Gradient - Revision history

DavidB4: new link

2020-09-08T21:19:16Z

new link

FredericBernard: Added category and see also section

2017-09-11T16:19:00Z

Added category and see also section

: Minor formatting

2016-11-08T17:55:47Z

Minor formatting

DavidB4-bot: /* Properties of the Gradient */clean up & uniformity

2016-07-13T13:35:56Z

Properties of the Gradient: clean up & uniformity

MarkGall: put in an example at the beginning -- hopefully this clears things up, but a bit of restructuring might help

2009-10-31T22:24:19Z

put in an example at the beginning -- hopefully this clears things up, but a bit of restructuring might help

Aschlafly: Stated another way, a gradient is a vector that has orthogonal (coordinate) components that consist of the partial derivatives of a function with respect to each of its variables.

2009-10-31T22:14:42Z

Stated another way, a gradient is a vector that has orthogonal (coordinate) components that consist of the partial derivatives of a function with respect to each of its variables.

RoyL: I am sure some university teaches it in freshman/sophomore level

2009-07-10T03:36:50Z

I am sure some university teaches it in freshman/sophomore level

Foxtrot: slightly reworked intro, links, bold, subcat

2008-07-14T16:54:34Z

slightly reworked intro, links, bold, subcat

DanielB at 02:09, July 4, 2008

2008-07-04T02:09:13Z

DanielB at 04:06, July 3, 2008

2008-07-03T04:06:51Z

@@ Line 1: / Line 1: @@
 {{math-h}}
-In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[derivative]] to functions with more than one [[variable]].
+In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[Derivative (calculus)|derivative]] to functions with more than one [[variable]].
 Stated another way, a gradient is a vector that has coordinate components that consist of the partial derivatives of a function with respect to each of its variables.  For example, if <math>f(x,y) = x^2+y^2</math>, then

← Older revision		Revision as of 16:19, September 11, 2017
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	Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.		Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.
		+
		+	==See also==
		+	*[[Curl]]
		+	*[[Divergence]]
		+	*[[Laplacian]]

	[[Category:Calculus]]		[[Category:Calculus]]
		+	[[Category:Vector Analysis]]

@@ Line 4: / Line 4: @@
 Stated another way, a gradient is a vector that has coordinate components that consist of the partial derivatives of a function with respect to each of its variables.  For example, if <math>f(x,y) = x^2+y^2</math>, then
-::<math>\nabla f(x,y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (2x,2y)</math>.  Observe that in this case, the gradient vector <math>(2x,2y)</math> is orthogonal to the "level curve" defined by <math>x^2+y^2=r^2</math>, which here is a circle: the gradient points outward from the origin, which is the direction of steepest increase of <math>f</math>, and vectors outward from the origin are perpendicular to circles centered at the origin.  We'll see later that this is a case of a more general property of the gradient.
+::<math>\nabla f(x,y) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right) = (2x,2y)</math>.
 More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the [[vector field]]:

← Older revision		Revision as of 13:35, July 13, 2016
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	Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.		Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.

−	[[Category:~~calculus~~]]	+	[[Category:Calculus]]

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 In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[derivative]] to functions with more than one [[variable]].
-Stated another way, a gradient is a vector that has orthogonal (coordinate) components that consist of the partial derivatives of a function with respect to each of its variables.
+Stated another way, a gradient is a vector that has coordinate components that consist of the partial derivatives of a function with respect to each of its variables.  For example, if <math>f(x,y) = x^2+y^2</math>, then
 More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the [[vector field]]:

← Older revision		Revision as of 22:14, October 31, 2009
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	In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[derivative]] to functions with more than one [[variable]].		In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[derivative]] to functions with more than one [[variable]].
		+
		+	Stated another way, a gradient is a vector that has orthogonal (coordinate) components that consist of the partial derivatives of a function with respect to each of its variables.

	More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the [[vector field]]:		More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the [[vector field]]:

← Older revision		Revision as of 03:36, July 10, 2009
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−	{{math-a}}	+	{{math-h}}

	In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[derivative]] to functions with more than one [[variable]].		In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[derivative]] to functions with more than one [[variable]].

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 {{math-a}}
-In [[mathematics]], the gradient of a real-valued differentiable function <math>f(x_1,...,x_n)</math> at a point <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum directional derivative of <math>f</math> at <math>p</math>.
+In [[mathematics]], the '''gradient''' is a [[vector]] associated to a point <math>p</math> of a [[differentiable]] [[function]] <math>f(x_1,...,x_n)</math> which takes [[real]] values. Specifically, the gradient at <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum [[directional derivative]] of <math>f</math> at <math>p</math>. The gradient is an extension of the idea of [[derivative]] to functions with more than one [[variable]].
-More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the vector-field:
+More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the [[vector field]]:
 <math>
@@ Line 9: / Line 9: @@
 </math>
-If <math>u</math> is a unit vector in <math>R^n</math>, then, by the chain rule, the directional derivative of <math>f</math> in the direction of <math>u</math> is simply the dot product:
+consisting of the various [[partial derivative]]s of <math>f</math>. If <math>u</math> is a [[unit vector]] in <math>R^n</math>, then, by the [[chain rule]], the directional derivative of <math>f</math> in the direction of <math>u</math> is simply the [[dot product]]:
 <math>
@@ Line 15: / Line 15: @@
 </math>
-Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction <math>u</math> is maximal in the direction of the gradient, and equal to <math>||\nabla f||</math> for <math>u</math> a unit vector in the direction of the gradient.
+Evidently by the [[Cauchy-Schwartz inequality]], the directional derivative in the direction <math>u</math> is maximal in the direction of the gradient, and equal to <math>||\nabla f||</math> for <math>u</math> a unit vector in the direction of the gradient.
 ==Properties of the Gradient==
-If <math>f</math> is a differentiable function with smooth level sets <math>f^{-1}(c)</math>, then the gradient vector field <math>\nabla f</math> is perpendicular to the level sets of <math>f</math>. For fix a level set <math>S = f^{-1}(c)</math>, and let <math>v</math> be a vector tangent to <math>S</math> at <math>p</math>. Then we can find a curve <math>\gamma(t)</math> on <math>S</math> with <math>\gamma'(0) = v</math>. Now
+If <math>f</math> is a differentiable function with smooth [[level set]]s <math>f^{-1}(c)</math>, then the gradient vector field <math>\nabla f</math> is [[perpendicular]] to the level sets of <math>f</math>. For fix a level set <math>S = f^{-1}(c)</math>, and let <math>v</math> be a vector [[tangent]] to <math>S</math> at <math>p</math>. Then we can find a [[curve]] <math>\gamma(t)</math> on <math>S</math> with <math>\gamma'(0) = v</math>. Now
 <math>
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 Thus, <math>\nabla f</math> is perpendicular to <math>v</math> at <math>p</math>, i.e., the gradient of <math>f</math> is perpendicular to the level sets of <math>f</math>.
-[[Category:mathematics]]

← Older revision		Revision as of 02:09, July 4, 2008
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	{{math-a}}		{{math-a}}
−	~~:''This article deals with the advanced concept of gradient in a vector field''~~
−	~~:''For the simplied gradient used in early high school see: [[Gradient (two points)]]''~~

	In [[mathematics]], the gradient of a real-valued differentiable function <math>f(x_1,...,x_n)</math> at a point <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum directional derivative of <math>f</math> at <math>p</math>.		In [[mathematics]], the gradient of a real-valued differentiable function <math>f(x_1,...,x_n)</math> at a point <math>p</math> is a vector in <math>R^n</math> which points in the direction in which <math>f</math> increases most rapidly at <math>p</math>. The magnitude of the gradient at <math>p</math> is equal to the maximum directional derivative of <math>f</math> at <math>p</math>.