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Let $d$ denote the Cartan differential, and let $\delta$ denote the codifferential. The underlying domain is not important for what follows.

The canonical generalization of the Laplace-operator $-\tilde\Delta = \operatorname{grad}\operatorname{div}$ is the Hodge Laplacian $\Delta = d\delta + \delta d$, such that we can regard $\Delta u = f$ as generalized second-order partial differential equation.

I would like to learn, which is the "correct" generalization of first-order partial differentials equations. The canonical candidate for this seems to be the Lie-differential

$\mathcal L_X(\alpha) = d i_X \alpha + i_X d \alpha$

where $i_X$ denotes the interior product. Still, I do not see why this should be natural, and I would learn to learn whether there are alternative first-order formulations in the language of differential geometry available (which are not a specialization of the Lie-differential). Maybe someone can help me to fill this gap.

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A vector field is itself already a differential operator. Why needlessly complicate things by introducing interior products? –  Zhen Lin Feb 22 '12 at 22:49
    
Because that is the way the Lie differential is defined. –  shuhalo Mar 1 '12 at 18:22
    
Where have you heard the term "Cartan differential"? A google search results in a hit in exactly one reference, a work of E. Zeidler. –  Christopher A. Wong Jul 9 at 7:06

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