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Grad Operator

It is useful to define the vector operator
\begin{displaymath}
\nabla \equiv \left( \frac{\partial}{\partial x},\, \frac{\partial}{\partial y},\,
\frac{\partial }{\partial z}\right),
\end{displaymath} (1362)

which is usually called the grad or del operator. This operator acts on everything to its right in a expression, until the end of the expression or a closing bracket is reached. For instance,
\begin{displaymath}
{\bf grad}\,f = \nabla f \equiv \left(\frac{\partial f}{\par...
...partial f}{\partial y},\,\frac{\partial f}{\partial z}\right).
\end{displaymath} (1363)

For two scalar fields $\phi$ and $\psi$,
\begin{displaymath}
{\bf grad}\,(\phi \,\psi) = \phi\,\, {\bf grad}\,\psi +\psi\,\, {\bf grad}\,\phi
\end{displaymath} (1364)

can be written more succinctly as
\begin{displaymath}
\nabla(\phi\, \psi) = \phi \,\nabla\psi + \psi\, \nabla \phi.
\end{displaymath} (1365)

Suppose that we rotate the coordinate axes through an angle $\theta $ about $Oz$. By analogy with Equations (A.1277)-(A.1279), the old coordinates ($x$, $y$, $z$) are related to the new ones ($x'$, $y'$, $z'$) via

$\displaystyle x$ $\textstyle =$ $\displaystyle x'\,\cos\theta - y'\,\sin \theta,$ (1366)
$\displaystyle y$ $\textstyle =$ $\displaystyle x\,'\sin\theta +y'\,\cos\theta,$ (1367)
$\displaystyle z$ $\textstyle =$ $\displaystyle z'.$ (1368)

Now,
\begin{displaymath}
\frac{\partial}{\partial x'} = \left(\frac{\partial x}{\part...
...l z}{\partial x'} \right)_{y',z'}
\frac{\partial}{\partial z},
\end{displaymath} (1369)

giving
\begin{displaymath}
\frac{\partial}{\partial x'} = \cos\theta \,\frac{\partial}{\partial x} +
\sin\theta \,\frac{\partial}{\partial y},
\end{displaymath} (1370)

and
\begin{displaymath}
\nabla_{x'} = \cos\theta\, \nabla_x + \sin\theta \,\nabla_y.
\end{displaymath} (1371)

It can be seen, from Equations (A.1280)-(A.1282), that the differential operator $\nabla$ transforms in an analogous manner to a vector. This is another proof that $\nabla f$ is a good vector.

Figure A.113: Cylindrical polar coordinates.
\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{AppendixA/figA.25.eps}}
\end{figure}


next up previous
Next: Curvilinear Coordinates Up: Vector Algebra and Vector Previous: Gradient
Richard Fitzpatrick 2011-03-31