy:::::::::xAx
IZYL:vYxI:<:;Zyuy:y^YL:rYry:jY;;ZYlyyI:::::Z:B:xiy=::::jyB:xIyA::rY:xI
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;:ZyL:rYry:jyB:vMvY:rYZylYsYvyy=::::B:<Zyuy:::::>:yI:vYyA::B:>:xiy=::>
:<:yqy;:::::::::jIsYry::jyB:vQxyy;vQ<:y]yyY:ra>:xCyyyxCy=yA::::<Z:ryxI
:::::;jy=:yqy;::<:;Zyuy:::;Z:jywY::::::::Z:jIsyyAxCy=yA::xA;Zyuy:y?>:x
Eyyy:xC;ZYlyyIvIxCxI:::Z:B:xiy=::::J:vy:jywY::Z:J:ryxI::J:B:vYyA::::::
::ra>:<::rYvyy=::y]ykYkywI<:;ZYlyyIZyL:rYvyy=B:vQxyy;::Z:B:xiy=::::J:v
y:jywY::Z:J:ryxI::J:B:vYyA::::::::ri>:<:::y=yA::xCy]::xC;:vyy=ri:ry:xE
ryxI:::<Z:Zy=::::J:vy:jywY::Z:J:ryxI::J:B:vYyA::::::::::::::::::;Z:jyw
Y:>:<:yqy;jIsYry:::<Z:Zy=::::J:vyy[y=::B::yA::J::xI::::::::::::::::::v
I<:y_yyY:;Z:jywY:ZykyryyA::<Z:Zy=::::jyrYvyy=::rYvIxyy;::vIxAyyy::::::
::::::::::::jIC:xyIsyyAxE;Z:jIky;ZYL:rYvQxyy;Z:B:xiy=::::jyrYvyy=::rYv
Ixyy;::vIxAyyy::::::::::::::::::jYC:xY:ryxyyB:>::jywYvMJ::ryxI:<Z:Zy=:
:::jyravyy=::rY:yA::jy:xI:::::::::::::::::Zy<yI:Zyuy::::::::B:<:xI::::
:y=ry:::xAjy;::vIZy=::::::::::::::::::::::::::::B:Zy=::::jyrivyy=::rY:
yA::jy:xI::::::::::::::::::::::::::::Z::xI:::::y^ylIC:xyyryyA:xCy?>:Zy
=:y^y<<:jy;::::::::::::::::::::::::::Z:B:xiy=:::::<ZyA::xiy=:>:<::yqy;
:<:;:Zyuy:::::::::::::::::::::::::::B:<:xI:::::ri>:yI:vMxyy;:vQJ:rary:
:xEB:vMvY:::::::::::::::::::::::::::B:<Zyuy:::::::::::::::::::::::::::
:::::::::::::::::Z::xI::::::::::::::::::::::::::::::::::::::::::::B:Zy
=::::::::::::::::::::::::::::::::::::::::::::<:ry:::::::::::::::::::::
:::::::::::::::::::::::Z::xI::::::::::::::::::::::::::::::::::::::::::
::B:Zy=::::::::::::::::::::::::::::::::::::::::::::<:ry:::::::::::::::
:::::::::::::::::::::::::::::Z::xI::::::::::::::::::::::::::::::::::::
::::::::B:Zy=::::::::::::::::::::::::::::::::::::::::::::<:ry:::::::::
:::::::::::::::::::::::::::::::::::Z::xI::::::::::::::::::::::::::::::
::::::::::::::B:3:" }{TEXT 636 7 "\nwhere " }{TEXT 634 1 "U" }{TEXT 
635 65 "  is the solid half-ball that is described by the inequalities
   " }{XPPEDIT 639 1 "x^2+y^2+z^2<=1" "6#1,(*$%\"xG\"\"#\"\"\"*$%\"yGF
'F(*$%\"zGF'F(F(" }{TEXT 638 9 "   and   " }{XPPEDIT 640 1 "z>=0" "6#1
\"\"!%\"zG" }{TEXT 637 7 ". \na)  " }{XPPEDIT 645 1 "0" "6#\"\"!" }
{TEXT 622 15 "           b)  " }{XPPEDIT 646 1 "Pi/12" "6#*&%#PiG\"\"
\"\"#7!\"\"" }{TEXT 623 19 "               c)  " }{XPPEDIT 647 1 "Pi/8
" "6#*&%#PiG\"\"\"\"\")!\"\"" }{TEXT 624 19 "               d)  " }
{XPPEDIT 648 1 "Pi/6" "6#*&%#PiG\"\"\"\"\"'!\"\"" }{TEXT 625 18 "     \+
          e) " }{XPPEDIT 649 1 "Pi/4" "6#*&%#PiG\"\"\"\"\"%!\"\"" }
{TEXT 626 11 "     \nf)   " }{XPPEDIT 650 1 "Pi/3" "6#*&%#PiG\"\"\"\"
\"$!\"\"" }{TEXT 627 19 "              g)   " }{XPPEDIT 651 1 "Pi/2" "
6#*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT 628 17 "             h)  " }
{XPPEDIT 652 1 "Pi" "6#%#PiG" }{TEXT 629 18 "              i)  " }
{XPPEDIT 653 1 "2*Pi" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT 630 18 "        \+
      j)  " }{XPPEDIT 654 1 "3*Pi" "6#*&\"\"$\"\"\"%#PiGF%" }{TEXT 
631 5 "    \n" }}{PARA 3 "" 0 "" {TEXT 632 8 "Solution" }{TEXT 633 6 "
: (e)\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "int(int(int(rho*cos(phi)*rho^2*sin(phi),rho=0..1),phi
=0..Pi/2),theta=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG#
\"\"\"\"\"%" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 
"" {TEXT 262 16 "11. Calculate   " }{XPPEDIT 295 1 "int( y*dx+2*x*dy+(
z-x*y+3),z=C..``)" "6#-%$intG6$,(*&%\"yG\"\"\"%#dxGF)F)*(\"\"#F)%\"xGF
)%#dyGF)F),(%\"zGF)*&F-F)F(F)!\"\"\"\"$F)F)/F0;%\"CG%!G" }{TEXT 296 
11 ".   where  " }{TEXT 343 1 "C" }{TEXT 344 42 "   is the oriented cu
rve parameterized by " }{XPPEDIT 346 1 "r(t)=[3*t,t^2,t^3]" "6#/-%\"rG
6#%\"tG7%*&\"\"$\"\"\"F'F+*$F'\"\"#*$F'F*" }{TEXT 345 7 ",      " }
{XPPEDIT 594 1 "0 <=t" "6#1\"\"!%\"tG" }{TEXT 593 1 " " }{XPPEDIT 596 
1 "``<=1" "6#1%!G\"\"\"" }{TEXT 595 1 " " }{TEXT 597 3 ". \n" }{TEXT 
598 152 "\na)  1            b)   2           c)   3          d)  4    \+
      e)   5  \nf)   6            g)   7           h)   8          i) \+
  9          j)   10\n  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "
" 0 "" {TEXT 341 8 "Solution" }{TEXT 342 6 ": (g)\n" }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "r := t -> [3*t,t^2,t^3];" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"rGf*6#%\"tG6\"6$%)operatorG%&arrowGF(7%,$9$\"\"$*
$F.\"\"#*$F.F/F(F(6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "I
nt( r(t)[2]*diff(r(t)[1],t)+ 2*r(t)[1]*diff(r(t)[2],t)+(r(t)[3]-r(t)[1
]*r(t)[2]+3)*diff(r(t)[3],t),t=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%$IntG6$,&*$%\"tG\"\"#\"#:*&,&*$F(\"\"$!\"#F.\"\"\"F0F(F)F./F(;\"
\"!F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
358 38 "12. The vector field  \n\n              " }{TEXT -1 3 "   " }
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:::::::::::::Z::yA::::::::;:xI:::::::::::::::::::::::::::::;:xIJ:Zy=::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:Z::yAB:jy;:::::::J:Zy=:::::::::::::::::::::::::::::<:;:vYyA::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::;Z::x
iy=::::::::>:ry:::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::>:ry:::
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::>:ry:::::::::::::::::::::
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::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::>:1:" }{TEXT -1 1 "\n" }{TEXT 362 39 "is the gradient field o
f some function " }{TEXT 363 2 "f(" }{TEXT 365 3 "x,y" }{TEXT 366 1 ")
" }{TEXT 364 13 ".  Calculate " }{TEXT 361 1 " " }{TEXT 359 15 "f(2,1)
 - f(0,1)" }{TEXT 360 3 ".\n\n" }{TEXT 367 79 "a)  2              b)  \+
4             c)   6            d)  8          e)  10   " }{TEXT 372 
4 "  \nf" }{TEXT 368 34 ")  12            g)  14           " }{TEXT 
557 1 " " }{TEXT 371 80 "h)  16           i) 18          j)  Cannot be
 calculated from given information." }{TEXT -1 2 "  " }}{PARA 3 "" 0 "
" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 369 8 "Solution" }{TEXT 370 6 
": (h)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart: with(li
nalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for nor
m" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "F := vector([y^2+7,2*x*y+3
*y^2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'VECTORG6#7$,&*$%\"
yG\"\"#\"\"\"\"\"(F-,&*&%\"xGF-F+F-F,F*\"\"$" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 63 "curl(vector([y^2+7,2*x*y+3*y^2,0]),[x,y,z]); #F \+
is conservative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%\"\"!
F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "eqn := u(x,y) = int
(y^2+7,x) + g(y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/-%\"uG6$%
\"xG%\"yG,&*&,&*$F*\"\"#\"\"\"\"\"(F0F0F)F0F0-%\"gG6#F*F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff(eqn,y);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%%diffG6$-%\"uG6$%\"xG%\"yGF+,&*&F*\"\"\"F+F.\"\"#-F%
6$-%\"gG6#F+F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F[2] = \+
rhs(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"xG\"\"\"%\"yGF'\"
\"#*$F(F)\"\"$,&F%F)-%%diffG6$-%\"gG6#F(F(F'" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 37 "diff(g(y),y) = solve(\",diff(g(y),y));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"gG6#%\"yGF*,$*$F*\"\"#\"\"$" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn2 := subs(g(y)=y^3+C,e
qn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"uG6$%\"xG%\"yG,(*
&,&*$F*\"\"#\"\"\"\"\"(F0F0F)F0F0*$F*\"\"$F0%\"CGF0" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 57 "subs( \{x=2,y=1\}, rhs(eqn2)) - subs(\{x=
0,y=1\}, rhs(eqn2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#;" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 263 17 "13.  \+
Calculate   " }{XPPEDIT 347 1 "int( (y+z)*dx+(x+1)*dy+(2*z+x),z=C..``)
" "6#-%$intG6$,(*&,&%\"yG\"\"\"%\"zGF*F*%#dxGF*F**&,&%\"xGF*F*F*F*%#dy
GF*F*,&*&\"\"#F*F+F*F*F/F*F*/F+;%\"CG%!G" }{TEXT 348 11 "    where  " 
}{TEXT 349 1 "C" }{TEXT 350 42 "   is the oriented curve parameterized
 by " }{XPPEDIT 352 1 "r(t)=[sqrt(t),cos(Pi*sqrt(t)),t*ln(t^2-t+1)]" "
6#/-%\"rG6#%\"tG7%-%%sqrtG6#F'-%$cosG6#*&%#PiG\"\"\"-F*6#F'F1*&F'F1-%#
lnG6#,(*$F'\"\"#F1F'!\"\"F1F1F1" }{TEXT 351 8 ",       " }{XPPEDIT 
353 1 "0 <=t" "6#1\"\"!%\"tG" }{TEXT 278 1 " " }{XPPEDIT 559 1 "``<=1
" "6#1%!G\"\"\"" }{TEXT 558 1 " " }{TEXT 560 6 ".   \n\n" }{TEXT 266 
72 "a)  1           b)  -1             c)   2          d)  -2         \+
  e)  " }{TEXT 356 6 "3    \n" }{TEXT 279 35 "f)  -3            g)  4 \+
           " }{TEXT 561 1 " " }{TEXT 357 38 "h)  -4           i) 5    \+
         j)  " }{TEXT 562 2 "-5" }{TEXT -1 2 "  " }}{PARA 3 "" 0 "" 
{TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT 354 8 "Solution" }{TEXT 355 6 ":
 (f)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "r := (t) -> [sqrt(
t), cos(Pi*sqrt(t)),t*ln(t^2-t+1)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"rGf*6#%\"tG6\"6$%)operatorG%&arrowGF(7%-%%sqrtG6#9$-%$cosG6#*&%#P
iG\"\"\"F-F6*&F0F6-%#lnG6#,(*$F0\"\"#F6F0!\"\"F6F6F6F(F(6\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "P := r(0);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"PG7%\"\"!\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "Q := r(1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"QG7
%\"\"\"!\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "F := vector([y+z,x+1,2*z+x]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"FG-%'VECTORG6#7%,&%\"yG\"\"\"%\"zGF+,&%\"xGF+F+F+,&
F,\"\"#F.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "curl(F,[x,y,
z]); #F is conservative" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6
#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "eqn := u(x,
y,z) = int(y+z,x) + g(y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG
/-%\"uG6%%\"xG%\"yG%\"zG,&*&,&F*\"\"\"F+F/F/F)F/F/-%\"gG6$F*F+F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "diff(eqn,y);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"uG6%%\"xG%\"yG%\"zGF+,&F*\"\"\"-F%6
$-%\"gG6$F+F,F+F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x + 1 \+
= rhs(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"F&F&,&F%F&
-%%diffG6$-%\"gG6$%\"yG%\"zGF.F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "diff(g(y,z),y) = solve(\",diff(g(y,z),y));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"gG6$%\"yG%\"zGF*\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn2 := subs(g(y,z)=y+h(z),e
qn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/-%\"uG6%%\"xG%\"yG%\"
zG,(*&,&F*\"\"\"F+F/F/F)F/F/F*F/-%\"hG6#F+F/" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 26 "2*z+x = diff(rhs(eqn2),z);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/,&%\"zG\"\"#%\"xG\"\"\",&F'F(-%%diffG6$-%\"hG6#F%F%F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "diff(h(z),z) = solve( \+
\" , diff(h(z),z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%\"
hG6#%\"zGF*,$F*\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn
3 := subs(h(z)=z^2 + C, eqn2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e
qn3G/-%\"uG6%%\"xG%\"yG%\"zG,**&,&F*\"\"\"F+F/F/F)F/F/F*F/*$F+\"\"#F/%
\"CGF/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "evalm( grad(rhs(e
qn3),[x,y,z]) - F );  #Check" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VEC
TORG6#7%\"\"!F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "subs( \+
\{x=1,y=-1,z=0\}, rhs(eqn3)) - subs(\{x=0,y=1,z=0\}, rhs(eqn3));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 264 49 "14.  What  is the surface area o
f the part of    " }{XPPEDIT 566 1 "z=1+x^2/2+y^2/2" "6#/%\"zG,(\"\"\"
F&*&%\"xG\"\"#F)!\"\"F&*&%\"yGF)F)F*F&" }{TEXT 563 35 "   that lies ov
er the unit disk    " }{XPPEDIT 565 1 "x^2+y^2<=1" "6#1,&*$%\"xG\"\"#
\"\"\"*$%\"yGF'F(F(" }{TEXT 564 10 "  ? \n\na)  " }{XPPEDIT 582 1 "sqr
t(2)/3*Pi" "6#*(-%%sqrtG6#\"\"#\"\"\"\"\"$!\"\"%#PiGF(" }{TEXT 568 22 
"                  b)  " }{XPPEDIT 583 1 "(sqrt(2)-1)/3*Pi" "6#*(,&-%%
sqrtG6#\"\"#\"\"\"F)!\"\"F)\"\"$F*%#PiGF)" }{TEXT 569 11 "        c) \+
" }{XPPEDIT 584 1 "(2-sqrt(2))/3*Pi" "6#*(,&\"\"#\"\"\"-%%sqrtG6#F%!\"
\"F&\"\"$F*%#PiGF&" }{TEXT 572 12 "         d) " }{XPPEDIT 585 1 "4*sq
rt(2)/3*Pi" "6#**\"\"%\"\"\"-%%sqrtG6#\"\"#F%\"\"$!\"\"%#PiGF%" }
{TEXT 573 20 "                e)  " }{XPPEDIT 586 1 "2*(2*sqrt(2)-1)*P
i/3" "6#**\"\"#\"\"\",&*&F$F%-%%sqrtG6#F$F%F%F%!\"\"F%%#PiGF%\"\"$F+" 
}{TEXT 570 14 "             \n" }{TEXT 567 4 "f)  " }{XPPEDIT 587 1 "(
4*(sqrt(2)-1))/3*Pi" "6#**\"\"%\"\"\",&-%%sqrtG6#\"\"#F%F%!\"\"F%\"\"$
F+%#PiGF%" }{TEXT 578 3 "   " }{TEXT 579 6 "  g)  " }{XPPEDIT 588 1 "2
*(4-sqrt(2))/3*Pi" "6#**\"\"#\"\"\",&\"\"%F%-%%sqrtG6#F$!\"\"F%\"\"$F+
%#PiGF%" }{TEXT 580 3 "   " }{TEXT 581 8 "   h)   " }{TEXT 642 2 "  " 
}{XPPEDIT 643 1 "(4-sqrt(2))*Pi/3" "6#*(,&\"\"%\"\"\"-%%sqrtG6#\"\"#!
\"\"F&%#PiGF&\"\"$F+" }{TEXT 641 2 "  " }{TEXT 644 6 "   i) " }{TEXT 
589 1 " " }{XPPEDIT 590 1 "2*(2-sqrt(2) )*Pi/3" "6#**\"\"#\"\"\",&F$F%
-%%sqrtG6#F$!\"\"F%%#PiGF%\"\"$F*" }{TEXT 574 2 "  " }{TEXT 575 7 "   \+
j)  " }{TEXT 591 2 "  " }{XPPEDIT 592 1 "4*(2*sqrt(2)-1)*Pi/3" "6#**\"
\"%\"\"\",&*&\"\"#F%-%%sqrtG6#F(F%F%F%!\"\"F%%#PiGF%\"\"$F," }{TEXT 
571 2 "  " }}{PARA 262 "" 0 "" {TEXT 297 5 "     " }}{PARA 3 "" 0 "" 
{TEXT 576 8 "Solution" }{TEXT 577 6 ": (e)\n" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "2*Pi*int(r*(1+r^2)
^(1/2),r=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%#PiG\"\"\",&*$
\"\"##F&F)#F)\"\"$#!\"\"F,F&F&F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 194 "answers := [sqrt(2)/3*Pi,(sqrt(2)-1)/3*Pi,(2-sqrt(2)
)/3*Pi,4*sqrt(2)/3*Pi,2*(2*sqrt(2)-1)*Pi/3,(4*(sqrt(2)-1))/3*Pi,2*(4-s
qrt(2))/3*Pi,(4-sqrt(2))*Pi/3,2*(2-sqrt(2) )*Pi/3,4*(2*sqrt(2)-1)*Pi/3
]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "for i from 1 to 9 do
\nfor j from i+1 to 10 do\nif (answers[i]=answers[j]) then print(`repe
at`);\nfi;\nod;\nod;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "int(int(sqrt(1+r^2
)*r,r=0..1),t=0..2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$-F
$6$*&,&\"\"\"F**$%\"rG\"\"#F*#F*F-F,F*/F,;\"\"!F*/%\"tG;F1,$%#PiGF-" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "int(r*sqrt(1+r^2),r=0..1);
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(#F(F+F*F(/F*;\"\"!F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ev
alf(Int(r*sqrt(1+r^2),r=0..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+#3dZ4'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "testeq(diff((
1+r^2)^(3/2)/3,r) = sqrt(1+r^2)*r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "subs(r=1,(1+r^2
)^(3/2)/3) - subs(r=0,(1+r^2)^(3/2)/3);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&*$\"\"##\"\"\"F%#F%\"\"$#!\"\"F)F'" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "evalf( \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"
+\"3dZ4'!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT 280 15 "15.  Calculate " }{XPPEDIT 388 1 "Int(
-5*y*dx+11*x,y=C..``)" "6#-%$IntG6$,&*(\"\"&\"\"\"%\"yGF)%#dxGF)!\"\"*
&\"#6F)%\"xGF)F)/F*;%\"CG%!G" }{TEXT 387 8 "  where " }{TEXT 385 1 "C
" }{TEXT 386 48 " is the unit circle traversed counter-clockwise." }
{TEXT 384 2 " \n" }{TEXT 373 4 "a)  " }{XPPEDIT 389 1 "0" "6#\"\"!" }
{TEXT 374 22 "                  b)  " }{XPPEDIT 390 1 "Pi" "6#%#PiG" }
{TEXT 375 23 "                  c)   " }{XPPEDIT 391 1 "2*Pi" "6#*&\"
\"#\"\"\"%#PiGF%" }{TEXT 376 20 "                d)  " }{XPPEDIT 392 
1 "3*Pi" "6#*&\"\"$\"\"\"%#PiGF%" }{TEXT 377 22 "                 e)  \+
 " }{XPPEDIT 393 1 "4*Pi" "6#*&\"\"%\"\"\"%#PiGF%" }{TEXT 378 11 "    \+
 \nf)   " }{XPPEDIT 394 1 "6*Pi" "6#*&\"\"'\"\"\"%#PiGF%" }{TEXT 379 
18 "             g)   " }{XPPEDIT 395 1 "8*Pi" "6#*&\"\")\"\"\"%#PiGF%
" }{TEXT 380 19 "               h)  " }{XPPEDIT 396 1 "12*Pi" "6#*&\"#
7\"\"\"%#PiGF%" }{TEXT 381 19 "               i)  " }{XPPEDIT 397 1 "1
6*Pi" "6#*&\"#;\"\"\"%#PiGF%" }{TEXT 382 22 "                 j)   " }
{XPPEDIT 398 1 "18*Pi" "6#*&\"#=\"\"\"%#PiGF%" }{TEXT 383 5 "    \n" }
}{PARA 3 "" 0 "" {TEXT 399 8 "Solution" }{TEXT 400 6 ": (i)\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r := t -> [cos(t),sin(t)]:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Int( -5*r(t)[2]*diff(r(t)
[1],t) + 11*r(t)[1]*diff(r(t)[2],t),t=0..2*Pi);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$,&*$-%$sinG6#%\"tG\"\"#\"\"&*$-%$cosGF*F,\"#6/
F+;\"\"!,$%#PiGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(
\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"#;" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "More easily by Green's \+
Theorem:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Int(Int( r*(diff(11*x
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{XPPMATH 20 "6#-%$IntG6$-F$6$,$%\"rG\"#;/F);\"\"!\"\"\"/%&thetaG;F-,$%
#PiG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,$%#PiG\"#;" }}}{PARA 0 "" 0 "" {TEXT 
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403 1 "G" }{TEXT -1 57 "  be a solid region in space. Suppose that the
 boundary  " }{TEXT 402 1 "S" }{TEXT -1 6 "  of  " }{TEXT 404 1 "G" }
{TEXT -1 59 "  is an orientable closed surface with outward unit norma
l " }{TEXT 401 1 "n" }{TEXT -1 46 ". Vector calculus tells us that\n\n
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::::::::::::::::5:" }{TEXT 405 16 " \nthen what is  " }{TEXT 407 2 "f \+
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}{TEXT 412 71 "a)  0           b)  1             c)   2          d)  3
           e)  4" }{TEXT 414 3 "   " }{TEXT 413 35 "\nf)  5           \+
 g)  6            " }{TEXT -1 1 " " }{TEXT 415 38 "h)  7           i) \+
8             j)  9" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 3 "" 0 "" {TEXT 416 8 "Solution" }{TEXT 417 6 ": (e)\n" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "diverg
e([x*y,2*y*z,x^2+x^2*y^3-y*z],[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$%\"zG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "subs(
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{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 300 20 
"17. Suppose that    " }{XPPEDIT 333 1 "f(x,y,z)=3*x^2 - 2*y^2 + 3*z^3
" "6#/-%\"fG6%%\"xG%\"yG%\"zG,(*&\"\"$\"\"\"*$F'\"\"#F-F-*&F/F-*$F(F/F
-!\"\"*&F,F-*$F)F,F-F-" }{TEXT 332 14 " .  Calculate " }{TEXT 336 1 " \+
" }{TEXT 334 11 "div(grad(f(" }{TEXT 337 5 "x,y,z" }{TEXT 338 23 ")) a
t the point (3,2,1)" }{TEXT 335 2 ". " }{TEXT 301 2 " \n" }{TEXT 299 
142 "\na)  1           b)  2            c)  3           d)  4         \+
  e) 5 \nf)   8          g)   12         h)  15          i)  16       \+
   j)  20" }{TEXT 411 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
3 "" 0 "" {TEXT 339 8 "Solution" }{TEXT 340 6 ": (j)\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "f := (x,y,z) -> 3*x^2 - 2*y^2 + 3*z
^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%%\"xG%\"yG%\"zG6\"6$%
)operatorG%&arrowGF*,(*$9$\"\"#\"\"$*$9%F1!\"#*$9&F2F2F*F*6\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grad(f(x,y,z),[x,y,z]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%,$%\"xG\"\"',$%\"yG!\"%,
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ge(\",[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"#\"\"\"%\"zG
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{TEXT 302 38 "18. Calculate the surface integral of " }{TEXT 305 1 " \+
" }{XPPEDIT 306 1 "f(x,y,z)=2*x+z" "6#/-%\"fG6%%\"xG%\"yG%\"zG,&*&\"\"
#\"\"\"F'F-F-F)F-" }{TEXT 304 31 "  over that part of  the plane " }
{TEXT 307 1 " " }{XPPEDIT 308 1 "2*x+2*y+z=2" "6#/,(*&\"\"#\"\"\"%\"xG
F'F'*&F&F'%\"yGF'F'%\"zGF'F&" }{TEXT 303 4 "   \n" }{TEXT 328 30 "that
 lies in the first octant." }{TEXT 329 1 "\n" }{TEXT 600 1 "\n" }
{TEXT 599 144 "\na)  1           b)  2            c)  3           d)  \+
4          e)  5 \nf)   6           g)  7           h)  8            i
)  9           j)  10" }{TEXT 601 1 "\n" }}{PARA 3 "" 0 "" {TEXT -1 0 
"" }}{PARA 3 "" 0 "" {TEXT 330 8 "Solution" }{TEXT 331 6 ": (b)\n" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
13 "with(linalg):" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 55 "vectorLength := v -> sqrt(sum(v[i]^2,i=1..ve
ctdim(v)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-vectorLengthGf*6#%\"
vG6\"6$%)operatorG%&arrowGF(-%%sqrtG6#-%$sumG6$*$&9$6#%\"iG\"\"#/F6;\"
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64 "r := (x,y) -> vector([x,y,1-2*(x+y)]); #surface parameterization" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rGf*6$%\"xG%\"yG6\"6$%)operatorG
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rx := map(z->diff(z,x),r(x,y));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#rxG-%'VECTORG6#7%\"\"\"\"\"!!\"#" }
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,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ryG-%'VECTORG6#7%\"\"!\"\"
\"!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "crossprod(rx,ry);
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "surfaceAreaFactor := vectorL
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\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := (x,y,z) -> 2*x \+
+ z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6%%\"xG%\"yG%\"zG6\"6$
%)operatorG%&arrowGF*,&9$\"\"#9&\"\"\"F*F*6\"" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 53 "Int(Int(f(x,y,z)*surfaceAreaFactor,y=0..1-x),x=0
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zG\"\"$/%\"yG;\"\"!,&\"\"\"F2F)!\"\"/F);F0F2" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "subs(z=2-2*(x+y),\");" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%$IntG6$-F$6$,&\"\"'\"\"\"%\"yG!\"'/F+;\"\"!,&F*F*%\"x
G!\"\"/F1;F/F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(\");
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3 "U  " }{TEXT 603 97 " be the cube with equal sides of length one and
 vertices at ( 0, 0, 0 )  and  ( 1, 1, 1 ).  Let  " }{TEXT 315 1 "S" }
{TEXT 316 63 "  be the surface of that part of the cube that lies abov
e the  " }{XPPEDIT 605 1 "xy" "6#%#xyG" }{TEXT 604 27 " plane.  (In ot
her words,  " }{TEXT 606 1 "S" }{TEXT 607 53 "  has the shape of a box
 without its bottom.)   Let  " }{TEXT 319 1 "n" }{TEXT 320 46 "  denot
e the outward-pointing unit normal on  " }{TEXT 317 1 "S" }{TEXT 318 
21 " .  Calculate  \n\n    " }{BITMAP 169 43 43 1 "?TMOVOB:n@>?FOV?F:k
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:::::::::J:3:" }{TEXT 610 4 "  \n\n" }{TEXT 311 1 "a" }{TEXT 309 154 "
)  1             b)  -1             c)  2            d)  -2           \+
e)  3 \nf)   -3           g)   4             h)  -4            i)  5  \+
           j)  " }{TEXT 321 5 "-5\n  " }}{PARA 3 "" 0 "" {TEXT 322 8 "
Solution" }{TEXT 323 6 ": (a)\n" }}{PARA 0 "" 0 "" {TEXT -1 91 "Using \+
Stokes's Theorem the required surface integral can be calculated by th
e line integral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 3 "   " }{TEXT 324 5 "     " }{XPPEDIT 325 1 "Int(x*y*z^4*dx+
2*x*y*dy+y^3,z)" "6#-%$IntG6$,(**%\"xG\"\"\"%\"yGF)%\"zG\"\"%%#dxGF)F)
**\"\"#F)F(F)F*F)%#dyGF)F)*$F*\"\"$F)F+" }}{PARA 0 "" 0 "" {TEXT -1 
86 "\nintegrated counterclockwise over the boundary of the square in t
he xy plane. Since   " }{TEXT 326 10 "z = dz = 0" }{TEXT -1 24 "  on a
ll four sides and " }{TEXT 327 6 "dy = 0" }{TEXT -1 48 " on the two si
des parallel to the x-axis we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "int(2*y,y=0..1);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 263 "" 0 "" {TEXT 312 
11 "20.  Let   " }{TEXT 611 3 "U  " }{TEXT 612 144 " be the cube with \+
equal sides of length one and vertices at ( 0, 0, 0 )  and  ( 1, 1, 1 \+
)  (as in the preceding problem). In this problem, let  " }{TEXT 613 
1 "S" }{TEXT 614 51 "  be the entire surface of the cube. \n\n        \+
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::::::::::::::::::<5:" }{TEXT 615 12 "          \n\n" }{TEXT 618 1 "a
" }{TEXT 617 154 ")  1             b)  -1             c)  2           \+
 d)  -2           e)  3 \nf)   -3           g)   4             h)  -4 \+
           i)  5             j)  " }{TEXT 619 3 "-5\n" }{TEXT 620 1 "
\n" }}{PARA 263 "" 0 "" {TEXT 616 2 "\n\n" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 0 
"" }}{PARA 3 "" 0 "" {TEXT 313 8 "Solution" }{TEXT 314 6 ": (f)\n" }}
{PARA 0 "" 0 "" {TEXT -1 28 "Using the Divergence Theorem" }}{PARA 0 "
" 0 "" {TEXT -1 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wit
h(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "F := vector([
y*z^5,12*z^3*y,y^3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%'VECT
ORG6#7%*&%\"yG\"\"\"%\"zG\"\"&,$*&F,\"\"$F*F+\"#7*$F*F0" }}}{PARA 3 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "p := \+
diverge(F,[x,y,z]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG,$*$%\"zG
\"\"$\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "int(int(int(p,
z=0..1),y=0..1),x=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"$" }}}
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0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "5" 
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