# Chapter 13Existence of Primitives

This chapter is dedicated to obtaining conditions for a distributions field

q= (q_{1}, . . . , q) to have a primitive_{d}f, i.e. for ∇f=q. The main conditions are the following.

- — On an arbitrary open set Ω, it suffices that
qis orthogonal to divergence-free test fields, namely that 〈q, ψ〉 = 0 for everyψsuch that ∇·ψ= 0. This is theorthogonality theoremfor distributions (Theorem 13.5).- — When Ω is simply connected, it suffices that
qsatisfies Poincaré’s condition∂=_{i}q_{j}∂for all_{j}q_{i}iandj. This isPoincaré’s theoremgeneralized to distributions (Theorem 13.7).These conditions are necessary and sufficient.

We prove these results as follows.

- — First of all, we reduce the existence of a primitive on Ω to the existence of a primitive on each of its subsets Ω
_{1}_{/n}= {x:B(x, 1/n) Ω}, i.e. on Ω with a neighborhood of its boundary of width 1/nremoved, by a method of gluing. This is theperipheral gluing theorem(Theorem 13.1).- — Next, we reduce the existence of a primitive of a distribution field
qto the existence of a primitive of a function field, namelyq ◊ ηon Ω_{n}_{1/n}, whenqsatisfies Poincaré’s condition. This is thetheorem on reducing to the function case(Theorem 13.2).- — The orthogonality theorem for distributions then easily follows from the orthogonality theorem for functions (Theorem 11.4).
- — Obtaining Poincaré’s theorem for distributions is more delicate, because the Ω
_{1}_{/n}are not necessarily simply connected, so we cannot ...

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