Gradient

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

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

For general Curvilinear Coordinates, the gradient is given by

$$\nabla\phi = {1\over h_1} {\partial\phi\over\partial u_1} ... ...+ {1\over h_3} {\partial\phi\over\partial u_3} \hat {\bf u}_3,$$ (2)

which simplifies to

$$\nabla\phi(x,y,z)={\partial\phi\over\partial x}\hat{\bf x} +... ...partial y}\hat{\bf y}+{\partial\phi\over\partial z}\hat{\bf z}$$ (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

$$ds^2=g_\mu\,{dx_\mu}^2$$ (4)

be the Line Element in principal form. Then

$$\nabla_{{\vec e}_\alpha} {\vec e}_\beta=\nabla_\alpha {\vec ... ...{g_\alpha}} {\partial \over \partial x_\alpha} {\vec e}_\beta.$$ (5)

For a Matrix ${\hbox{\sf A}}$,

$$\nabla \vert{\hbox{\sf A}}{\bf x}\vert={({\hbox{\sf A}}{\bf x})^{\rm T} {\hbox{\sf A}}\over \vert{\hbox{\sf A}}{\bf x}\vert}.$$ (6)

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

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

References

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.