Chapter 16, Problem 26E

(a) Find an equation of the tangent plane at the point $(4,-2,1)$ to the parametric surface $S$ given by $\begin{array}{l}r(u,v)=v^{2}i-uvj+u^{2}k\\ 0⩽u⩽3,-3⩽v⩽3\end{array}$

(b) Graph the surface $S$ and the tangent plane found in part (a).

(c) Set up, but do not evaluate, an integral for the surface area of $S$ .

(d) If $F(x,y,z)=\frac{z^2}{1+x^2}i+\frac{x^2}{1+y^2}j+\frac{y^2}{1+z^2}k$ use a computer algebra system to find ${\displaystyle {\iint }_{S}F}\cdot dS$ correct to four decimal places.