Research/Etc_Research2007.01.08 10:39
Gradient
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The gradient 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

del phi==1/(h_1)(partialphi)/(partialu_1)u_1^^+1/(h_2)(partialphi)/(partialu_2)u_2^^+1/(h_3)(partialphi)/(partialu_3)u_3^^, (2)

which simplifies to

del phi(x,y,z)==(partialphi)/(partialx)x^^+(partialphi)/(partialy)y^^+(partialphi)/(partialz)z^^ (3)

in Cartesian coordinates.

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

del _(e^->_alpha)e^->_beta==del _alphae^->_beta==1/(sqrt(g_alpha))partial/(partialx_alpha)e^->_beta. (5)

For a matrix A,

del |Ax|==((Ax)^(T)A)/(|Ax|). (6)

For expressions giving the gradient in particular coordinate systems, see curvilinear coordinates.

SEE ALSO: Convective Derivative, Curl, Divergence, Laplacian, Vector Derivative. [Pages Linking Here]

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.



CITE THIS AS:

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



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