Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Helmholtz-Hodge Theorems: Unification of Integration and Decomposition
  Perspectives   We develop a Helmholtz-like theorem for differential forms in Euclidean space
$E_{n}$ using a uniqueness theorem similar to the one for vector fields. We
then apply it to Riemannian manifolds, $R_{n}$, which, by virtue of the
Schlaefli-Janet-Cartan theorem of embedding, are here considered as
hypersurfaces in $E_{N}$ with $N\geq n(n+1)/2$. We obtain a Hodge decomposition
theorem that includes and goes beyond the original one, since it specifies the
terms of the decomposition. We then view the same issue from a perspective of
integrability of the system ($d\alpha =\mu ,$ $\delta \alpha =\nu $), relating
boundary conditions to solutions of ($d\alpha =0,$ $\delta \alpha =0$),
[$\delta $ is what goes by the names of divergence and co-derivative,
inappropriate for the Kaehler calculus, with which we obtained the foregoing).

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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: hep-ex