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I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple.

  1. Can some one tell me some reference to study about the invertibility of Divergence operator $\operatorname{div}\colon C^1(\omega)\to G$ where $G$ is a space of real valued function and $\omega$ is a subset of $\Bbb R^2$. Here I assume a Dirichlet type condition on the boundary of $\omega$ is specified and all boundary and domain have nice smoothness.
  2. In above context can someone give me some reference on the Null space structure of the divergence operator operating on differentiable maps defined on $\Bbb R^2$ ?

Ariwn

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    "curl"? There is only such a thing (as a map from vector fields to vector fields) in $n=3$.2012-09-06
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    By curl you mean the exterior derivative acting on 1-form? And why should $G$ be a space of vector valued function?2012-09-06
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    Divergence takes vector fields to scalar fields.2012-09-06
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    Sorry for my typo error and horrible english.I mean it maps vector valued maps to scalar valued and the curl of a map case in question 2 was for $R^3 (poincare type )$ but , here I need in $R^2$2012-09-06
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    Perhaps you want the analogue of [Helmholtz decomposition](https://en.wikipedia.org/wiki/Helmholtz_decomposition) in two dimensions. I think this is dealt with in fluid mechanics in terms of the stream function and velocity potential, but I'm afraid I don't have a reference handy.2012-09-06

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