Chapter 11Primitives of Functions
This chapter provides three results on the existence of primitives of fields q = (q1, . . . , qd) of functions, to which we will come back to construct primitives of distributions, in Chapter 13.
The first (Theorem 11.1) gives the existence of a primitive of every field whose line integral is zero along closed paths.
The second is the orthogonality theorem for functions (Theorem 11.4), which gives the existence of a primitive of every field which is orthogonal to test fields, namely such that
for all ψ in
(Ω;
d) such that ∇ . ψ = 0.
This result is based on the concentration theorem (Theorem 11.3), which says that, for any field q, the integral
associated with an incompressible tubular flow Ψ with support in a tube of axis Γ is equal to the integral
concentrated on the closed path Γ.
The third is Poincaré’s theorem (Theorem 11.5), which gives the existence of a primitive on a ball of every field which satisfies Poincaré’s condition ∂iqj = ∂jqi.
11.1. ...
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