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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.

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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.

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