Multiplicatively separable function

Definition

For a function of two variables

Suppose $G$ is a function of two variables $x$ and $y$ . We say that $G$ is multiplicatively separable if there exist functions $f, g$ of one variable such that:

$G (x, y) = f (x) g (y)$

on the entire domain of $G$ .

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 $G$ is a function of $n$ variables $x_{1}, x_{2}, \dots, x_{n}$ . We say that $G$ is completely multiplicatively separable if there exist functions $g_{1}, g_{2}, \dots, g_{n}$ , each a function of one variable, such that:

$G (x_{1}, x_{2}, \dots, x_{n}) = g_{1} (x_{1}) g_{2} (x_{2}) \dots g_{n} (x_{n})$

(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 ${1, 2, \dots, n}$ as a union of two disjoint subsets $A, B$ , $G$ is multiplicatively separable with respect to the partition if there exist functions $g_{A}, g_{B}$ such that:

$G (x_{1}, x_{2}, \dots, x_{n}) = g_{A} (only the variables x_{i}, i \in A) g_{B} (only the variables x_{i}, i \in B)$

Partial derivatives

For a function of two variables

Consider the case $G (x, y) = f (x) g (y)$ .

Then, if $f$ is $m$ times differentiable and $g$ is $n$ times differentiable, then $G_{x x \dots x y y \dots y}$ makes sense where $x$ occurs $m$ times and $y$ occurs $n$ times, and:

$G_{x x \dots x y y \dots y} = f^{(m)} (x) g^{(n)} (y)$

Further, any partial derivative of $G$ that uses $m$ occurrences of $x$ and $n$ occurrences of $y$ will have the same derivative as the above.

In particular, we have that:

$G_{x} (x, y) = f^{'} (x) g (y)$
$G_{y} (x, y) = f (x) g^{'} (y)$
$G_{x y} (x, y) = G_{y x} (x, y) = f^{'} (x) g^{'} (y)$

For a function of many variables

Consider the case $G (x_{1}, x_{2}, \dots, x_{n}) = g_{1} (x_{1}) g_{2} (x_{2}) \dots g_{n} (x_{n})$

Suppose $m_{1}, m_{2}, \dots, m_{n}$ are (possibly equal, possibly distinct) nonnegative integers such that each $f_{i}$ is $m_{i}$ times differentiable. Now, consider a partial derivative of $G$ that involves $m_{1}$ differentiations in $x_{1}$ , $m_{2}$ differentiations in $x_{2}$ , and so on, with $m_{i}$ differentiations in each $x_{i}$ . The order of the differentiations does not matter. Then, this partial derivative equals:

$g_{1}^{(m_{1})} (x_{1}) g_{2}^{(m_{2})} (x_{2}) \dots g_{n}^{(m_{n})} (x_{n})$

Integration on rectangular regions

For a function of two variables

Suppose $G (x, y) = f (x) g (y)$ is a function of two variables. Consider a rectangular region $R$ of the form $[a, b] \times [p, q]$ where $a, b, p, q$ are numbers. Then:

$\int_{R} G (x, y) d A = (\int_{a}^{b} f (x) d x) (\int_{p}^{q} g (y) d y)$

For a function of many variables

Suppose $G (x_{1}, x_{2}, \dots, x_{n}) = g_{1} (x_{1}) g_{2} (x_{2}) \dots g_{n} (x_{n})$ is a function of $n$ variables. Consider a rectangular region $R$ of the form $[a_{1}, b_{1}] \times [a_{2}, b_{2}] \times \dots \times [a_{n}, b_{n}]$ . Then:

$\int_{R} G (x, y) d A = \prod_{i = 1}^{n} (\int_{a_{i}}^{b_{i}} g_{i} (x_{i}) d x_{i})$