Mathematical Methods for Physics and Engineering - Matematica.NET

10.7 VECTOR OPERATORS∇φaθQPdφdsin the direction aφ =constantFigure 10.5 Geometrical properties of ∇φ. PQ gives the value of dφ/ds inthe direction a.then the total derivative of φ with respect to u along the curve is simplydφ dr= ∇φ ·du du . (10.28)In the particular case where the parameter u is the arc length s along the curve,the total derivative of φ with respect to s along the curve is given bydφds = ∇φ · ˆt, (10.29)where ˆt is the unit tangent to the curve at the given point, as discussed insection 10.3.In general, the rate of change of φ with respect to the distance s in a particulardirection a is given bydφ= ∇φ · â (10.30)dsand is called the directional derivative. Since â is a unit vector we havedφ= |∇φ| cos θdswhere θ is the angle between â and ∇φ as shown in figure 10.5. Clearly ∇φ liesin the direction of the fastest increase in φ, and|∇φ| is the largest possible valueof dφ/ds. Similarly, the largest rate of decrease of φ is dφ/ds = −|∇φ| in thedirection of −∇φ.349