81
edits
Changes
Jump to navigation
Jump to search
Line 1:
Line 1: − + − + + + + + + + +
no edit summary
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>.
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:
More precisely, we define the gradient, <math>\nabla f</math> of <math>f</math> to be the vector-field:
<math>
<math>
(\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n})
\nabla f = (\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n})
</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:
<math>
\nabla f \cdot u
</math>
Evidently by the Cauchy-Schwartz inequality, the directional derivative 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.
[[Category:mathematics]]
[[Category:mathematics]]