How to convert $r(t,s)=\langle t+s,t-s,t^2+s^2\rangle$ to Cartesian coordinates?

Question

Convert $r(t,s)=\langle t+s,t-s,t^2+s^2\rangle$ to Cartesian coordinates?

I really have no idea how to do this. I tried using sphere coordinates but that didn't lead to anywhere. I checked online that the surface is actually a paraboloid.

Answer 1

If $x=t+s$ and $y=t-s$, you can solve these as a system of equations for $t$ and $s$. This gives $t=\frac12(x+y)$ and $s=\frac12(x-y)$. Then you can substitute these into $z=t^2+s^2$ to get $z$ as a function of $x$ and $y$.

Answer 2

I don't understand the question? Is it this?

If $(x,y,z)=(t+s,t-s,t^2+s^2)$ then write $z$ as an explicit function of $x$ and $y$.

If so then $$z=\frac{x^2+y^2}{2}.$$