Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.7 VECTOR OPERATORS

φ=constant

∇φ

a

P

Q

dφ

ds

in the directiona

θ

Figure 10.5 Geometrical properties of∇φ.PQgives the value ofdφ/dsin

the directiona.

then the total derivative ofφwith respect toualong the curve is simply

dφ

du

=∇φ·

dr

du

. (10.28)

In the particular case where the parameteruis the arc lengthsalong the curve,

the total derivative ofφwith respect tosalong the curve is given by

dφ

ds

=∇φ·ˆt, (10.29)

where ˆtis the unit tangent to the curve at the given point, as discussed in

section 10.3.

In general, the rate of change ofφwith respect to the distancesin a particular

directionais given by

dφ

ds

=∇φ·aˆ (10.30)

and is called the directional derivative. Sinceaˆis a unit vector we have

dφ

ds

=|∇φ|cosθ

whereθis the angle betweenˆaand∇φas shown in figure 10.5. Clearly∇φlies

in the direction of the fastest increase inφ,and|∇φ|is the largest possible value

ofdφ/ds. Similarly, the largest rate of decrease ofφisdφ/ds=−|∇φ|in the

direction of−∇φ.