Multiplicatively separable function
Contents
Definition
For a function of two variables
Suppose is a function of two variables
and
. We say that
is multiplicatively separable if there exist functions
of one variable such that:
on the entire domain of .
Note that the concept of multiplicatively separable is sensitive to the coordinate system, i.e., if we change the coordinate system, a function that was originally multiplicatively separable need not remain multiplicatively separable.
For a function of many variables
Suppose is a function of
variables
. We say that
is completely multiplicatively separable if there exist functions
, each a function of one variable, such that:
(note that the subscripts here are not to be confused with subscripts used for partial derivatives).
There is a weaker notion of partially multiplicatively separable: if we express the set as a union of two disjoint subsets
,
is multiplicatively separable with respect to the partition if there exist functions
such that:
Partial derivatives
For a function of two variables
Consider the case .
Then, if is
times differentiable and
is
times differentiable, then
makes sense where
occurs
times and
occurs
times, and:
Further, any partial derivative of that uses
occurrences of
and
occurrences of
will have the same derivative as the above.
In particular, we have that:
For a function of many variables
Consider the case
Suppose are (possibly equal, possibly distinct) nonnegative integers such that each
is
times differentiable. Now, consider a partial derivative of
that involves
differentiations in
,
differentiations in
, and so on, with
differentiations in each
. The order of the differentiations does not matter. Then, this partial derivative equals:
Integration on rectangular regions
For a function of two variables
Suppose is a function of two variables. Consider a rectangular region
of the form
where
are numbers. Then:
For a function of many variables
Suppose is a function of
variables. Consider a rectangular region
of the form
. Then: