Question about regular surface patches

Question

I’m studying Differential Geometry from Pressley’s book. Two questions:

1.- Shouldn’t the $U$ in the last sentence be $\Bbb{R^3}$?

2.- How does one obtain equation 1? Why is there a cross product on the right-hand side of equation $(1)$?

Answer 1

Let's just write it out, but using some less cumbersome notation.

Let $\vec V = a\vec v + b\vec w$ and $\vec W = c\vec v + d\vec w$, where $a,b,c,d$ are scalars. Then \begin{align*} \vec V\times\vec W &= (a\vec v + b\vec w)\times (c\vec v + d\vec w)\\ &=ac(\vec v\times \vec v) + ad(\vec v\times\vec w) + bc(\vec w\times\vec v) + bd(\vec w\times \vec w) \\ &= (ad-bc)(\vec v\times \vec w). \end{align*} We used only the basic algebraic properties of the cross product.

Answer 2

On your first question, you are right, instead of $U$ it should say $\Bbb{R^3}$.

On the second question, it is a common manipulation of differential forms. As you probably know, $\sigma_u \times \sigma_u=0$ and $\sigma_v \times \sigma_v=0$, you can use the traditional definition for the cross product in $\Bbb{R^3}$ of the determinant. Using this definition as well, you can see that $\sigma_u \times \sigma_v=-\sigma_v \times \sigma_u$. Now you only have to use these two facts and operate the left hand side in $(1)$.