Suppose I have a parametric surface $\mathbf{r}(u,v)=x(u,v)\mathbf{\hat e}_x+y(u,v)\mathbf{\hat e}_y+z(u,v)\mathbf{\hat e}_z$, which is a function $\mathbb{R}^2 \rightarrow \mathbb{R}^3$. And I also have a vector field $\mathbf{A}(u,v)=A_x(u,v)\mathbf{\hat e}_x+A_y(u,v)\mathbf{\hat e}_y+A_z(u,v)\mathbf{\hat e}_z$, such as $\mathbb{R}^2 \rightarrow \mathbb{R}^3$.
What are the differences between a parametric surface and a vector field (or a vector function)? The notation is the same.
Thanks!
A vector field associates a single vector to each point; you have one vector based at each point.
For a parametric surface, all of the vectors are based at some origin. All of the vectors give the position vectors of the points on the surface relative to some origin.