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is called the `gradient'or `grad'or `del'
of and is a simple generalisation of the one dimensional
gradient, . It behaves in a similar manner but also obeys both
vector laws (adding, subtracting and multiplying vectors) and operator
laws (such as differentiating a product, quotient and so on). The
basic properties of the gradient operator are listed below.
Properties



gives the rate of change of in the direction .


(1.14) 
is the rate of change of in the direction of , i.e.
. It is also called the directional derivative.
From the above properties, we see that is a vector whose
magnitude and direction gives the maximum rate of change of or,
equivalently, the direction of the maximum gradient.
For the pressure, , if we plot contours of constant pressure ( =
constant), these contours are called ISOBARS and, as shown in Figure
1.7, crosses the isobars at right angles.
Figure 1.7:
A sketch of the isobars and the associated direction of
. Note that the force, however, acts from high pressure to
low pressure whereas points from low to high.

The pressure force is and so in Figure 1.7 the
air would flow from high pressure to low pressure. However, the
Earth's rotation makes the flow spiral as seen on weather maps.
Two final properties of are
 points towards larger values of .
 is invariant. This means that it is independent of
the location and orientation of the coordinate axes. Thus,
expressing in both cartesian and cylindrical coordinates
produces the same vector. Obviously, the components in each
coordinate system will be different.
Two examples illustrate the force generated by a nonuniform pressure.
Example 1. .4Consider the pressure . The isobars are shown in Figure
1.8 The force is
.
Figure 1.8:
The isobars and associated direction of the pressure force.

Example 1. .5Consider the pressure
. If is constant, then
the isobars are given by , as shown in Figure
1.9.
Figure 1.9:
The isobars and associated direction of the pressure force.

The force is given by
Next: The Flux and Divergence
Up: Introduction
Previous: Equation of Motion
Prof. Alan Hood
20001106