Gradient - online puzzles
Gradient
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.
