Is a vector field a mathematical field?

Question

I would like to ask if a vector field is mathematical field which is defined to be a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

I can see that a vector field satisfies addition, subtraction, multiplication but I am not sure about division. Is this just terminology or is a vector field a field in the mathematical sense?

Answer 1

No, these are distinct concepts. A field (in Algebra) is what you think a field is. But a vector field is, roughly speaking, an assignment of a vector to each point in a space.

Answer 2

I believe by field they are referring to space(manifold), think of a paddy field.

Vector field is assigning a vector to each element of this field.

Answer 3

A vector field in differential geometry is a smooth section of the tangent bundle. A field in abstact algebra is a commutative ring in which every nonzero element is a unit. Two entirely different notions.