What is the difference between a vector field and vector space?

[harjyot]

I'm unable to understand this generalization of vectors from a quality having a magnitude and direction, to the more mathematical approach.
what is the difference between vector space and vector field? more of an intuitive example?

 

[D H]

A vector space V over a field F is a mathematical space that obeys some very simple and generic requirements. (A space is a set with some additional structure; a field is (oversimplified) a set for which addition, subtraction, multiplication, and division are defined.) Elements of the space V are called vectors. The requirements on a vector space are

• There is a commutative and associative operation "+" by which two element of the space can be added to form another element in the space.
• There exists a special member of the set V, the zero vector $\vec 0$, such that $\vec v + \vec 0 = \vec 0 + \vec v = \vec v$ for all members $\vec v$ in V.
• For every vector $\vec v$ in V there exists another vector $-\vec v$ such that $\vec v + -\vec v = \vec 0$
• Multiplication by a scalar: Every member of the vector space V can be scaled (multiplied) by a member of the field F, yielding a member of the space.
• Scaling is consistent. Scaling any element $\vec v$ in the vector space V by the multiplicative identity 1 of F yields the vector $\vec v$, and [itex]a(b\vec v) = (ab)\vec v[/tex] for any scalars a and b and any vector v.

That's all there is to vector spaces. Nothing about magnitude, nothing about direction (or the angle between two vectors). That requires something extra, the concept of a norm for magnitude, of an inner product for angle.


A vector field is something different from a vector space. Let's start with the concept of a function. A function is something that maps members of one space to members of some other space. If that other space is a vector space, well, that's a vector field.

 

[mathwonk]

given a point p on a sphere, the set of all arrows starting from p and tangent to the sphere, forms a vector space, the space of all tangent vectors to S at p.

Each point of the sphere has its own tangent space, and the family of all these vector spaces is called a bundle of vector spaces.

If we choose one tangent vector at each point of the sphere, this collection of vectors, one from each vector space in the bundle, is called a (tangent) vector field, on the sphere.

so a vector field occurs when you have a collection of vector spaces, and it means you choose one vector from each space.

so a vector field is analogous to a vector. I.e. a vector bundle is a collection of vector spaces, and a vector field is a collection of vectors, one from each space in the bundle.

a vector is a choice of one element of a vector space, and if you have a collection of vector spaces, and you choose one element from each space, that is a vector field. so a vector bundle is a family of vector spaces, and a vector field is a family of vectors.

 

[HallsofIvy]

A "vector field" is a function that assigns a vector at each point of a set, usually a manifold or smooth subset of Rⁿ. In order that we have the concept of a vector at each point, we must have a vector space defined at each point, typically, though not necessarily, the "tangent space" to the manifold at that point. The assemblage of a manifold together with a vector space at each point is a "vector bundle", specifically the "tangent bundle" if the vector space is the tangent space.

 

[danzibr]

Right, it's already been said, but in short, heuristically speaking a vector space is a set equipped with an underlying field and two operations, while a vector field is a vector-valued function.