# Calculate the Jacobian matrix $\frac{\partial(x, y)}{\partial(s, t)}(-1,2)$

Consider the next transformations $$S:\left\{\begin{array}{ll} x=u^{2}-v^{2}\\y=3uv & and \quad T:\left\{\begin{array}{l} u=t^{2}-s+s t \\ v=s^{2}-\frac{2 s}{t}-3 \end{array}\right. \end{array}\right.$$ Calculate the Jacobian matrix $$\frac{\partial(x, y)}{\partial(s, t)}(-1,2)$$

What should I do?

do $$x=(t^2-s+st)^{2}-(s^2-\frac{2s}{t}-2)^2$$ and

$$y=3(t^2-s+st)(s^2-\frac{2s}{t}-2)^2$$

and then build the matrix $$2\times2$$ with the partials respect $$s$$ and $$t$$ of $$x=(t^2-s+st)^{2}-(s^2-\frac{2s}{t}-2)^2$$ and $$y=3(t^2-s+st)(s^2-\frac{2s}{t}-2)^2$$ ?

what I mean

$$$$\begin{pmatrix} \frac{\partial x}{\partial s} & \frac{\partial y}{\partial s}\\ \frac{\partial x}{\partial t} & \frac{\partial y}{\partial t} \end{pmatrix}$$$$ in $$(-1,2)=$$

$$$$\begin{pmatrix} 0 & 17\\ -30 & \frac{-27}{2} \end{pmatrix}$$$$

is right?

$$\frac{\partial (x,y)}{\partial (s,t)} = \frac{\partial (x,y)}{\partial (u,v)}\frac{\partial (u,v)}{\partial (s,t)} = \begin{bmatrix} \frac{\partial x}{\partial u} \frac{\partial u}{\partial s} + \frac{\partial x}{\partial v} \frac{\partial v}{\partial s} & \frac{\partial x}{\partial u} \frac{\partial u}{\partial t} + \frac{\partial x}{\partial v} \frac{\partial v}{\partial t} \\ \frac{\partial y}{\partial u} \frac{\partial u}{\partial s} + \frac{\partial y}{\partial v} \frac{\partial v}{\partial s} & \frac{\partial y}{\partial u} \frac{\partial u}{\partial t} + \frac{\partial y}{\partial v} \frac{\partial v}{\partial t}\end{bmatrix}\; .$$