Vector fields

  • Thread starter yoghurt54
  • Start date
  • #1
19
0

Main Question or Discussion Point

Hey - I'm stuck on a concept:

Are ALL vector fields expressable as the product of a scalar field [tex]\varphi[/tex] and a constant vector [tex]\vec{c}[/tex]?

i.e. Is there always a [tex]\varphi[/tex] such that

[tex]\vec{A}[/tex] = [tex]\varphi[/tex] [tex]\vec{c}[/tex] ?

for ANY field [tex]\vec{A}[/tex]?

I ask because there are some derivations from Stokes' theorem that follow from this idea, and I'm not sure these rules apply to all vector fields, because surely there are some vector fields that can't be expressed as such a product.
 

Answers and Replies

  • #2
mathman
Science Advisor
7,800
430
What is the definition of vector field that you are using?
 
  • #3
19
0
I'm not sure exactly what you mean, but my understanding of a vector field in this context is that it's a field in a coordinate system where each component is a function of the coordinates of that point, e.g.
[tex]\vec{A}(x,y,z) = (x^2 - y^2, xz, y^3 + xz^2)[/tex]
 
Last edited:
  • #4
mathman
Science Advisor
7,800
430
The answer to your original question is obviously no. Vector fields would have many different vectors which are not scalar multiples of each other.
 
  • #5
202
0
The vector field as you've described it would consist of a field of parallel vectors, each perhaps having a different length, as constituted by your scalar field phi. Clearly not all vector fields are of this type (ie. parallel).
 

Related Threads on Vector fields

  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
16
Views
4K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
6
Views
4K
Replies
3
Views
1K
Top