Mathematical Methods for Physics and Engineering - Matematica.NET

25.1 COMPLEX POTENTIALSyQxPˆnFigure 25.2 A curve joining the points P and Q. Also shown is ˆn, the unitvector normal to the curve.the difference in the values of ψ at any two points P and Q connected by somepath C, as shown in figure 25.2. This difference is given by∫ Q ∫ Q( )∂ψ ∂ψψ(Q) − ψ(P )= dψ = dx +PP ∂x ∂y dy ,which, on using the Cauchy–Riemann relations, becomes∫ Q(ψ(Q) − ψ(P )= − ∂φ)∂φdx +P ∂y ∂x dy∫ Q∫ Q∂φ= ∇φ · ˆn ds =PP ∂n ds,where ˆn is the vector unit normal to the path C and s is the arc length along thepath; the last equality is written in terms of the normal derivative ∂φ/∂n ≡∇φ· ˆn.Now suppose that in an electrostatics application, the path C is the surface ofa conductor; then∂φ∂n = − σ ,ɛ 0where σ is the surface charge density per unit length normal to the xy-plane.Therefore −ɛ 0 [ψ(Q) − ψ(P )] is equal to the charge per unit length normal to thexy-plane on the surface of the conductor between the points P and Q. Similarly,in fluid mechanics applications, if the density of the fluid is ρ and its velocity isV then∫ Q∫ Qρ[ψ(Q) − ψ(P )] = ρ ∇φ · ˆn ds = ρ V · ˆn dsPis equal to the mass flux between P and Q per unit length perpendicular to thexy-plane.875P