Download We use Theorem 11.4, regarding the - Full

11.4
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Two-Dimensional Mathematical Models
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We use Theorem 11.4, regarding the orthogonality of the families of level curves
(Equations (11-16) and (11-17)), to develop ideas concerning the physical applications that we will consider.
◗ Theorem 11.4 (Orthogonal families of level curves) Let φ (x, y) be harmonic in a domain D, let ψ (x, y) be the harmonic conjugate, and let
F (z) = φ (x, y) + iψ (x, y) be the complex potential. Then the two families of level curves given in Equations (11-16) and (11-17), respectively, are
orthogonal in the sense that if (a, b) is a point common to the two curves
φ (x, y) = K1 and ψ (x, y) = K2 and if F (a + ib) = 0, then these two curves
intersect orthogonally.
Proof Since φ (x, y) = K1 is an implicit equation of a plane curve, the
gradient vector grad φ, evaluated at (a, b), is perpendicular to the curve at
(a, b). This vector is given by
N1 = φx (a, b) + iφy (a, b) .
Similarly, the vector N2 defined by
N2 = ψx (a, b) + iψy (a, b)
is orthogonal to the curve ψ (x, y) = K2 at (a, b). Using the Cauchy–
Riemann equations, φx = ψy and φy = −ψx , we have
N1 · N2 = φx (a, b) ψx (a, b) + φy (a, b) ψy (a, b)
= −φx (a, b) φy (a, b) + φy (a, b) φx (a, b) = 0.
(11-18)
In addition, F (a + ib) = 0, so we have
φx (a, b) + iψx (a, b) = 0.
The Cauchy–Riemann equations and the facts φx (a, b) = 0 and ψx (a, b) = 0
imply that both N1 and N2 are nonzero. Therefore, Equation (11-18) implies
that N1 is perpendicular to N2 , and hence the curves are orthogonal.