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**Unformatted text preview: **D g Ω Figure 7.14. Mapping to the Unit Disk. Remark : In this section, we have focused on the fluid mechanical roles of a harmonic function and its conjugate. An analogous interpretation applies when ϕ ( x, y ) represents an electromagnetic potential function; the level curves of its harmonic conjugate ψ ( x, y ) are the paths followed by charged particles under the electromotive force field v = ∇ ϕ . Similarly, if ϕ ( x, y ) represents the equilibrium temperature distribution in a planar domain, its level lines represent the isotherms — curves of constant temperature, while the level lines of its harmonic conjugate are the curves along which heat energy flows. Finally, if ϕ ( x, y ) represents the height of a deformed membrane, then its level curves are the contour lines of elevation. The level curves of its harmonic conjugate are the curves of steepest descent along the membrane, i.e., the paths followed by, say, water flowing down the membrane. 7.4. Conformal Mapping. As we now know, complex functions provide an almost inexhaustible supply of har- monic functions, i.e., solutions to the the two-dimensional Laplace equation. Thus, to solve an associated boundary value problem, we “merely” find the complex function whose real part matches the prescribed boundary conditions. Unfortunately, even for relatively simple domains, this remains a daunting task. The one case where we do have an explicit solution is that of a circular disk, where the Poisson integral formula (4.116) provides a complete solution to the Dirichlet boundary value problem. (See also Exercise for the Neumann problem.) Thus, one evident solution strategy for the corresponding boundary value problem on a more complicated domain is to transform it into a solved case by an inspired change of variables.to transform it into a solved case by an inspired change of variables....

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