Difference between revisions of "Gradient"
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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. 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> | + | 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]] | ||
Revision as of 15:26, July 2, 2008
In mathematics, the gradient of a real-valued differentiable function 
at a point 
is a vector in 
which points in the direction in which 
increases most rapidly. The magnitude of the gradient at is equal to the maximum directional derivative of at .
More precisely, we define the gradient, 
of to be the vector-field:
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 is maximal in the direction of the gradient, and equal to 
for a unit vector in the direction of the gradient.
