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 $$x=u\qquad y=v\qquad z=5-u-v^{2}$$

Figure 43 $\mathbf{r}(u,v) =u\mathbf{i}+v\mathbf{j}+( 5-u-v^{2}) \mathbf{k},\\ 0\leq u\leq 1, -1\leq v\leq 1$

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