# Identify the surface with the given vector equation. r(s,t)=<s,t,t^2-s^2>

Identify the surface with the given vector equation.
$$\displaystyle{r}{\left({s},{t}\right)}={<}{s},{t},{t}^{{2}}-{s}^{{2}}{>}$$

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Archie Griffin
The idea here is to first write down the parametric equations of the given surface. Then we try to identify some rule or regularity between them to find out what the given surface is. The parametric equations are:
$$\displaystyle{x}={s}$$
$$\displaystyle{y}={t}$$
$$\displaystyle{z}={t}^{{2}}-{s}^{{2}}$$
The first thing one should notice is that the equality $$\displaystyle{z}={y}^{{2}}-{x}^{{2}}$$ holds for any point on the surface. But we know for the face that the given equation corresponds to a quadratic surface which is hyperbolic paraboloid.