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).

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^^,

which simplifies to

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

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


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.

For a matrix A,

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

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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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.

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Cite this as:

Weisstein, Eric W. "Gradient." From MathWorld--A Wolfram Web Resource.

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