On the smoothness condition in Euler's theorem on homogeneous functions
-
David E. Dobbs [1]
-
[1] University of Tennessee
-
- Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 49, Nº. 8, 2018, págs. 1250-1259
- Idioma: inglés
- DOI: 10.1080/0020739x.2018.1452303
- Texto completo no disponible (Saber más ...)
-
Resumen
For a function f : Rn \ {(0, ... , 0)} → R with continuous first partial derivatives, a theorem of Euler characterizes when f is a homogeneous function. This note determines whether the conclusion of Euler’s theorem holds if the smoothness of f is not assumed. An example is given to show that if n 2, a homogeneous function (of any degree) need not be differentiable (and so the conclusion of Euler’s theorem would fail for such a function). By way of contrast, it is shown that if n = 1, a homogeneous function (of any degree) must be differentiable (and so Euler’s theorem does not need to assume the smoothness of f if n = 1). Additional characterizations of homogeneous functions, remarks and examples illustrate the theory, emphasizing differences in behaviour between the contexts n 2 and n = 1. This note could be used as enrichment material in calculus courses and possibly some science courses.
