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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?

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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

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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:

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y=(-1/2)x+6 12 6 y=2x+3

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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.

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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

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

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