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) |
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.
" and "Successive Applications of 
." §1.6 and 1.9 in