Form Fields and Vector Fields do not form a Vector Space

Form fields or vector fields over a manifold (as opposed to forms and vectors) do not form a vector space. They form a module.

The difference is that the scalars of a vector space form an algebraic field while the scalars of a module form a ring. For us humans (as opposed to “those higher beings that I do not understand (a.k.a. mathematicians)”) this means that the scalars in the vector field can divide each other while the scalars in the spaces spanned by fields (i.e. a module) can not.

And just so we all can become even more confused: This has nothing to do with the fact that the “components” of each form field or vector field in certain basis are functions, i.e. themselves elements of a vector space with infinite number of dimensions.

The first way to see this module-not-a-vector-space characteristic is by showing directly that the scalars that form the “coordinate components” of a vector field can not always be divided, even if they are not identically zero. Take, for instance the, manifold \( \mathbb{R}^2\) with the polar coordinate system and look at the vector \(

\). The “scalars” are \( r\) and \( r\cos(\theta)\). Obviously we can not divide the former by the latter because it will be undefined at \( \theta=\frac{\pi}{2}+n\pi\).

Another, more amusing way to show that the space spanned by these fields is not a vector space is to explicitly show that a property expected from vector spaces is not fulfilled. Namely, that in \( n\) dimensions an \( n\)-uple of linearly independent elements forms a basis. However, in the case of fields over a manifold we can easily have a number of fields that are linearly independent over the manifold as a whole, and are at the same time linearly dependent (or simply equal to zero) on a subdomain. Hence, we have an \( n\)-uple of linearly independent fields that can not be linearly combined to represent another arbitrary field.