Relation between gradient vector and partial derivatives - Revision history

Vipul at 16:30, 12 May 2012

2012-05-12T16:30:12Z

← Older revision		Revision as of 16:30, 12 May 2012
Line 12:		Line 12:

	<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.

	==Related facts==		==Related facts==

Vipul at 16:08, 12 May 2012

2012-05-12T16:08:50Z

← Older revision		Revision as of 16:08, 12 May 2012
Line 1:		Line 1:
	==Statement==		==Statement==

	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:		{\| class="sortable" border="1"
			! Version type !! Statement
	<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>		\|-
			\| 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>

Vipul: Created page with "==Statement== Suppose $f$ is a real-valued function of $n$ variables $x_1,x_2,\dots,x_n$ . Suppose $(a_1,a_2,\dots,a_n)$ is a point..."

2012-05-12T16:04:33Z

Created page with "==Statement== 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..."

New page

==Statement==

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:

<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>

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

==Related facts==

* [[Existence of partial derivatives not implies differentiable]]: It is possible that all the partial derivatives exist at a point but the gradient vector doesn't.
* [[Relation between gradient vector and directional derivatives]]

Relation between gradient vector and partial derivatives - Revision history

Vipul at 16:30, 12 May 2012

Vipul at 16:08, 12 May 2012

Vipul: Created page with "==Statement== Suppose f is a real-valued function of n variables x_1,x_2,\dots,x_n. Suppose (a_1,a_2,\dots,a_n) is a point..."

Vipul: Created page with "==Statement== Suppose $f$ is a real-valued function of $n$ variables $x_1,x_2,\dots,x_n$ . Suppose $(a_1,a_2,\dots,a_n)$ is a point..."