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)
|
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.
Arfken, G. "Gradient, 
" and "Successive Applications
of 
." §1.6 and 1.9 in Mathematical
Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 33-37
and 47-51, 1985.
Kaplan, W. "The Gradient Field." §3.3 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 183-185, 1991.
Morse, P. M. and Feshbach, H. "The Gradient." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 31-32, 1953.
Schey, H. M. Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, 3rd ed. New York: W. W. Norton, 1997.
Weisstein, Eric W. "Gradient." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Gradient.html
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