gradients

In vector calculus, the gradient of a scalar-valued differentiable function



f


{\displaystyle f}
of several variables is the vector field (or vector-valued function)



∇
f


{\displaystyle \nabla f}
whose value at a point



p


{\displaystyle p}
is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point



p


{\displaystyle p}
, the direction of the gradient is the direction in which the function increases most quickly from



p


{\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



f
(

r

)


{\displaystyle f(\mathbf {r} )}
may be defined by:

where



d
f


{\displaystyle df}
is the total infinitesimal change in



f


{\displaystyle f}
for an infinitesimal displacement



d

r



{\displaystyle d\mathbf {r} }
, and is seen to be maximal when



d

r



{\displaystyle d\mathbf {r} }
is in the direction of the gradient



∇
f


{\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



f


{\displaystyle f}
at



p


{\displaystyle p}
. That is, for



f
:


R


n


→

R



{\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }
, its gradient



∇
f
:


R


n


→


R


n




{\displaystyle \nabla f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}
is defined at the point



p
=
(

x

1


,
…
,

x

n


)


{\displaystyle p=(x_{1},\ldots ,x_{n})}
in n-dimensional space as the vector
The gradient is dual to the total derivative



d
f


{\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



f


{\displaystyle f}
at a point



p


{\displaystyle p}
with another tangent vector




v



{\displaystyle \mathbf {v} }
equals the directional derivative of



f


{\displaystyle f}
at



p


{\displaystyle p}
of the function along




v



{\displaystyle \mathbf {v} }
; that is,



∇
f
(
p
)
⋅

v

=



∂
f


∂

v




(
p
)
=
d

f

p


(

v

)


{\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.

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