Gradient vector - Revision history

Vipul at 22:08, 8 May 2016

2016-05-08T22:08:36Z

← Older revision		Revision as of 22:08, 8 May 2016
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	Suppose <math>f</math> is a function of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{c}</math> is a point in the interior of the [[domain]] of <math>f</math>, i.e., <math>f</math> is defined in an open ball centered at <math>\overline{c}</math>. The gradient vector of <math>f</math> at <math>\overline{c}</math>, denoted <math>(\nabla f)(\overline{c})</math>, is a vector <math>\overline{v}</math> satisfying the following:		Suppose <math>f</math> is a function of a vector variable <math>\overline{x}</math>. Suppose <math>\overline{c}</math> is a point in the interior of the [[domain]] of <math>f</math>, i.e., <math>f</math> is defined in an open ball centered at <math>\overline{c}</math>. The gradient vector of <math>f</math> at <math>\overline{c}</math>, denoted <math>(\nabla f)(\overline{c})</math>, is a vector <math>\overline{v}</math> satisfying the following:

	* For every <~~matH~~>\~~epsilon~~ > 0</math>		* For every <math>\varepsilon > 0</math>
	* there exists <math>\delta > 0</math> such that		* there exists <math>\delta > 0</math> such that
	* for every <math>\overline{x}</math> satisfying <math>0 < \|\overline{x} - \overline{c}\| < \delta</math> (in other words, <math>\overline{x}</math> is in an open ball of radius <math>\delta</math> centered at <math>\overline{c}</math>, but not qual to <math>\overline{c}</math>)		* for every <math>\overline{x}</math> satisfying <math>0 < \|\overline{x} - \overline{c}\| < \delta</math> (in other words, <math>\overline{x}</math> is in an open ball of radius <math>\delta</math> centered at <math>\overline{c}</math>, but not qual to <math>\overline{c}</math>)
	* we have <math>\|f(\overline{x}) - f(\overline{c}) - \overline{v} \cdot (\overline{x} - \overline{c})\| < \~~epsilon~~\|\overline{x} - \overline{c}\|</math>		* we have <math>\|f(\overline{x}) - f(\overline{c}) - \overline{v} \cdot (\overline{x} - \overline{c})\| < \varepsilon\|\overline{x} - \overline{c}\|</math>

	===Note on why the epsilon-delta definition is necessary===		===Note on why the epsilon-delta definition is necessary===
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	<math>\lim_{\overline{x} \to \overline{c}} \frac{f(\overline{x}) - f(\overline{c})}{\overline{x} - \overline{c}}</math>		<math>\lim_{\overline{x} \to \overline{c}} \frac{f(\overline{x}) - f(\overline{c})}{\overline{x} - \overline{c}}</math>

	Unfortunately, the above notation does not make direct sense because it is not permissible to divide a scalar by a vector. To rectify this, we revisit what the <math>\~~epsilon~~-\delta</math> definition of the derivative says. It turns out that that <math>\~~epsilon~~-\delta</matH> definition can more readily be generalized to functions of vector variables. The key insight is to use the [[dot product of vectors]].		Unfortunately, the above notation does not make direct sense because it is not permissible to divide a scalar by a vector. To rectify this, we revisit what the <math>\varepsilon-\delta</math> definition of the derivative says. It turns out that that <math>\varepsilon-\delta</matH> definition can more readily be generalized to functions of vector variables. The key insight is to use the [[dot product of vectors]].

	<center>{{#widget:YouTube\|id=XFda15ReMQ8}}</center>		<center>{{#widget:YouTube\|id=XFda15ReMQ8}}</center>

Vipul: /* Graphical interpretation */

2013-01-27T04:27:23Z

Graphical interpretation

← Older revision		Revision as of 04:27, 27 January 2013
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	Note that it is possible that the partial derivatives both exist but the function is not differentiable. In this case, the surface does not have a well defined tangent plane at the point. Even though we can define a plane by the equation above, this is not the tangent plane, because the tangent plane does not exist.		Note that it is possible that the partial derivatives both exist but the function is not differentiable. In this case, the surface does not have a well defined tangent plane at the point. Even though we can define a plane by the equation above, this is not the tangent plane, because the tangent plane does not exist.

	<center>{{#widget:YouTube\|id=~~eiGgAcIrLSY~~}}</center>		<center>{{#widget:YouTube\|id=sICGShQ7CHE}}{{#widget:YouTube\|id=hX5KhF_Xog0}}</center>
	===For a function of multiple variables===		===For a function of multiple variables===

Vipul: /* Relation with partial derivatives */

2013-01-27T04:22:43Z

Relation with partial derivatives

← Older revision		Revision as of 04:22, 27 January 2013
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	\|}		\|}

	<center>{{#widget:YouTube\|id=~~yINihD_bYzA~~}}</center>		<center>{{#widget:YouTube\|id=84hww4kkoq8}}</center>

	===Note on continuous partials===		===Note on continuous partials===

Vipul: /* Relation with directional derivatives and partial derivatives */

2013-01-27T04:22:05Z

Relation with directional derivatives and partial derivatives

← Older revision		Revision as of 04:22, 27 January 2013
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	\|}		\|}

	<center>{{#widget:YouTube\|id=~~TvvSB2q5L1E~~}}</center>		<center>{{#widget:YouTube\|id=TAeK0MVuzJU}}</center>
	<center>{{#widget:YouTube\|id=~~RvponqyVtFU~~}}</center>		<center>{{#widget:YouTube\|id=T8WGcg_T-DM}}</center>
	===Relation with partial derivatives===		===Relation with partial derivatives===

Vipul: /* Definition as a function */

2013-01-27T04:20:42Z

Definition as a function

← Older revision		Revision as of 04:20, 27 January 2013
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	If the gradient vector of <math>f</math> exists at all points of the domain of <math>f</math>, we say that <math>f</math> is differentiable everywhere on its domain.		If the gradient vector of <math>f</math> exists at all points of the domain of <math>f</math>, we say that <math>f</math> is differentiable everywhere on its domain.

	<center>{{#widget:YouTube\|id=~~cg7z5auWG30~~}}</center>		<center>{{#widget:YouTube\|id=jyC5wEgBipg}}</center>

	==Relation with directional derivatives and partial derivatives==		==Relation with directional derivatives and partial derivatives==

Vipul: /* Note on why the epsilon-delta definition is necessary */

2013-01-27T04:20:16Z

Note on why the epsilon-delta definition is necessary

← Older revision		Revision as of 04:20, 27 January 2013
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	Unfortunately, the above notation does not make direct sense because it is not permissible to divide a scalar by a vector. To rectify this, we revisit what the <math>\epsilon-\delta</math> definition of the derivative says. It turns out that that <math>\epsilon-\delta</matH> definition can more readily be generalized to functions of vector variables. The key insight is to use the [[dot product of vectors]].		Unfortunately, the above notation does not make direct sense because it is not permissible to divide a scalar by a vector. To rectify this, we revisit what the <math>\epsilon-\delta</math> definition of the derivative says. It turns out that that <math>\epsilon-\delta</matH> definition can more readily be generalized to functions of vector variables. The key insight is to use the [[dot product of vectors]].

	<center>{{#widget:YouTube\|id=~~0a9NEdMHSpI~~}}</center>		<center>{{#widget:YouTube\|id=XFda15ReMQ8}}</center>

	==Definition as a function==		==Definition as a function==

Vipul: /* Graphical interpretation */

2012-05-13T02:42:39Z

Graphical interpretation

← Older revision		Revision as of 02:42, 13 May 2012
Line 94:		Line 94:
	Note that it is possible that the partial derivatives both exist but the function is not differentiable. In this case, the surface does not have a well defined tangent plane at the point. Even though we can define a plane by the equation above, this is not the tangent plane, because the tangent plane does not exist.		Note that it is possible that the partial derivatives both exist but the function is not differentiable. In this case, the surface does not have a well defined tangent plane at the point. Even though we can define a plane by the equation above, this is not the tangent plane, because the tangent plane does not exist.

			<center>{{#widget:YouTube\|id=eiGgAcIrLSY}}</center>
	===For a function of multiple variables===		===For a function of multiple variables===

Vipul: /* Relation with directional derivatives */

2012-05-12T16:46:32Z

Relation with directional derivatives

← Older revision		Revision as of 16:46, 12 May 2012
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	<center>{{#widget:YouTube\|id=cg7z5auWG30}}</center>		<center>{{#widget:YouTube\|id=cg7z5auWG30}}</center>

	==Relation with directional derivatives==		==Relation with directional derivatives and partial derivatives==

	===~~Statement of relation~~===		===Relation with directional derivatives===

	{{further\|[[Relation between gradient vector and directional derivatives]]}}		{{further\|[[Relation between gradient vector and directional derivatives]]}}

Vipul: /* Relation with partial derivatives */

2012-05-12T16:30:49Z

Relation with partial derivatives

← Older revision		Revision as of 16:30, 12 May 2012
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	<center>{{#widget:YouTube\|id=yINihD_bYzA}}</center>		<center>{{#widget:YouTube\|id=yINihD_bYzA}}</center>

			===Note on continuous partials===

			{{further\|[[Continuous partials implies differentiable]]}}

			This says that if all the [[partial derivative]]s of a function are continuous at and around a point in the domain, then the function is in fact differentiable, hence the gradient vector is described in terms of the partial derivatives as described above.

			In particular, if all the partials exist and are continuous everywhere, the gradient vector exists everywhere and is given as described above.

			Note that this is significant because, ''a priori'' (i.e., without checking continuity), knowledge of the partials tells us what the gradient vector should be if it exists, but it doesn't tell us whether the gradient vector does exist. Continuity helps bridge that knowledge gap.

	==Graphical interpretation==		==Graphical interpretation==

Vipul: /* Relation with directional derivatives */

2012-05-12T16:08:35Z

Relation with directional derivatives

← Older revision		Revision as of 16:08, 12 May 2012
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	{{further\|[[Relation between gradient vector and partial derivatives]]}}		{{further\|[[Relation between gradient vector and partial derivatives]]}}
			{\| class="sortable" border="1"
	If the gradient vector exists, ~~then its~~ coordinates are ~~given by~~ the partial derivatives.		! Version type !! Statement
			\|-
			\| at a point, in multivariable notation \|\| Suppose <math>f</math> is a real-valued function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. Suppose <math>(a_1,a_2,\dots,a_n)</math> is a point in the domain of <math>f</math> such that the [[gradient vector]] of <math>f</math> at <math>(a_1,a_2,\dots,a_n)</math>, denoted <math>(\nabla f)(a_1,a_2,\dots,a_n)</math>, exists. Then, the [[partial derivative]]s of <math>f</math> with respect to all variables exist, and the coordinates of the gradient vector are the partial derivatives. In other words:<br><math>(\nabla f)(a_1,a_2,\dots,a_n) = \langle f_{x_1}(a_1,a_2,\dots,a_n), f_{x_2}(a_1,a_2,\dots,a_n), \dots f_{x_n}(a_1,a_2,\dots,a_n)\rangle </math>
			\|-
			\| generic point, in multivariable notation \|\| Suppose <math>f</math> is a real-valued function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. Then, we have <br><math>(\nabla f)(x_1,x_2,\dots,x_n) = \langle f_{x_1}(x_1,x_2,\dots,x_n), f_{x_2}(x_1,x_2,\dots,x_n), \dots f_{x_n}(x_1,x_2,\dots,x_n)\rangle </math>.<br>Equality holds [[concept of equality conditional to existence of one side\|wherever the left side makes sense]].
			\|-
			\| generic point, point-free notation \|\| Suppose <math>f</math> is a function of <math>n</math> variables <math>x_1,x_2,\dots,x_n</math>. Then, we have <br><math>\nabla f = \langle f_{x_1}, f_{x_2}, \dots f_{x_n} \rangle</math>. Equality holds [[concept of equality conditional to existence of one side\|wherever the left side makes sense]].
			\|}

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

	~~===Proof of relation===~~

	~~{{fillin}}~~

	==Graphical interpretation==		==Graphical interpretation==