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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)?

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    On wikipedia: http://en.wikipedia.org/wiki/Pushforward_(differential). In general, a good place to start would be any of the older textbooks on tensor calculus.2011-09-09
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    Thanks for pointing out where to start. Since I have absolutely no background in differential geometry yet, your advise helped me search for what I need. I'm having some problems with the notation, but I think I've got it.2011-09-11

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