Second-order mixed partial derivative - Revision history

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

2013-01-25T22:49:17Z

For a function of many variables

← Older revision		Revision as of 22:49, 25 January 2013
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	In general, for a function of <math>n</math> variables, there are <matH>n(n - 1)</math> many second-order mixed partials that we can construct.		In general, for a function of <math>n</math> variables, there are <matH>n(n - 1)</math> many second-order mixed partials that we can construct.

	<center>{{#widget:YouTube\|id=~~ePbvOUNPt1E~~}}</center>		<center>{{#widget:YouTube\|id=OeBm21RJBz0}}</center>

	===Definition as a double limit at a point===		===Definition as a double limit at a point===

Vipul: /* Definition as a double limit at a point */

2013-01-25T17:41:42Z

Definition as a double limit at a point

← Older revision		Revision as of 17:41, 25 January 2013
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	As [[Clairaut's theorem on equality of mixed partials]] shows, we can, under reasonable assumptions of existence and continuity, show that these two second-order mixed partials are the same.		As [[Clairaut's theorem on equality of mixed partials]] shows, we can, under reasonable assumptions of existence and continuity, show that these two second-order mixed partials are the same.

	<center>{{#widget:YouTube\|id=~~ANWyGlWw4Ig~~}}</center>		<center>{{#widget:YouTube\|id=zhXsLX9DAbc}}</center>

	==Domain considerations==		==Domain considerations==

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

2013-01-25T02:24:44Z

For a function of two variables

← Older revision		Revision as of 02:24, 25 January 2013
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	* Building on this, a ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f</math> should exist if we perturb <math>y</math> a bit and ''then'' perturb <math>x</math> a bit. It's tempting to believe that it is necessary that <math>f</math> should be defined in an open neighborhood of the point <math>(x_0,y_0)</math>. However, this is not necessarily the case because it is not necessary that there exist a positive lower bound on the radius of the <math>x</math>-neighborhoods for definition of <math>f</math> for <math>y</math> close to <math>y_0</math>.		* Building on this, a ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f</math> should exist if we perturb <math>y</math> a bit and ''then'' perturb <math>x</math> a bit. It's tempting to believe that it is necessary that <math>f</math> should be defined in an open neighborhood of the point <math>(x_0,y_0)</math>. However, this is not necessarily the case because it is not necessary that there exist a positive lower bound on the radius of the <math>x</math>-neighborhoods for definition of <math>f</math> for <math>y</math> close to <math>y_0</math>.

	<center>{{#widget:YouTube\|id=~~1VDORdMjiPw~~}}</center>		<center>{{#widget:YouTube\|id=4S1uattqonE}}</center>

	===For a function of more than two variables===		===For a function of more than two variables===

Vipul: /* Domain considerations */

2012-04-22T04:06:04Z

Domain considerations

← Older revision		Revision as of 04:06, 22 April 2012
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	* A ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f_x(x_0,y)</math> exist for <math>y</math> everywhere in an open interval containing <math>y_0</math>. In other words, <math>f_x</math> exists at and near <math>y_0</math> on the line <math>x =x_0</math>. Another way of saying this is that <math>f_x</math> needs to exist not only at the point, but also if we perturb <math>y</math> slightly.		* A ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f_x(x_0,y)</math> exist for <math>y</math> everywhere in an open interval containing <math>y_0</math>. In other words, <math>f_x</math> exists at and near <math>y_0</math> on the line <math>x =x_0</math>. Another way of saying this is that <math>f_x</math> needs to exist not only at the point, but also if we perturb <math>y</math> slightly.
	* Building on this, a ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f</math> should exist if we perturb <math>y</math> a bit and ''then'' perturb <math>x</math> a bit. It's tempting to believe that it is necessary that <math>f</math> should be defined in an open neighborhood of the point <math>(x_0,y_0)</math>. However, this is not necessarily the case because it is not necessary that there exist a positive lower bound on the radius of the <math>x</math>-neighborhoods for definition of <math>f</math> for <math>y</math> close to <math>y_0</math>.		* Building on this, a ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f</math> should exist if we perturb <math>y</math> a bit and ''then'' perturb <math>x</math> a bit. It's tempting to believe that it is necessary that <math>f</math> should be defined in an open neighborhood of the point <math>(x_0,y_0)</math>. However, this is not necessarily the case because it is not necessary that there exist a positive lower bound on the radius of the <math>x</math>-neighborhoods for definition of <math>f</math> for <math>y</math> close to <math>y_0</math>.

			<center>{{#widget:YouTube\|id=1VDORdMjiPw}}</center>

	===For a function of more than two variables===		===For a function of more than two variables===

Vipul: /* Definition as a double limit at a point */

2012-04-22T04:05:02Z

Definition as a double limit at a point

← Older revision		Revision as of 04:05, 22 April 2012
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	As [[Clairaut's theorem on equality of mixed partials]] shows, we can, under reasonable assumptions of existence and continuity, show that these two second-order mixed partials are the same.		As [[Clairaut's theorem on equality of mixed partials]] shows, we can, under reasonable assumptions of existence and continuity, show that these two second-order mixed partials are the same.

			<center>{{#widget:YouTube\|id=ANWyGlWw4Ig}}</center>

	==Domain considerations==		==Domain considerations==

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

2012-04-22T01:03:35Z

For a function of more than two variables

← Older revision		Revision as of 01:03, 22 April 2012
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	* A ''necessary'' (though not sufficient) condition for this second-order mixed partial to exist is that <math>f_{x_i}</math> be defined at points close to <math>(a_1,a_2,\dots,a_n)</math> on the line where we fix all coordinates other than <math>x_j</math> and allow <math>x_j</math> to vary. Another way of saying this is that <math>f_{x_i}</math> needs to exist not just at the point, but also if we perturb <math>x_j</math> slightly.		* A ''necessary'' (though not sufficient) condition for this second-order mixed partial to exist is that <math>f_{x_i}</math> be defined at points close to <math>(a_1,a_2,\dots,a_n)</math> on the line where we fix all coordinates other than <math>x_j</math> and allow <math>x_j</math> to vary. Another way of saying this is that <math>f_{x_i}</math> needs to exist not just at the point, but also if we perturb <math>x_j</math> slightly.
	* A ''necessary'' (though not sufficient) condition for this second-order mixed partial to exist is that <math>f</math> be defined ~~in a small neighborhood of~~ <math>(a_1,a_2,\dots,a_n)</math> in the plane parallel to the <math>x_ix_j</math>-plane that passes through the point. ~~Another way of saying this is that~~ <math>f</math> needs to exist not just at the point, but at all points close to it obtained by perturbing <math>~~x_i~~</math> and <math>~~x_j~~</math> slightly.		* A ''necessary'' (though not sufficient) condition for this second-order mixed partial to exist is that <math>f</math> be defined at points close to <math>(a_1,a_2,\dots,a_n)</math> in the plane parallel to the <math>x_ix_j</math>-plane that passes through the point. Explicitly, <math>f</math> needs to exist not just at the point, but at all points close to it obtained by perturbing <math>x_j</math> slightly and then perturbing <math>x_i</math> slightly.

	==Facts==		==Facts==

	* [[Clairaut's theorem on equality of mixed partials]] states that under assumption of continuity (on an open set) of both the second-order mixed partials of a function of two variables, the two mixed partials are equal.		* [[Clairaut's theorem on equality of mixed partials]] states that under assumption of continuity (on an open set) of both the second-order mixed partials of a function of two variables, the two mixed partials are equal.

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

2012-04-22T01:02:42Z

For a function of two variables

← Older revision		Revision as of 01:02, 22 April 2012
Line 67:		Line 67:

	* A ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f_x(x_0,y)</math> exist for <math>y</math> everywhere in an open interval containing <math>y_0</math>. In other words, <math>f_x</math> exists at and near <math>y_0</math> on the line <math>x =x_0</math>. Another way of saying this is that <math>f_x</math> needs to exist not only at the point, but also if we perturb <math>y</math> slightly.		* A ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f_x(x_0,y)</math> exist for <math>y</math> everywhere in an open interval containing <math>y_0</math>. In other words, <math>f_x</math> exists at and near <math>y_0</math> on the line <math>x =x_0</math>. Another way of saying this is that <math>f_x</math> needs to exist not only at the point, but also if we perturb <math>y</math> slightly.
	* Building on this, a ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f</math> should exist if we perturb <math>y</math> a bit and ''then'' perturb <math>x</math> a bit. It's tempting to believe that it is necessary that <math>f</math> should be defined in an open neighborhood of the point <math>(x_0,y_0)</math>. However, this is not necessarily the case because it is not necessary that there exist a positive lower bound on the radius of the <math>x</math>-neighborhoods for definition of <math>f</math for <math>y</math> close to <math>y_0</math>.		* Building on this, a ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f</math> should exist if we perturb <math>y</math> a bit and ''then'' perturb <math>x</math> a bit. It's tempting to believe that it is necessary that <math>f</math> should be defined in an open neighborhood of the point <math>(x_0,y_0)</math>. However, this is not necessarily the case because it is not necessary that there exist a positive lower bound on the radius of the <math>x</math>-neighborhoods for definition of <math>f</math> for <math>y</math> close to <math>y_0</math>.

	===For a function of more than two variables===		===For a function of more than two variables===

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

2012-04-22T01:00:10Z

For a function of two variables

← Older revision		Revision as of 01:00, 22 April 2012
Line 67:		Line 67:

	* A ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f_x(x_0,y)</math> exist for <math>y</math> everywhere in an open interval containing <math>y_0</math>. In other words, <math>f_x</math> exists at and near <math>y_0</math> on the line <math>x =x_0</math>. Another way of saying this is that <math>f_x</math> needs to exist not only at the point, but also if we perturb <math>y</math> slightly.		* A ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f_x(x_0,y)</math> exist for <math>y</math> everywhere in an open interval containing <math>y_0</math>. In other words, <math>f_x</math> exists at and near <math>y_0</math> on the line <math>x =x_0</math>. Another way of saying this is that <math>f_x</math> needs to exist not only at the point, but also if we perturb <math>y</math> slightly.
	* Building on this, a ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f</math> exist ~~in an open neighborhood of the point. The reason is that for each~~ <math>~~f_x(x_0,~~y)</math> ~~to exist, it is necessary that~~ <math>f</math> ~~exist for~~ <math>x</math> ~~close to~~ <math>x_0</math>, ~~and for~~ that ~~value~~ of <math>y</math>~~. Thus, the upshot is that~~ <math>f</math~~> should exist if we perturb both~~ <math>x</math> ~~and~~ <math>y</math>.		* Building on this, a ''necessary'' (though not sufficient) condition for <math>f_{xy}(x_0,y_0)</math> to exist is that <math>f</math> should exist if we perturb <math>y</math> a bit and ''then'' perturb <math>x</math> a bit. It's tempting to believe that it is necessary that <math>f</math> should be defined in an open neighborhood of the point <math>(x_0,y_0)</math>. However, this is not necessarily the case because it is not necessary that there exist a positive lower bound on the radius of the <math>x</math>-neighborhoods for definition of <math>f</math for <math>y</math> close to <math>y_0</math>.

	===For a function of more than two variables===		===For a function of more than two variables===

Vipul at 00:22, 21 April 2012

2012-04-21T00:22:26Z

← Older revision		Revision as of 00:22, 21 April 2012
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	<center>{{#widget:YouTube\|id=ePbvOUNPt1E}}</center>		<center>{{#widget:YouTube\|id=ePbvOUNPt1E}}</center>

			===Definition as a double limit at a point===

			We consider again the case of a function of two variables. In this case, the partial derivatives <math>f_{xy}</math> and <math>f_{yx}</math> ''at a point'' <math>(x_0,y_0)</math> can be expressed as double limits:

			<math>f_{xy}(x_0,y_0) = \lim_{y \to y_0} \frac{f_x(x_0,y) - f_x(x_0,y_0)}{y - y_0} \qquad \mbox{(1)}</math>

			We now use that:

			<math>f_x(x_0,y) = \lim_{x \to x_0)} \frac{f(x,y) - f(x_0,y)}{x- x_0} \qquad \mbox{(2)}</math>

			and:

			<math>f_x(x_0,y) = \lim_{x \to x_0)} \frac{f(x,y_0) - f(x_0,y_0)}{x- x_0} \qquad \mbox{(3)}</math>

			Plugging (2) and (3) back into (1), we obtain that:

			<math>f_{xy}(x_0,y_0) = \lim_{y \to y_0} \frac{\lim_{x\to x_0} \frac{f(x,y) - f(x_0,y)}{x- x_0} - \lim_{x \to x_0} \frac{f(x,y_0) - f(x_0,y_0)}{x- x_0}}{y - y_0}</math>

			A similar calculation yields that:

			<math>f_{yx}(x_0,y_0) = \lim_{x \to x_0} \frac{\lim_{y\to y_0} \frac{f(x,y) - f(x,y_0)}{y- y_0} - \lim_{y \to y_0} \frac{f(x_0,y) - f(x_0,y_0)}{y- y_0}}{x - x_0}</math>

			As [[Clairaut's theorem on equality of mixed partials]] shows, we can, under reasonable assumptions of existence and continuity, show that these two second-order mixed partials are the same.
	==Domain considerations==		==Domain considerations==

Vipul at 00:08, 21 April 2012

2012-04-21T00:08:09Z

@@ Line 36: / Line 36: @@
 <center>{{#widget:YouTube|id=ePbvOUNPt1E}}</center>
 ==Facts==
 * [[Clairaut's theorem on equality of mixed partials]] states that under assumption of continuity (on an open set) of both the second-order mixed partials of a function of two variables, the two mixed partials are equal.