Difference between revisions of "Gradient"

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(I am sure some university teaches it in freshman/sophomore level)
(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.)
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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]].
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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]]:  

Revision as of 16:14, 31 October 2009

This article/section deals with mathematical concepts appropriate for late high school or early college.

In mathematics, the gradient is a vector associated to a point of a differentiable function which takes real values. Specifically, the gradient at is a vector in which points in the direction in which increases most rapidly at . The magnitude of the gradient at is equal to the maximum directional derivative of at . 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, of to be the vector field:

consisting of the various partial derivatives of . If is a unit vector in , then, by the chain rule, the directional derivative of in the direction of is simply the dot product:

Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction is maximal in the direction of the gradient, and equal to for a unit vector in the direction of the gradient.

Properties of the Gradient

If is a differentiable function with smooth level sets , then the gradient vector field is perpendicular to the level sets of . For fix a level set , and let be a vector tangent to at . Then we can find a curve on with . Now

since is a level set. Taking derivatives of both sides and applying the chain rule, we get that

Thus, is perpendicular to at , i.e., the gradient of is perpendicular to the level sets of .