Applications of the Epimorphism Theorem

Abstract

We denote by Rⁿ the n-dimensional Euclidean space, by x=(x_1,..., x _n ) the variable point in Rⁿ, by Nⁿ the subset of Rⁿ consisting of the n-tuples p=(p₁,...,p _n ) where the p _j are integers ≥0. We set |p|=p₁+...+p _n (whereas $\left| x \right| = {(x_{1}^{2} + ... + x_{n}^{2})^{{1/2}}}$. If ƒis a C∞ functions in Rⁿ, say with complex values, we write

$$ {f^{{(p)}}}for{(\partial /\partial {x_{1}})^{{p1}}}...{(\partial /\partial {x_{n}})^{{Pn}}}f $$