## Proof That the Set of Conservative Fields is a Vector Space

Theorem
The set of conservative vector fields defined on a domain
$D$
is a vector space.
Proof
Let the set of conservative vector fields be represented by
$V$
then
$\mathbf{F}, \: \mathbf{G} \in V$
imply there exist functions
$f,g$
operating on elements of
$D$
such that
$\mathbf{F} = \mathbf{\nabla} f, \: \mathbf{G} = \mathbf{\nabla} g$
.
Then
$\mathbf{\nabla} (f+g) =\mathbf{F} + \mathbf{G} \in V$

Let
$\alpha \in \mathbb{R}$
then
$\mathbf{ \nabla} \alpha f = \alpha \mathbf{\nabla} f = \alpha \mathbf{F}$

The zero field is conservative, taking the value zero everywhere.
Hence the set of conservative vector fields is a vector space.