Exercise $5.($a$)$ Consider the change of variables u$=$x$^(2)-$y$^(2),$v$=$x$+$y$.$ Calculate the Jacobian

factor $($del$($u$,$v$))/($del$($x$,$y$)).$

$($b$)$ Solve for x and y in terms of u and v$,$ and use this to calculate the inverse Jacobian factor

$($del$($x$,$y$))/($del$($u$,$v$)).$

$($c$)$ In general $($i$.$e$.$ not for the variables in part $($a$)),$ if x$=$x$($u$,$v$),$y$=$y$($u$,$v$)$ is a change of

variables, and u$=$u$($x$,$y$),$v$=$v$($x$,$y$)$ is the inverse change of variables. Use chain rule to

show that

$[[($delx$)/($delu$),($delx$)/($delv$)],[($dely$)/($delu$),($dely$)/($delv$)]][[($delu$)/($delx$),($delu$)/($dely$)],[($delv$)/($delx$),($delv$)/($dely$)]]=[[1,0],[0,1]]$

$($d$)$ Using the identity

detAB$=$detAdetB

show that

$($del$($x$,$y$))/($del$($u$,$v$))*($del$($u$,$v$))/($del$($x$,$y$))=1$

You do not need to prove the determinant identity.

$($e$)$ Verify that the result in part $($d$)$ is consistent with your computations from parts $($a$)$ and $($b$).$

$($f$)$ Use the change of variables from part $($a$)$ to find the area of the region bounded by the

following four curves: the line x$+$y$=1,$ the line x$+$y$=2,$ the hyperbola x$^(2)-$y$^(2)=1,$ and

the hyperbola y$^(2)-$x$^(2)=1.$