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{{math-a}}
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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 (calculus)|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:
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.
More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the [[vector field]]:
<math>
<math>
</math>
</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>
<math>
</math>
</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==
==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>
<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>.
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]]
==See also==
*[[Curl]]
*[[Divergence]]
*[[Laplacian]]
[[Category:Calculus]]
[[Category:Vector Analysis]]