Multiplicatively separable function - Revision history

Vipul: /* For a function of two variables */

2012-04-10T23:46:07Z

For a function of two variables

← Older revision		Revision as of 23:46, 10 April 2012
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	===For a function of two variables===		===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>. Then:		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 ~~\int~~ G(x,y) \, dA = \left(\int_a^b f(x) \, dx\right) \left(\int_p^q g(y) \, dy \right)</math>		<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>

Vipul: /* Partial derivatives */

2012-04-10T23:43:41Z

Partial derivatives

← Older revision		Revision as of 23:43, 10 April 2012
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	* <math>G_y(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>		* <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>. Then:

			<math>\int_R \int G(x,y) \, dA = \left(\int_a^b f(x) \, dx\right) \left(\int_p^q g(y) \, dy \right)</math>

Vipul at 23:38, 10 April 2012

2012-04-10T23:38:57Z

← Older revision		Revision as of 23:38, 10 April 2012
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	<center>{{#widget:YouTube\|id=9pXmMkHG248}}</center>		<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>

Vipul at 23:35, 10 April 2012

2012-04-10T23:35:34Z

← Older revision		Revision as of 23:35, 10 April 2012
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	===For a function of two variables===		===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.		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.
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	===For a function of many variables===		===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).		(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>

Vipul at 23:30, 10 April 2012

2012-04-10T23:30:28Z

← Older revision		Revision as of 23:30, 10 April 2012
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	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>F</matH> is multiplicatively separable with respect to the partition if there exist functions <math>f_A,f_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>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>

Vipul: Created page with "==Definition== ===For a function of two variables=== Suppose $F$ is a function of two variables $x$ and $y$ . We say that $F$ is '..."

2012-04-10T23:29:56Z

Created page with "==Definition== ===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 '..."

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==Definition==

===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:

<math>F(x,y) = f(x)g(y)</math>

on the entire domain of <math>F</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.

===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:

<math>F(x_1,x_2,\dots,x_n) = f_1(x_1)f_2(x_2) \dots f_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:

<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>