Representing a Surface Parametrically

Find a parametrization of the surface \(S\) given by \[ x+y^{2}+z=5\qquad 0\leq x\leq 1\qquad -1\leq y\leq 1\]

Solution We begin by expressing \(S\) as an explicit function of \(x\) and \(y.\) \[ z=f(x,y) =5-x-y^{2} \]

Since \(z\) is a function of \(x\) and \(y,\) we let \(x=u\) and \(y=v.\) Then the parametric equations of \(S\) are \[ x=u\qquad y=v\qquad z=5-u-v^{2} \]

Using these parametric equations, we obtain \[ \mathbf{r}(u,v) =u\mathbf{i}+v\mathbf{j}+( 5-u-v^{2}) \mathbf{k}\qquad 0\leq u\leq 1\qquad -1\leq v\leq 1 \]