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 .

See also Convective Derivative ,

Curl ,

Derivative ,

Divergence ,

Laplacian ,

Relative
Rate of Change ,

Slope ,

Vector
Derivative
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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 Gradient
Cite this as:
Weisstein, Eric W. "Gradient." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/Gradient.html

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