Take the tour ×  Let $X:\mathbb{R}^3\rightarrow T\mathbb{R}^3$
be a vector field, what is the definition of its dual ? I know
that the set of vector fields on $\mathbb{R}^3$ forms an
$\mathbb{R}$-vector space.  share|improve this question add
comment  1 Answer  What you want is probably the metric dual. If
$g$ is a Riemannian metric on $\mathbb{R}^3$ the metric dual is
the 1-form $\alpha=g(X,\cdot)$, which is a section
$\alpha:\mathbb{R}^3\to T^* \mathbb{R}^3$ of the cotangent
bundle. When $g$ is the standard metric (i.e. the dot product
$g(v,w)=v\cdot w$ from vector calculus) this is the map induced
by $\mathbf{i}\to dx$, $\mathbf{j}\to dy$, $\mathbf{k}\to dz$ in
each fiber.  share|improve this answer add comment  Your Answer