Gradient

In mathematics, the gradient of a real-valued differentiable function $f(x_{1},...,x_{n})$ at a point $p$ is a vector in $R^{n}$ which points in the direction in which $f$ increases most rapidly at $p$. The magnitude of the gradient at $p$ is equal to the maximum directional derivative of $f$ at $p$.

More precisely, we define the gradient, $\nabla f$ of $f$ to be the vector-field:

$\nabla f=({\frac {\partial f}{\partial x_{1}}},...,{\frac {\partial f}{\partial x_{n}}})$

If $u$ is a unit vector in $R^{n}$, then, by the chain rule, the directional derivative of $f$ in the direction of $u$ is simply the dot product:

$\nabla f\cdot u$

Evidently by the Cauchy-Schwartz inequality, the directional derivative in the direction $u$ is maximal in the direction of the gradient, and equal to $||\nabla f||$ for $u$ a unit vector in the direction of the gradient.