Rotation of axes

One significant way in which calculations in Rn differ from those in R1 is that the axes can be chosen fairly freely. Furthermore one often wishes to calculate multiple integrals by means of spherical polars or cylindrical polars instead of Cartesians. In order to provide similar facilities for generalised functions it is necessary to see what effect a change of variable has.

We commence by examining the effect of choosing a different set of Cartesian axes with the same origin. (A change of origin without alteration of the directions of the axes is covered by Definition 7.9.) Regarding x as a column matrix we can obtain any other Cartesian set with the same origin by a linear transformation y = Lx where L is an orthogonal matrix, i.e. LTL = I where LT is the transpose of L. The determinant of L, det L, is either 1 or – 1. If det L = 1 the new axes are derived from the old by a proper rotation; if det L = – 1 an improper rotation, i.e. a proper rotation together with a reflection, is involved.

Definition 8.1.If {γm} is a regular sequence defining g, the sequence {γm(Lx)} is regular and defines a generalised function which is denoted by g (Lx).