Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be defined by $$f(x, y) = (\text{cos}(x)(\text{cos}(y)+4), \text{sin}(x)(\text{cos}(y) +4), \text{sin}(y)).$$
How would I go about computing the shape operator $S_\eta :T_pN \rightarrow T_pN$ for the outward pointing unit normal vector $\eta$ at a point $p = (3, 0, 0)$.
I computed $$f_x(x, y) = (-\text{sin}(x)(\text{cos}(y)+4), \text{cos}(x)(\text{cos}(y)+4), 0)$$ $$f_y(x, y) = (-\text{cos}(x)\text{sin}(y), -\text{sin}(x)\text{sin}(y), \text{cos}(y))$$ $$n(x, y) = \frac{f(x, y)}{\sqrt{8cos(y)+17}}$$
I don't exactly know how to put these all together, or if I'm even on the right track here.