A function u(x, y) is said to be a harmonic function if uxx + Uyy 0. That is, u is a solution of the Laplace equation. If u is a harmonic function, then -uy dx +ux dy = 0 is an exact equation (you should verify this). By solving this exact equation, one finds the so-called harmonic conjugate function v(x, y) such that vx = -Uy and Vy == ux. If u = 3xy (a harmonic function), then the corresponding exact equation for vis dx+ |dy = 0 and a harmonic conjugate of u is v= help (formulas) If u = 9e cos y (a harmonic function), then the corresponding exact equation for vis dx+ |dy = 0 and a harmonic conjugate of u is v= help (formulas) If u = = 5x³ — 15xy² (a harmonic function), then the corresponding exact equation for vis dx+ - |dy = 0 and a harmonic conjugate of u is v = help (formulas) Book: Section 1.8 of Notes on Diffy Qs

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