The edges of a rectangular box are $10$ cm, $12$ cm, and $16$ cm. The measurements are accurate to within $1/2$ cm.

Consider the transformation ${\mathbb R}^2 \to {\mathbb R}^3$ given by $${\bf f}(x,y) = (x e^y + \cos(\pi y), 2x^2, x-e^y)$$

Consider the transformation ${\bf f}: {\mathbb R}^2 \to {\mathbb R}^3$ given by $${\bf f}(x,y) = (x^2 - y^2, 2xy, x^2 + y^2)$$ and the transformation${\bf g}: {\mathbb R}^3 \to {\mathbb R}^2$ given by $${\bf g}(u,v,w) = (u+v, v+w)$$