34
DIFFERENTIAL AND INTEGRAL OPERATIONS
(2.70)
Finally, the right-hand side of Equation 2.70 is converted back into vector notation
to obtain the result
v
XVXV=V(V-5)
-v2V.
(2.7
1)
Notice that the Laplacian operator can act on both vector and scalar fields.
In
Equation
2.68, the Laplacian operates on a scalar field to product a scalar.
In
Equation 2.71,
it
operates on a vector field to produce a vector.
-
2.4
INTEGRAL DEFJNITIONS
OF
THE
DIFFERENTLAL
OPERATORS
In Equations 2.25,2.27, and 2.29, we provided expressions for calculating the gradi-
ent, divergence, and curl. Each of these relations is valid only in a Cartesian coordinate
system, and all are in terms of spatial derivatives
of
the field. Integal definitions of
each of the differential operators
also
exist. We already derived one such definition
for the curl in Equation 2.63.
In
this
section,
we
present similar definitions for the
gradient and divergence,
as
well
as
an alternate definition for the curl. Their deriva-
tions, which are all similar to the derivation
of
Equation 2.63, are in most introductory
calculus texts. We present only the results here.
The gradient of a scalar field at a particular point can be generated
from
(2.72)
where
V
is a volume that includes the point of interest, and
S
is the closed surface
that surrounds
V.
Both
S
and
V
must be
shrunk
to infinitesimal size for
this
equation
to hold.
To get the divergence of a vector field at a point, we integrate the vector field over
an
infinitesimal surface
S
that encloses that point, and divide by the infinitesimal
volume:
We already derived the integral definition
(2.73)
(2.74)
which generates the curl.
This
definition is a bit clumsy because it requires the
calculation of three different integrals, each with a different orientation of
S,
to get
all three components of the curl. The following integral definition does not have this
