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 del and sometimes also called del or nabla. It is most often applied to a real function of three variables f(u_1,u_2,u_3), and may be denoted
| del f=grad(f). |
(1)
|
For general curvilinear coordinates, the gradient is given by
which simplifies to
| [画像: del phi(x,y,z)=(partialphi)/(partialx)x^^+(partialphi)/(partialy)y^^+(partialphi)/(partialz)z^^ ] |
(3)
|
The direction of del f is the orientation in which the directional derivative has the largest value and |del f| is the value of that directional derivative. Furthermore, if del f!=0, then the gradient is perpendicular to the level curve through (x_0,y_0) if z=f(x,y) and perpendicular to the level surface through (x_0,y_0,z_0) if F(x,y,z)=0.
In tensor notation, let
| ds^2=g_mudx_mu^2 |
(4)
|
be the line element in principal form. Then
For a matrix A,
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 DerivativeExplore with Wolfram|Alpha
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References
Arfken, G. "Gradient, del " and "Successive Applications of del ." §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
GradientCite this as:
Weisstein, Eric W. "Gradient." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Gradient.html