question gives
$\displaystyle x=t^2-s^2, y=ts,u=x,v=-y$
a) compute derivative matrix (jacobian matrix)
$\displaystyle f(u,v)=f(t^2-s^2,-(ts))$
$\displaystyle \vec{D}f(u,v) = \left[ \begin{array}{cc}2t&-2s\\-s&-t\end{array}\right]$
b) express (u,v) in terms of (t,s)
$\displaystyle (u,v) = (t^2-s^2, -(ts))$
c) calculate $\displaystyle \vec{D}(u,v)$
$\displaystyle \left[\begin{array}{cc}1&-1\end{array}\right] \left[\begin{array}{cc}2t&-2s\\s&t\end{array}\right] = \left[\begin{array}{cc}2t-s&-2s-t\end{array}\right]$
d) verify chain rule holds
$\displaystyle f(t^2-s^2,-(ts))$ not sure how to go about verifying this when im not sure if the rest is correct