Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.1 COMPLEX POTENTIALS

P

Q

y

x

nˆ

Figure 25.2 A curve joining the pointsPandQ. Also shown isˆn, the unit

vector normal to the curve.

the difference in the values ofψat any two pointsPandQconnected by some

pathC, as shown in figure 25.2. This difference is given by

ψ(Q)−ψ(P)=

∫Q

P

dψ=

∫Q

P

(

∂ψ

∂x

dx+

∂ψ

∂y

dy

)

,

which, on using the Cauchy–Riemann relations, becomes

ψ(Q)−ψ(P)=

∫Q

P

(

−

∂φ

∂y

dx+

∂φ

∂x

dy

)

=

∫Q

P

∇φ·nˆds=

∫Q

P

∂φ

∂n

ds,

whereˆnis the vector unit normal to the pathCandsis the arc length along the

path; the last equality is written in terms of the normal derivative∂φ/∂n≡∇φ·ˆn.

Now suppose that in an electrostatics application, the pathCis the surface of

a conductor; then

∂φ

∂n

=−

σ

 0

,

whereσis the surface charge density per unit length normal to thexy-plane.

Therefore− 0 [ψ(Q)−ψ(P)] is equal to the charge per unit length normal to the

xy-plane on the surface of the conductor between the pointsPandQ. Similarly,

in fluid mechanics applications, if the density of the fluid isρand its velocity is

Vthen

ρ[ψ(Q)−ψ(P)] =ρ

∫Q

P

∇φ·nˆds=ρ

∫Q

P

V·ˆnds

is equal to the mass flux betweenPandQper unit length perpendicular to the

xy-plane.