Grad Operator
It is useful to define the vector operator
![$\displaystyle \nabla \equiv \left( \frac{\partial}{\partial x},\, \frac{\partial}{\partial y},\,
\frac{\partial }{\partial z}\right),$](img4805.png) |
(1.119) |
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,
![$\displaystyle {\bf grad}\,f = \nabla f \equiv \left(\frac{\partial f}{\partial x},\,
\frac{\partial f}{\partial y},\,\frac{\partial f}{\partial z}\right).$](img4806.png) |
(1.120) |
For two scalar fields
and
,
![$\displaystyle {\bf grad}\,(\phi \,\psi) = \phi\,\, {\bf grad}\,\psi +\psi\,\, {\bf grad}\,\phi$](img4807.png) |
(1.121) |
can be written more succinctly as
![$\displaystyle \nabla(\phi\, \psi) = \phi \,\nabla\psi + \psi\, \nabla \phi.$](img4808.png) |
(1.122) |
Suppose that we rotate the coordinate axes through an angle
about
.
By analogy with Equations (A.17)–(A.19), the old coordinates (
,
,
) are related
to the new ones (
,
,
) via
Now,
![$\displaystyle \frac{\partial}{\partial x'} = \left(\frac{\partial x}{\partial x...
...eft(\frac{\partial z}{\partial x'} \right)_{y',z'}
\frac{\partial}{\partial z},$](img4812.png) |
(1.126) |
giving
![$\displaystyle \frac{\partial}{\partial x'} = \cos\theta \,\frac{\partial}{\partial x} +
\sin\theta \,\frac{\partial}{\partial y},$](img4813.png) |
(1.127) |
and
![$\displaystyle \nabla_{x'} = \cos\theta\, \nabla_x + \sin\theta \,\nabla_y.$](img4814.png) |
(1.128) |
It can be seen, from Equations (A.20)–(A.22), that
the differential operator
transforms in an analogous manner to
a vector.
This is another proof that
is a good vector.