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