Difference between revisions of "Gradient"

Revision as of 15:22, July 2, 2008

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. The magnitude of the gradient at <math>p</math> is equal to the maximum directional derivative of <math>f</math> at <math>p</math>.

More precisely, the gradient of <math>f</math> is the vector-field:

<math> (\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n}) </math>

@@ Line 1: / Line 1: @@
-In [[mathematics]], a '''gradient''' is the rate a [[function]] increases.
+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. The magnitude of the gradient at <math>p</math> is equal to the maximum directional derivative of <math>f</math> at <math>p</math>.
-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,
+More precisely, the gradient of <math>f</math> is the vector-field:
-:<math>\frac{F(b)-F(a)}{b-a}</math>.
+<math>
+(\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n})
-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>,
-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 equation]]s. The equation of a straight line in Cartesian coordinates is given by,
-:<math>y=mx+c</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]].
 [[Category:mathematics]]

Difference between revisions of "Gradient"

Revision as of 15:22, July 2, 2008

Navigation menu

Search