Given ~A $=(2$xy $3)$x $+$ x$2$y$.$ Evaluate $$

C ~A $$ d~r along the two paths, blue and red:$($a$,0)$

$(0,$a$)$

$>$

$>$

$>$

$($a$,0)$

$(0,$b$)$

$>$

$>$

$>$

$>$

yAAAB$6$HicbVBNS$8$NAEJ$3$Ur$1$q$/$qh$69$LBbBU$0$mqoMeiF$48$t$2$FpoQ$9$lsJ$+3$azSbsboQS$+$gu$8$eFDEqz$/$Jm$//$GbZuDtj$4$YeLw$3$w$8$y$8$IBFcG$9$f$9$dgpr$6$xubW$8$Xt$0$s$7$u$3$v$5$B$+$fCoreNUMWyxWMSqE$1$CNgktsGW$4$EdhKFNAoEPgTj$25$n$/8$IRK$81$jem$0$mCfkSHkoecUWOl$5$qRfrrhVdw$6$ySrycVCBHo$1/+6$g$1$ilkYoDRNU$667$nJsbPqDKcCZyWeqnGhLIxHWLXUkkj$1$H$42$P$3$RKzqwyIGGsbElD$5$urviYxGWk$+$iwHZG$1$Iz$0$sjcT$//$O$6$qQmv$/$YzLJDUo$2$WJRmApiYjL$7$mgy$4$QmbExBLKFLe$3$EjaiijJjsynZELzll$1$dJu$1$b$1$Lqq$15$mWlfpPHUYQTOIVz$8$OAK$6$nAHDWgBA$4$RneIU$359$F$5$cd$6$dj$0$VrwclnjuEPnM$8$f$6$QuNAQ$==$

xAAAB$6$HicbVDLTgJBEOzFF$+$IL$9$ehlIjHxRHbRRI$9$ELx$4$hkUcCGzI$79$MLI$7$OxmZtZICF$/$gxYPGePWTvPk$3$DrAHBSvppFLVne$6$uIBFcG$9$f$9$dnJr$6$xubW$/$ntws$7$u$3$v$5$B$8$fCoqeNUMWywWMSqHVCNgktsGG$4$EthOFNAoEtoLR$7$cxvPaLSPJb$3$ZpygH$9$GB$5$CFn$1$Fip$/$tQrltyyOwdZJV$5$GSpCh$1$it$+$dfsxSyOUhgmqdcdzE$+$NPqDKcCZwWuqnGhLIRHWDHUkkj$1$P$5$kfuiUnFmlT$8$JY$2$ZKGzNXfExMaaT$2$OAtsZUTPUy$95$M$/$M$/$rpCa$89$idcJqlByRaLwlQQE$5$PZ$16$TPFTIjxpZQpri$9$lbAhVZQZm$03$BhuAtv$7$xKmpWyd$1$Gu$1$C$9$L$1$ZssjjycwCmcgwdXUIU$7$qEEDGCA$8$wyu$8$OQ$/$Oi$/$PufCxac$042$cwx$/4$Hz$+$AOeHjQA$=$

yAAAB$6$HicbVBNS$8$NAEJ$3$Ur$1$q$/$qh$69$LBbBU$0$mqoMeiF$48$t$2$FpoQ$9$lsJ$+3$azSbsboQS$+$gu$8$eFDEqz$/$Jm$//$GbZuDtj$4$YeLw$3$w$8$y$8$IBFcG$9$f$9$dgpr$6$xubW$8$Xt$0$s$7$u$3$v$5$B$+$fCoreNUMWyxWMSqE$1$CNgktsGW$4$EdhKFNAoEPgTj$25$n$/8$IRK$81$jem$0$mCfkSHkoecUWOl$5$qRfrrhVdw$6$ySrycVCBHo$1/+6$g$1$ilkYoDRNU$667$nJsbPqDKcCZyWeqnGhLIxHWLXUkkj$1$H$42$P$3$RKzqwyIGGsbElD$5$urviYxGWk$+$iwHZG$1$Iz$0$sjcT$//$O$6$qQmv$/$YzLJDUo$2$WJRmApiYjL$7$mgy$4$QmbExBLKFLe$3$EjaiijJjsynZELzll$1$dJu$1$b$1$Lqq$15$mWlfpPHUYQTOIVz$8$OAK$6$nAHDWgBA$4$RneIU$359$F$5$cd$6$dj$0$VrwclnjuEPnM$8$f$6$QuNAQ$==$

xAAAB$6$HicbVDLTgJBEOzFF$+$IL$9$ehlIjHxRHbRRI$9$ELx$4$hkUcCGzI$79$MLI$7$OxmZtZICF$/$gxYPGePWTvPk$3$DrAHBSvppFLVne$6$uIBFcG$9$f$9$dnJr$6$xubW$/$ntws$7$u$3$v$5$B$8$fCoqeNUMWywWMSqHVCNgktsGG$4$EthOFNAoEtoLR$7$cxvPaLSPJb$3$ZpygH$9$GB$5$CFn$1$Fip$/$tQrltyyOwdZJV$5$GSpCh$1$it$+$dfsxSyOUhgmqdcdzE$+$NPqDKcCZwWuqnGhLIRHWDHUkkj$1$P$5$kfuiUnFmlT$8$JY$2$ZKGzNXfExMaaT$2$OAtsZUTPUy$95$M$/$M$/$rpCa$89$idcJqlByRaLwlQQE$5$PZ$16$TPFTIjxpZQpri$9$lbAhVZQZm$03$BhuAtv$7$xKmpWyd$1$Gu$1$C$9$L$1$ZssjjycwCmcgwdXUIU$7$qEEDGCA$8$wyu$8$OQ$/$Oi$/$PufCxac$042$cwx$/4$Hz$+$AOeHjQA$=$

$(2)$

Should you expect the same result for both paths? The blue path traverses two sides of a square

with side a and the red path is an arc of a circle with radius a$.$ To evaluate the integral along the

red path you may want to change variables to x $$ a cos and y $$ a sin $,$ and use the result

$$

$2$

$0$

d$($cos $)$m$($sin $)$n $=[1+$m

$2][1+$n

$2]$

$2[2+$m$+$n

$2]$ F $[$m$,$ n$](3)$

where $[$n$]$ is the Gamma function. In particular, F $[3,0]=2/3,$ F $[1,2]=1/3,$ F $[0,1]=1.$