In vector calculus, the gradient of a scalar-valued differentiable function ${\displaystyle f}$ of several variables is the vector field (or vector-valued function) ${\displaystyle \nabla f}$ whose value at a point ${\displaystyle p}$ is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point ${\displaystyle p}$, the direction of the gradient is the direction in which the function increases most quickly from ${\displaystyle p}$, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function ${\displaystyle f(\mathbf {r} )}$ may be defined by:

${\displaystyle df=\nabla f\cdot d{\bf {r}}}$

where ${\displaystyle df}$ is the total infinitesimal change in ${\displaystyle f}$ for an infinitesimal displacement ${\displaystyle d{\bf {r}}}$, and is seen to be maximal when ${\displaystyle d{\bf {r}}}$ is in the direction of the gradient ${\displaystyle \nabla f}$. The nabla symbol ${\displaystyle \nabla }$, written as an upside-down triangle and pronounced "del", denotes the vector differential operator.

When a coordinate system is used in which the basis vectors are not functions of position, the gradient is given by the vector whose components are the partial derivatives of ${\displaystyle f}$ at ${\displaystyle p}$. That is, for ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$, its gradient ${\displaystyle \nabla f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}$ is defined at the point ${\displaystyle p=(x_{1},\ldots ,x_{n})}$ in n-dimensional space as the vector

${\displaystyle \nabla f(p)={\begin{bmatrix}{\frac {\partial f}{\partial x_{1}}}(p)\\\vdots \\{\frac {\partial f}{\partial x_{n}}}(p)\end{bmatrix}}.}$

The gradient is dual to the total derivative ${\displaystyle df}$: the value of the gradient at a point is a tangent vector – a vector at each point; while the value of the derivative at a point is a cotangent vector – a linear functional on vectors. They are related in that the dot product of the gradient of ${\displaystyle f}$ at a point ${\displaystyle p}$ with another tangent vector ${\displaystyle \mathbf {v} }$ equals the directional derivative of ${\displaystyle f}$ at ${\displaystyle p}$ of the function along ${\displaystyle \mathbf {v} }$; that is, ${\textstyle \nabla f(p)\cdot \mathbf {v} ={\frac {\partial f}{\partial \mathbf {v} }}(p)=df_{p}(\mathbf {v} )}$. The gradient admits multiple generalizations to more general functions on manifolds; see § Generalizations.