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.
See also
Convective Derivative, Curl, Derivative, Divergence, Laplacian, Relative Rate of Change, Slope, Vector Derivative
Explore with Wolfram|Alpha
References
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.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Gradient." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Gradient.html