Directional derivative - Revision history

Vipul: /* Graphical interpretation */

2013-01-27T04:27:06Z

Graphical interpretation

← Older revision		Revision as of 04:27, 27 January 2013
Line 112:		Line 112:
	This intersection can be thought of as the graph of a function of one variable, where the point <math>(x_0,y_0,0)</math> is treated as the origin, the direction <math>(u,v,0)</math> is the independent variable axis, and the <math>z</math>-axis direction is the dependent variable axis. Now, the directional derivative is the slope of this graph for dependent variable value of 0.		This intersection can be thought of as the graph of a function of one variable, where the point <math>(x_0,y_0,0)</math> is treated as the origin, the direction <math>(u,v,0)</math> is the independent variable axis, and the <math>z</math>-axis direction is the dependent variable axis. Now, the directional derivative is the slope of this graph for dependent variable value of 0.

	<center>{{#widget:YouTube\|id=~~qJyNoZcHmrk~~}}</center>		<center>{{#widget:YouTube\|id=sICGShQ7CHE}}{{#widget:YouTube\|id=hX5KhF_Xog0}}</center>

	==Relation with gradient vector==		==Relation with gradient vector==

Vipul at 04:25, 27 January 2013

2013-01-27T04:25:25Z

← Older revision		Revision as of 04:25, 27 January 2013
Line 20:		Line 20:
	\|}		\|}

	<center>{{#widget:YouTube\|id=~~y6nf~~-~~1NTznM~~}}</center>		<center>{{#widget:YouTube\|id=FhrHn_mA-mc}}</center>

	===For a function of multiple variables===		===For a function of multiple variables===
Line 50:		Line 50:
	\|}		\|}

	<center>{{#widget:YouTube\|id=~~MzILb6z-Ov0~~}}</center>		<center>{{#widget:YouTube\|id=3Xu9660Occs}}</center>

	==Definition as a function==		==Definition as a function==
Line 72:		Line 72:
	\|}		\|}

	<center>{{#widget:YouTube\|id=~~HVcTwTjqiIA~~}}</center>		<center>{{#widget:YouTube\|id=EIot8Fdi0EE}}</center>

	===For a function of multiple variables===		===For a function of multiple variables===
Line 125:		Line 125:
	\|}		\|}

	<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==
Line 144:		Line 144:
	<math>D_{\langle u_1,u_2,\dots,u_n \rangle}(f)(x_1,x_2,\dots,x_n) = \sum_{i=1}^n u_if_{x_i}(x_1,x_2,\dots,x_n) = u_1f_{x_1}(x_1,x_2,\dots,x_n) + u_2f_{x_2}(x_1,x_2,\dots,x_n) + \dots + u_nf_{x_n}(x_1,x_2,\dots,x_n)</math>		<math>D_{\langle u_1,u_2,\dots,u_n \rangle}(f)(x_1,x_2,\dots,x_n) = \sum_{i=1}^n u_if_{x_i}(x_1,x_2,\dots,x_n) = u_1f_{x_1}(x_1,x_2,\dots,x_n) + u_2f_{x_2}(x_1,x_2,\dots,x_n) + \dots + u_nf_{x_n}(x_1,x_2,\dots,x_n)</math>

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

	In particular, for a function <math>f(x,y)</math> of two variables and a unit vector <math>\langle u,v \rangle</math>, we have the following, assuming that the gradient vector exists (i.e., that the function is differentiable):		In particular, for a function <math>f(x,y)</math> of two variables and a unit vector <math>\langle u,v \rangle</math>, we have the following, assuming that the gradient vector exists (i.e., that the function is differentiable):

	<math>D_{\langle u,v \rangle}(f)(x,y) = uf_x(x,y) + vf_y(x,y)</math>		<math>D_{\langle u,v \rangle}(f)(x,y) = uf_x(x,y) + vf_y(x,y)</math>

Vipul: /* Relation with gradient vector */

2012-05-13T02:41:37Z

Relation with gradient vector

← Older revision		Revision as of 02:41, 13 May 2012
Line 102:		Line 102:
	\|}		\|}

			==Graphical interpretation==

			===For a function of two variables===

			{{further\|[[graph of a function of two variables]]}}

			The directional derivative <math>D_{\langle u,v \rangle} f(x_0,y_0)</math> in the direction of a unit vector <math>\langle u,v \rangle</math> at a point <math>(x_0,y_0)</math> can be determined as follows: first, intersect the graph of the function with the plane <math>v(x - x_0) = u(y - y_0)</math>. This plane is perpendicular to the <math>xy</math>-plane and its intersection with the <math>xy</math>-plane is the line through <math>(x_0,y_0)</math> in the direction of the unit vector <math>\langle u,v \rangle</math>.

			This intersection can be thought of as the graph of a function of one variable, where the point <math>(x_0,y_0,0)</math> is treated as the origin, the direction <math>(u,v,0)</math> is the independent variable axis, and the <math>z</math>-axis direction is the dependent variable axis. Now, the directional derivative is the slope of this graph for dependent variable value of 0.

			<center>{{#widget:YouTube\|id=qJyNoZcHmrk}}</center>
	==Relation with gradient vector==		==Relation with gradient vector==

Vipul at 15:34, 4 May 2012

2012-05-04T15:34:17Z

← Older revision		Revision as of 15:34, 4 May 2012
Line 1:		Line 1:
			{{multivariable analogue of\|derivative}}
	==Definition at a point==		==Definition at a point==

Vipul at 04:18, 22 April 2012

2012-04-22T04:18:14Z

← Older revision		Revision as of 04:18, 22 April 2012
Line 112:		Line 112:
	\| generic point, point-free notation (multiple variables) \|\| Suppose <math>f</math> is a function of a vector variable <math>\overline{x} = \langle x_1,x_2,\dots,x_n \rangle</math>. Suppose <math>\overline{u}</math> is a unit vector. We then have:<br><math>D_{\overline{u}}(f) = \overline{u} \cdot (\nabla f)</math><br>The right side here is a dot product of vector-valued functions (the constant function <math>\overline{u}</math> and the gradient vector of <math>f</math>). The equality holds [[concept of equality conditional to existence of one side\|whenever the right side makes sense]].		\| generic point, point-free notation (multiple variables) \|\| Suppose <math>f</math> is a function of a vector variable <math>\overline{x} = \langle x_1,x_2,\dots,x_n \rangle</math>. Suppose <math>\overline{u}</math> is a unit vector. We then have:<br><math>D_{\overline{u}}(f) = \overline{u} \cdot (\nabla f)</math><br>The right side here is a dot product of vector-valued functions (the constant function <math>\overline{u}</math> and the gradient vector of <math>f</math>). The equality holds [[concept of equality conditional to existence of one side\|whenever the right side makes sense]].
	\|}		\|}

			<center>{{#widget:YouTube\|id=TvvSB2q5L1E}}</center>
			<center>{{#widget:YouTube\|id=RvponqyVtFU}}</center>

	==Relation with partial derivatives==		==Relation with partial derivatives==
Line 128:		Line 131:

	<math>D_{\langle u_1,u_2,\dots,u_n \rangle}(f)(x_1,x_2,\dots,x_n) = \sum_{i=1}^n u_if_{x_i}(x_1,x_2,\dots,x_n) = u_1f_{x_1}(x_1,x_2,\dots,x_n) + u_2f_{x_2}(x_1,x_2,\dots,x_n) + \dots + u_nf_{x_n}(x_1,x_2,\dots,x_n)</math>		<math>D_{\langle u_1,u_2,\dots,u_n \rangle}(f)(x_1,x_2,\dots,x_n) = \sum_{i=1}^n u_if_{x_i}(x_1,x_2,\dots,x_n) = u_1f_{x_1}(x_1,x_2,\dots,x_n) + u_2f_{x_2}(x_1,x_2,\dots,x_n) + \dots + u_nf_{x_n}(x_1,x_2,\dots,x_n)</math>

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

	In particular, for a function <math>f(x,y)</math> of two variables and a unit vector <math>\langle u,v \rangle</math>, we have the following, assuming that the gradient vector exists (i.e., that the function is differentiable):		In particular, for a function <math>f(x,y)</math> of two variables and a unit vector <math>\langle u,v \rangle</math>, we have the following, assuming that the gradient vector exists (i.e., that the function is differentiable):

	<math>D_{\langle u,v \rangle}(f)(x,y) = uf_x(x,y) + vf_y(x,y)</math>		<math>D_{\langle u,v \rangle}(f)(x,y) = uf_x(x,y) + vf_y(x,y)</math>

Vipul at 03:49, 16 April 2012

2012-04-16T03:49:30Z

← Older revision		Revision as of 03:49, 16 April 2012
Line 18:		Line 18:
	\| Definition as an ordinary derivative \|\| <math>\frac{d}{dh}[f(x_0 + uh,y_0 + vh)]\|_{h = 0}</math>		\| Definition as an ordinary derivative \|\| <math>\frac{d}{dh}[f(x_0 + uh,y_0 + vh)]\|_{h = 0}</math>
	\|}		\|}

			<center>{{#widget:YouTube\|id=y6nf-1NTznM}}</center>

	===For a function of multiple variables===		===For a function of multiple variables===

Line 45:		Line 48:
	\| Definition as an ordinary derivative \|\| <math>\frac{d}{dh}[f(\overline{a} + h\overline{u})]\|_{h = 0}</math>		\| Definition as an ordinary derivative \|\| <math>\frac{d}{dh}[f(\overline{a} + h\overline{u})]\|_{h = 0}</math>
	\|}		\|}

			<center>{{#widget:YouTube\|id=MzILb6z-Ov0}}</center>

	==Definition as a function==		==Definition as a function==
Line 65:		Line 70:
	\| Definition as a partial derivative \|\| <math>\frac{\partial}{\partial h}[f(x + uh,y + vh)]\|_{h = 0}</math>. Note that we need to use a partial derivative because <math>x,y</matH> are now variable as we are not doing this at a single point.		\| Definition as a partial derivative \|\| <math>\frac{\partial}{\partial h}[f(x + uh,y + vh)]\|_{h = 0}</math>. Note that we need to use a partial derivative because <math>x,y</matH> are now variable as we are not doing this at a single point.
	\|}		\|}

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

	===For a function of multiple variables===		===For a function of multiple variables===

Vipul: /* Relation with partial derivatives */

2012-04-13T20:52:41Z

Relation with partial derivatives

@@ Line 107: / Line 107: @@
 ==Relation with partial derivatives==
 If the [[gradient vector]] at a point exists, then it is a vector whose coordinates are the corresponding partial derivatives of the function. Thus, ''conditional to the existence of the gradient vector'', we have that:
 <math>D_{\langle u_1,u_2,\dots,u_n \rangle}(f)(x_1,x_2,\dots,x_n) = \sum_{i=1}^n u_if_{x_i}(x_1,x_2,\dots,x_n) = u_1f_{x_1}(x_1,x_2,\dots,x_n) + u_2f_{x_2}(x_1,x_2,\dots,x_n) + \dots + u_nf_{x_n}(x_1,x_2,\dots,x_n)</math>

Vipul: /* Generic definition */

2012-04-13T02:19:42Z

Generic definition

← Older revision		Revision as of 02:19, 13 April 2012
Line 50:		Line 50:
	===Generic definition===		===Generic definition===

	Suppose <math>f</math> is a function of many variables. Consider the domain of <math>f</math> as a subset of Euclidean space. Fix a direction in this space. Then, the '''directional derivative'''in the direction is the function sending a point in the domain of <math>f</math> to the derivative of the function with respect to movement of the point along that direction.		Suppose <math>f</math> is a function of many variables. Consider the domain of <math>f</math> as a subset of Euclidean space. Fix a direction in this space. Then, the '''directional derivative''' in the direction is the function sending a point in the domain of <math>f</math> to the derivative of the function with respect to movement of the point along that direction.

	===For a function of two variables===		===For a function of two variables===

Vipul at 02:17, 13 April 2012

2012-04-13T02:17:52Z

← Older revision		Revision as of 02:17, 13 April 2012
Line 1:		Line 1:
	==Definition at a point==		==Definition at a point==

			===Generic definition===

			Suppose <math>f</math> is a function of many variables. Consider the domain of <math>f</math> as a subset of Euclidean space. Fix a direction in this space and a point in the domain. Then, the '''directional derivative''' at the point in the direction is the derivative of the function with respect to movement of the point along that direction, at the specific point.

	===For a function of two variables===		===For a function of two variables===
Line 43:		Line 47:

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

			===Generic definition===

			Suppose <math>f</math> is a function of many variables. Consider the domain of <math>f</math> as a subset of Euclidean space. Fix a direction in this space. Then, the '''directional derivative'''in the direction is the function sending a point in the domain of <math>f</math> to the derivative of the function with respect to movement of the point along that direction.

	===For a function of two variables===		===For a function of two variables===

Vipul: /* Relation with gradient vector */

2012-04-13T02:12:41Z

Relation with gradient vector

← Older revision		Revision as of 02:12, 13 April 2012
Line 88:		Line 88:
	==Relation with gradient vector==		==Relation with gradient vector==

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