Difference between revisions of "Gradient"

Revision as of 20:31, July 2, 2008

In mathematics, a gradient is the rate a function increases.

If a function has value <math>f(a)</math> at <math>x=a</math> and <math>f(b)</math> at <math>x=b</math> with <math>a<b</math> than the gradient of <math>[a,b]</math> is,

<math>\frac{F(b)-F(a)}{b-a}</math>.

In schooling, usually early high school, students are taught that the gradient is equal to "the rise over run" or more formally,

<math>\mathrm{gradient}=\frac{\mathrm{rise}}{\mathrm{run}}</math>,

the rise being defined as the difference between the highest point and the lowest point, negative if the highest is on the left and positive if the highest point in on the right. The run is defined as difference of the right <math>x</math>-value and the left <math>x</math>-value (see Cartesian coordinates).

This concept is usually first introduced with the introduction of linear equations. The equation of a straight line in Cartesian coordinates is given by,

where <math>m</math> is the gradient of the line and <math>c</math> is the value of the <math>y</math> coordinate when <math>x=0</math>, this is called the y-intercept, e.g, where the line intercepts the y-axis.

@@ Line 1: / Line 1: @@
-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]], a '''gradient''' is the rate a [[function]] increases.
-More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the vector-field:
+If a function has value <math>f(a)</math> at <math>x=a</math> and <math>f(b)</math> at <math>x=b</math> with <math>a<b</math> than the gradient of <math>[a,b]</math> is,
-<math>
+:<math>\frac{F(b)-F(a)}{b-a}</math>.
-\nabla f = (\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n})
-</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:
+In schooling, usually early high school, students are taught that the gradient is equal to "the rise over run" or more formally,
-<math>
+:<math>\mathrm{gradient}=\frac{\mathrm{rise}}{\mathrm{run}}</math>,
-\nabla f \cdot u
-</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.
+the rise being defined as the difference between the highest point and the lowest point, negative if the highest is on the left and positive if the highest point in on the right. The run is defined as difference of the right <math>x</math>-value and the left <math>x</math>-value (see [[Cartesian coordinates]]).
-==Properties of the Gradient==
+This concept is usually first introduced with the introduction of [[linear equation]]s. The equation of a straight line in Cartesian coordinates is given by,
-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
+:<math>y=mx+c</math>
-<math>
+where <math>m</math> is the gradient of the line and <math>c</math> is the value of the <math>y</math> coordinate when <math>x=0</math>, this is called the [[y-intercept]], e.g, where the line intercepts the [[y-axis]].
-f\circ\gamma(t) = c
-</math>
-since <math>S</math> is a level set. Taking derivatives of both sides and applying the chain rule, we get that
-<math>
-\nabla f\cdot \gamma'(0) = \nabla f\cdot v = 0
-</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]]

Difference between revisions of "Gradient"

Revision as of 20:31, July 2, 2008

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