Gradient Operator

The gradient operator is defined as

$$\nabla \triangleq \hat{\mathbf{i}}\frac{\partial }{\partial x}+\hat{\mathbf{j}}\frac{\partial }{\partial y}+\hat{\mathbf{k}}\frac{\partial }{\partial z}$$

where $\hat{\mathbf{i}}$, $\hat{\mathbf{j}}$, and $\hat{\mathbf{k}}$ are the unit vectors in the $x$, $y$, and $z$ directions, respectively.

Use

The gradient operator operates on scalar functions and returns a vector.

Thus, the gradient operator points in the direction of greatest increase and ha a magnitude equal to the slop of the function at that point.

For some function $f$

$$f:{\mathbb{R}}^{n-1}\to \mathbb{R},\nabla f\in {\mathbb{R}}^{n-1}$$

and points in the direction of greatest increase.

Related

• Computing the Normal to a Surface

Nomenclature

To specifically refer to the $x$ component of the gradient of the function $F$, we would write

$$(\nabla F{)}_x=\frac{\partial F}{\partial x}$$