Definition

Gradient

\nabla f = \frac{\partial f}{\partial x}  + \frac{\partial f}{\partial y}  + \frac{\partial f}{\partial z} k

Statement

A vector built from the partial derivatives of a scalar function, pointing towards its fastest increase.

Symbols and units

$f$	the scalar field	—
$\partial f / \partial x$	partial derivative along x, others fixed	—
$, , k$	unit vectors along x, y, z	—

When it applies

Scalar fields that are differentiable at the point in question.

Don't use when

Not defined at a point where the field is not differentiable (a kink or discontinuity in the scalar field).

Derivation

Each component is the partial derivative along that axis, holding the others fixed:

\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}

Collected into a single vector:

\nabla f = \frac{\partial f}{\partial x}  + \frac{\partial f}{\partial y}  + \frac{\partial f}{\partial z} k

Worked examples

Steepest ascent on a paraboloid

f(x,y) = x² + y², evaluated at (1, 2):

\nabla f = (2 x, 2 y) = (2, 4)

Points radially outward from the origin, exactly the direction a ball would roll away from the bottom of the bowl.

Watch out for

A field derived as a negative gradient (such as the electric field from potential) points towards decreasing, not increasing, values of the scalar.