Compute the Jacobian of the Transformation T Given By $x = \frac { 1 } { \sqrt { 2 } } ( u - v )$

Essay

Compute the Jacobian of the transformation T given by $x = \frac { 1 } { \sqrt { 2 } } ( u - v )$ , $y = \frac { 1 } { \sqrt { 2 } } ( u + v )$ , and find the image of $S = \{ ( u , v ) \mid 0 \leq u \leq 1,0 \leq v \leq 1 \}$ under T.