Question

what is the difference between a parametric surface and vector field?

Answer 1

parametric tends to mean that the coordinates are defined independantly.. not sure about a vector field tho

Answer 2

The two things are completely different. A parametric surface is a two-dimensional differentiable submanifold embedded in the three dimensional Euclidian space. A vector field on the other hand is a lifting from an arbitrary manifold into its tangent bundle.

Answer 3

i had my suspicions :)

Answer 4

can you explain that in simpler terms please? Sorry!

Answer 5

parametric is draping a towel over a beach ball..... not sure what the other means :/

Answer 6

maybe stretching a rubber plane up into 3d space?

Answer 7

...blowing up a balloon is what I get out of it :)

Answer 8

Tangent bundle means for every point of the manifold (e.g. a surface) you a vector space that has the same dimension as the manifold. Those vector spaces (one tangent space for each point) can geometrically be seen as the space of directions that curves over the manifold through that point can have. Then a vector field means for every point of the manifold you select one element of the tangent bundle and that element should lie in the tangent space corresponding to the point of the manifold. So basically you on every point of the manifold you choose a certain direction.