Multiplicatively separable function: Difference between revisions

Latest revision as of 23:46, 10 April 2012

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:

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

@@ Line 3: / Line 3: @@
 ===For a function of two variables===
-Suppose <math>F</math> is a function of two variables <matH>x</math> and <math>y</math>. We say that <math>F</matH> is '''multiplicatively separable''' if there exist functions <math>f,g</math> of one variable such that:
+Suppose <math>G</math> is a function of two variables <matH>x</math> and <math>y</math>. We say that <math>G</matH> is '''multiplicatively separable''' if there exist functions <math>f,g</math> of one variable such that:
-<math>F(x,y) = f(x)g(y)</math>
+<math>G(x,y) = f(x)g(y)</math>
-on the entire domain of <math>F</math>.
+on the entire domain of <math>G</math>.
 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.
@@ Line 13: / Line 13: @@
 ===For a function of many variables===
-Suppose <math>F</math> is a function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. We say that <math>F</math> is '''completely multiplicatively separable''' if there exist functions <math>f_1,f_2,\dots,f_n</math>, each a function of one variable, such that:
+Suppose <math>G</math> is a function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. We say that <math>G</math> is '''completely multiplicatively separable''' if there exist functions <math>g_1,g_2,\dots,g_n</math>, each a function of one variable, such that:
-<math>F(x_1,x_2,\dots,x_n) = f_1(x_1)f_2(x_2) \dots f_n(x_n)</math>
+<math>G(x_1,x_2,\dots,x_n) = g_1(x_1)g_2(x_2) \dots g_n(x_n)</math>
 (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 <math>\{ 1,2,\dots,n\}</math> as a union of two disjoint subsets <math>A,B</math>, <math>F</matH> is multiplicatively separable with respect to the partition if there exist functions <math>f_A,f_B</math> such that:
+There is a weaker notion of ''partially multiplicatively separable'': if we express the set <math>\{ 1,2,\dots,n\}</math> as a union of two disjoint subsets <math>A,B</math>, <math>G</matH> is multiplicatively separable with respect to the partition if there exist functions <math>g_A,g_B</math> such that:
-<math>F(x_1,x_2,\dots,x_n) = f_A(\mbox{only the variables} x_i, i \in A)f_B(\mbox{only the variables} x_i, i \in B)</math>
+<math>G(x_1,x_2,\dots,x_n) = g_A(\mbox{only the variables } x_i, i \in A)g_B(\mbox{only the variables } x_i, i \in B)</math>
+<center>{{#widget:YouTube|id=9pXmMkHG248}}</center>
+==Partial derivatives==
+===For a function of two variables===
+Consider the case <math>\! G(x,y) = f(x)g(y)</math>.
+Then, if <math>f</math> is <math>m</math> times differentiable and <math>g</math> is <math>n</math> times differentiable, then <math>G_{xx\dots xyy\dots y}</math> makes sense where <math>x</math> occurs <math>m</math> times and <math>y</math> occurs <math>n</math> times, and:
+<math>\! G_{xx\dots xyy\dots y} = f^{(m)}(x)g^{(n)}(y)</math>
+Further, ''any'' partial derivative of <math>G</matH> that uses <math>m</matH> occurrences of <math>x</math> and <math>n</math> occurrences of <math>y</math> will have the same derivative as the above.
+In particular, we have that:
+* <math>G_x(x,y) = f'(x)g(y)</math>
+* <math>G_y(x,y) = f(x)g'(y)</math>
+* <math>G_{xy}(x,y) = G_{yx}(x,y) = f'(x)g'(y)</math>
+===For a function of many variables===
+Consider the case <math>G(x_1,x_2,\dots,x_n) = g_1(x_1)g_2(x_2) \dots g_n(x_n)</math>
+Suppose <matH>m_1,m_2,\dots,m_n</math> are (possibly equal, possibly distinct) nonnegative integers such that each <math>f_i</math> is <math>m_i</matH> times differentiable. Now, consider a partial derivative of <math>G</math> that involves <math>m_1</math> differentiations in <math>x_1</math>, <math>m_2</math> differentiations in <math>x_2</math>, and so on, with <math>m_i</matH> differentiations in each <math>x_i</math>. The order of the differentiations does not matter. Then, this partial derivative equals:
+<math>\! g_1^{(m_1)}(x_1)g_2^{(m_2)}(x_2) \dots g_n^{(m_n)}(x_n)</math>
+==Integration on rectangular regions==
+===For a function of two variables===
+Suppose <matH>G(x,y) = f(x)g(y)</math> is a function of two variables. Consider a rectangular region <math>R</matH> of the form <math>[a,b] \times [p,q]</math> where <matH>a,b,p,q</math> are numbers. Then:
+<math>\int_R G(x,y) \, dA = \left(\int_a^b f(x) \, dx\right) \left(\int_p^q g(y) \, dy \right)</math>
+===For a function of many variables===
+Suppose <math>G(x_1,x_2,\dots,x_n) = g_1(x_1)g_2(x_2) \dots g_n(x_n)</matH> is a function of <math>n</math> variables. Consider a rectangular region <math>R</math> of the form <math>[a_1,b_1] \times [a_2,b_2] \times \dots \times [a_n,b_n]</math>. Then:
+<math>\int_R G(x,y) \, dA = \prod_{i=1}^n \left(\int_{a_i}^{b_i} g_i(x_i) \, dx_i \right)</math>