If $z = f ( x , y )$ Has Continuous Second Partial Derivatives

Essay

If $z = f ( x , y )$ has continuous second partial derivatives, $x = 2 u + v$ , and $y = u - 3 v$ , express $\frac { \delta ^ { 2 } z } { \delta u \delta v }$ in terms of $\frac { \delta ^ { 2 } z } { \delta x ^ { 2 } }$ , $\frac { \delta ^ { 2 } z } { \delta x \delta y }$ , and $\frac { \delta ^ { 2 } z } { \delta y ^ { 2 } }$ .