A parametrized surface describes a two-dimensional surface using two parameters:

$$\mathbf r(u,v)=(x(u,v),y(u,v),z(u,v)).$$

A region in the $(u,v)$ parameter plane is mapped onto the surface in space.

Tangent directions

Holding $v$ fixed and varying $u$ traces one family of curves on the surface; holding $u$ fixed and varying $v$ traces another. Their tangent vectors are

$$\mathbf r_u=\frac{\partial\mathbf r}{\partial u},\qquad \mathbf r_v=\frac{\partial\mathbf r}{\partial v}.$$

When these vectors are not parallel, they span the tangent plane to the surface.

Normal direction

The cross product

$$\mathbf r_u\times\mathbf r_v$$

is perpendicular to both tangent directions, so it provides an oriented normal to the surface.

Reversing the order of the cross product reverses the orientation:

$$\mathbf r_v\times\mathbf r_u =-(\mathbf r_u\times\mathbf r_v).$$

Local area scaling

A small rectangle $du,dv$ in parameter space becomes approximately a parallelogram on the surface. Its area is

$$dS=|\mathbf r_u\times\mathbf r_v|,du,dv.$$

Thus the same cross product supplies both the normal direction and the factor needed to convert parameter-area into actual surface-area.