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The gradient is a Vector operator denoted $\nabla$ 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

\nabla f \equiv \rm {grad}(f).
\end{displaymath} (1)

For general Curvilinear Coordinates, the gradient is given by
= {1\over h_1} {\partial\phi\over\partial u_1} ...
...+ {1\over h_3} {\partial\phi\over\partial u_3} \hat {\bf u}_3,
\end{displaymath} (2)

which simplifies to
\nabla\phi(x,y,z)={\partial\phi\over\partial x}\hat{\bf x}
...partial y}\hat{\bf y}+{\partial\phi\over\partial z}\hat{\bf z}
\end{displaymath} (3)

in Cartesian Coordinates.

The direction of $\nabla f$ is the orientation in which the Directional Derivative has the largest value and $\vert\nabla f\vert$ is the value of that Directional Derivative. Furthermore, if $\nabla f \not = 0$, 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

\end{displaymath} (4)

be the Line Element in principal form. Then
\nabla_{{\vec e}_\alpha} {\vec e}_\beta=\nabla_\alpha {\vec ...
...{g_\alpha}} {\partial \over \partial x_\alpha} {\vec e}_\beta.
\end{displaymath} (5)

For a Matrix ${\hbox{\sf A}}$,
\nabla \vert{\hbox{\sf A}}{\bf x}\vert={({\hbox{\sf A}}{\bf x})^{\rm T} {\hbox{\sf A}}\over \vert{\hbox{\sf A}}{\bf x}\vert}.
\end{displaymath} (6)

For expressions giving the gradient in particular coordinate systems, see Curvilinear Coordinates.

See also Convective Derivative, Curl, Divergence, Laplacian, Vector Derivative


Arfken, G. ``Gradient, $\nabla$'' and ``Successive Applications of $\nabla$.'' §1.6 and 1.9 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 33-37 and 47-51, 1985.

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© 1996-9 Eric W. Weisstein