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

In summary, a vector space is a mathematical space that follows certain requirements and has elements called vectors, while a vector field is a vector-valued function that assigns a vector at each point of a set. 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.
  • #1
harjyot
42
0
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?
 
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  • #2
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 [itex]\vec 0[/itex], such that [itex]\vec v + \vec 0 = \vec 0 + \vec v = \vec v[/itex] for all members [itex]\vec v[/itex] in V.
  • For every vector [itex]\vec v[/itex] in V there exists another vector [itex]-\vec v[/itex] such that [itex]\vec v + -\vec v = \vec 0[/itex]
  • 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 [itex]\vec v[/itex] in the vector space V by the multiplicative identity 1 of F yields the vector [itex]\vec v[/itex], 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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 

1. What is a vector field?

A vector field is a mathematical concept that assigns a vector to every point in a given space. This means that at each point in the space, there is a corresponding direction and magnitude represented by the vector. Vector fields are commonly used to represent physical quantities such as velocity, force, and electric fields.

2. What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and operations that can be performed on them, such as addition and multiplication. These operations follow specific rules, such as closure, associativity, and distributivity, and allow for the manipulation and combination of vectors to create new vectors.

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

The main difference between a vector field and a vector space is that a vector field is a function that maps points in a space to vectors, while a vector space is a mathematical structure that contains vectors and operations on those vectors. In other words, a vector field is a concept that describes the behavior of vectors in a specific space, whereas a vector space is a mathematical construct that defines the properties and rules of vectors.

4. Can a vector field exist without a vector space?

No, a vector field cannot exist without a vector space. A vector field is defined as a function that maps points in a space to vectors, meaning that it needs a space in which to assign these vectors. Without a vector space, there is no framework for the vectors to exist and be manipulated within.

5. How are vector fields and vector spaces used in science?

Vector fields and vector spaces are used in various fields of science, including physics, engineering, and computer science. In physics, vector fields are used to represent physical quantities, such as force and electric fields, while vector spaces are used to model the behavior of physical systems. In engineering, vector fields and spaces are used for calculations and simulations, and in computer science, they are used for data analysis and visualization.

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