Definition:Gradient

Definition

Let $\LL$ be a straight line embedded in a Cartesian plane.

The slope of $\LL$ is defined as the tangent of the angle that $\LL$ makes with the $x$-axis.

Let $\map f {x_1, x_2, \ldots, x_n}$ denote a real-valued function on $\R^n$.

Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$.

Let $\ds \mathbf u = u_1 \mathbf e_1 + u_2 \mathbf e_2 + \cdots + u_n \mathbf e_n = \sum_{k \mathop = 1}^n u_k \mathbf e_k$ be a vector in $\R^n$.

The gradient of $f$ (at $\mathbf u$) is defined as:

$\ds \grad f$

$:=$

$\ds \nabla f$

$\ds $

$=$

$\ds \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \map f {\mathbf u}$

Definition of Del Operator

$\ds $

$=$

$\ds \sum_{k \mathop = 1}^n \dfrac {\map {\partial f} {\mathbf u} } {\partial x_k} \mathbf e_k$

A gradient, in the sense of an inclination to the horizontal, is also known as a slope.

The word grade can sometimes be seen in this context, but this is discouraged as it has a number of meanings.