# 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: