Book contents
- Frontmatter
- Contents
- Preface
- 1 Convergence
- 2 Good functions
- 3 Generalised functions
- 4 Powers of x
- 5 Series
- 6 Multiplication and the convolution product
- 7 Several variables
- 8 Change of variables and related topics
- 9 Asymptotic behaviour of Fourier integrals
- 10 Some applications
- 11 Weak functions
- 12 The Laplace transform
- Table of Fourier transforms
- Table of Laplace transforms
- Index
8 - Change of variables and related topics
Published online by Cambridge University Press: 18 December 2009
- Frontmatter
- Contents
- Preface
- 1 Convergence
- 2 Good functions
- 3 Generalised functions
- 4 Powers of x
- 5 Series
- 6 Multiplication and the convolution product
- 7 Several variables
- 8 Change of variables and related topics
- 9 Asymptotic behaviour of Fourier integrals
- 10 Some applications
- 11 Weak functions
- 12 The Laplace transform
- Table of Fourier transforms
- Table of Laplace transforms
- Index
Summary
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).
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- Chapter
- Information
- The Theory of Generalised Functions , pp. 267 - 344Publisher: Cambridge University PressPrint publication year: 1982