Laplace’s Equation and Harmonic Functions (p.75) Laplace’s Equation. Theorem: If f(z)=u(x,y)+iv(x,y) is analytic in a domain D, then its components functions u and v are harmonic in D.  Flux lines in a charge free field satisfy Laplace’s Equation.  The potential & flux in a charge free 2-dimensional region are the real and imaginary parts respectively of an analytic function.  Their max. & min. are obtained on the boundary.  Constant potential lines and constant flux lines are perpendicular to each other.  How to determine if lines are perpendicular?

Supongamos que u y v son las componentes horizontal y vertical, respectivamente del campo de velocidades be un flujo estacionario incompresible (la densidad permanece aproximadamente constante a lo largo de todo el flujo) e irrotacional en dos dimensiones. La condición para que el flujo sea incompresible es: Y la condición para que el flujo sea irrotacional es: Fluidos

then the incompressibility condition is the integrability condition for this differential: the resulting function is called the stream function because it is constant along flow lines. The first derivatives of ψ are given bystream functionflow lines and the irrotationality condition implies that ψ satisfies the Laplace equation. The harmonic function φ that is conjugate to ψ is called the velocity potential. The Cauchy-Riemann equations imply thatvelocity potential Thus every analytic function corresponds to a steady incompressible, irrotational fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function. Si definimos la diferencial de la función  como: entonces la condición de incompresibilidad es la condición de integrabilidad para esta diferencial. La función resultanteSi definimos la diferencial de la función  como:

y=(-1/2)x y=2x+3

y=m’x+b’ y=mx+b 1    m = tan  tan  =  /m m’ = -m/D = -1/ tan  = -1/m For an analytic function f(z)=u(x,y)+iv(x,y), consider the curves: u(x,y) = constant = k v(x,y) = constant = k’ Tangents to these curves: dy/dx = -u x /u y dy/dx = -v x /v y = -u y /u x by Cauchy-Riemann equations. du = u x dx + u y dy = 0 dv = v x dx + v y dy = 0 The curves are orthogonal.

Consider the electrostatic potential between 2 infinite parallel conducting plates separated by a dielectric ( say air ) and 1 meter apart. Clearly f(z) = z = x + i y is analytic and has its real part u(x,y) = the electrostatic potential. The imaginary part v(x,y) = the flux. x=a constant y=a constant x y

If we consider the analytic function z 2 = (x + iy) 2 = (x 2  y 2 ) + i2xy, we get the following equipotential and flux lines: