Can ALL Vector Fields Be Expressed as a Product?

[yoghurt54]

Hey - I'm stuck on a concept:

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

i.e. Is there always a \varphi such that

\vec{A} = \varphi \vec{c} ?

for ANY field \vec{A}?

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.

 

[mathman]

What is the definition of vector field that you are using?

 

[yoghurt54]

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.
\vec{A}(x,y,z) = (x^2 - y^2, xz, y^3 + xz^2)

 

[mathman]

The answer to your original question is obviously no. Vector fields would have many different vectors which are not scalar multiples of each other.

 

[Apteronotus]

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).