Definition:Gradient Operator/Real Cartesian Space

Let:


 * $f: \R^n \to \R, \mathbf x \mapsto f\left({\mathbf x}\right)$ be a real-valued function where:


 * $\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$

is a vector in $\R^n$.

Let the partial derivative of $f$ with respect to $x_i$ exist for all $x_i$.

The gradient of $f$ is defined as the column matrix:

Also see

 * Directional Derivative