Gradient

The term "gradient" has several meanings in mathematics. The simplest is as a synonym for slope.

The more general gradient, called simply "the" gradient in vector analysis, is a vector operator denoted and sometimes also called del or nabla. It is most often applied to a real function of three variables , and may be denoted

(1)

For general curvilinear coordinates, the gradient is given by

(2)

which simplifies to

(3)

in Cartesian coordinates.

The direction of is the orientation in which the directional derivative has the largest value and is the value of that directional derivative. Furthermore, if , then the gradient is perpendicular to the level curve through if and perpendicular to the level surface through if .

In tensor notation, let

(4)

be the line element in principal form. Then

(5)

For a matrix ,

(6)

For expressions giving the gradient in particular coordinate systems, see curvilinear coordinates.