The gradient 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.

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.

CITE THIS AS:

Eric W. Weisstein. "Gradient." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Gradient.html