Verified answer

Hola

u = x/y

v = x y

Jacobiano

J(x;y/u;v) =

|....∂x/∂u....∂y/∂u....|

|....∂x/∂v....∂y/∂v....|

Multiplicamos y dividimos

para obtener x;y en función de u;v

v u = x^2

x = (v u)^(1/2)

x = u^(1/2) v^(1/2)

v / u = y^2

y = (v / u)^(1/2)

y = u^(-1/2) v^(1/2)

Derivadas parciales

∂x/∂u = (1/2) u^(-1/2) v^(1/2)

∂x/∂v = (1/2) u^(1/2) v^(-1/2)

∂y/∂u = (-1/2) u^(-3/2) v^(1/2)

∂y/∂v = (1/2) u^(1/2) v^(-1/2)

Jacobiano de x;y con respecto a u;v

J(x;y/u;v) =

|....(1/2) u^(-1/2) v^(1/2) ..(-1/2) u^(-3/2) v^(1/2)....|

|....(1/2) u^(1/2) v^(-1/2)....(1/2) u^(1/2) v^(-1/2)....|

Saludos