a$)$ If $u=x^2-y^2$ and $v=xy,$ determine:

i$)$ the Jacobian, J

$(5$ marks$)$

ii$)\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{u}},\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{v}},\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{u}},\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{v}}.$

$(10$ marks$)$

b$)$ i$)$ Determine the location of the stationary point of the function

$z=x^2+y^2+2x-4y+10$

$(5$ marks$)$

ii$)$ Identify if it is a maximum, minimum or saddle point.

iii$)$ Determine stationary values of $z$ when existing.

$(7$ marks$)$

$(5$ marks$)$