Symmetric derivative
From Encyclopedia of Mathematics
A generalization of the concept of derivative to the case of set functions on an
-dimensional Euclidean space. The symmetric derivative at a point
is the limit
![]() |
where is the closed ball with centre
and radius
, if this limit exists. The symmetric derivative of order
at a point
of a function
of a real variable is defined as the limit
![]() |
![]() |
A function of a real variable has a symmetric derivative of order
at a point
,
![]() |
if
![]() |
and one of order ,
![]() |
if
![]() |
If has an
-th order derivative
at a point
, then there is (in both cases) a symmetric derivative at
, and it is equal to
.
References
[1] | S. Saks, "Theory of the integral" , Hafner (1937) (Translated from French) |
[2] | R.D. James, "Generalized ![]() |
Comments
In [1] instead of derivative, "derivate" is used: symmetric derivate.
How to Cite This Entry:
Symmetric derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_derivative&oldid=48922
Symmetric derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetric_derivative&oldid=48922
This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article