Consider two open subsets $\Omega, \Omega^{\prime}\subset \mathbb{R}^n$. Now consider a (volume preserving) diffeomorphism \begin{align} \varphi:\Omega^{\prime}\to\Omega; \alpha\to \varphi(\alpha) \end{align} If we have a (scalar) function $u\in C^1(\Omega)$, it will be transformed via $\tilde{u}=u\circ \varphi$. By differentiation, one can prove that the gradient of $u$ will be transformed according to \begin{align} (\nabla u)\circ \varphi=(\nabla \tilde{u})(D\varphi)^{-1} \end{align} where $D\varphi$ denotes the Jacobian matrix of $\varphi$ and $\nabla \tilde{u}$ is taken to be a row vector.
Now consider an arbitrary vector field $v$ in $\Omega$ (i.e. an electric field or a velocity field, etc.), which is not necessarily a gradient field of some function, how does this transform under $\varphi$? Does it transform the same way? And if so, how can I prove it (or where can I look it up)?