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Previous: Gradient
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}](img3476.png) |
(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}](img3477.png) |
(1363) |
For two scalar fields
and
,
![\begin{displaymath}
{\bf grad}\,(\phi \,\psi) = \phi\,\, {\bf grad}\,\psi +\psi\,\, {\bf grad}\,\phi
\end{displaymath}](img3478.png) |
(1364) |
can be written more succinctly as
![\begin{displaymath}
\nabla(\phi\, \psi) = \phi \,\nabla\psi + \psi\, \nabla \phi.
\end{displaymath}](img3479.png) |
(1365) |
Suppose that we rotate the coordinate axes through an angle
about
.
By analogy with Equations (A.1277)-(A.1279), the old coordinates (
,
,
) are related
to the new ones (
,
,
) via
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}](img3482.png) |
(1369) |
giving
![\begin{displaymath}
\frac{\partial}{\partial x'} = \cos\theta \,\frac{\partial}{\partial x} +
\sin\theta \,\frac{\partial}{\partial y},
\end{displaymath}](img3483.png) |
(1370) |
and
![\begin{displaymath}
\nabla_{x'} = \cos\theta\, \nabla_x + \sin\theta \,\nabla_y.
\end{displaymath}](img3484.png) |
(1371) |
It can be seen, from Equations (A.1280)-(A.1282), that
the differential operator
transforms in an analogous manner to
a vector.
This is another proof that
is a good vector.
Figure A.113:
Cylindrical polar coordinates.
![\begin{figure}
\epsfysize =2.75in
\centerline{\epsffile{AppendixA/figA.25.eps}}
\end{figure}](img3487.png) |
Next: Curvilinear Coordinates
Up: Vector Algebra and Vector
Previous: Gradient
Richard Fitzpatrick
2011-03-31