Complex Analysis: Theory and Applications · Problem 3.3.14

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · orthogonal-equipotentials-streamlines

Problem

Assume that F(z)=ϕ(x,y)+iψ(x,y)F(z) = \phi(x,y) + i\psi(x,y) is analytic on the domain DD and that F′(z)≠0F'(z) \neq 0 on DD. Consider the families of level curves {ϕ(x,y)=constant}\{\phi(x,y) = \text{constant}\} and {ψ(x,y)=constant}\{\psi(x,y) = \text{constant}\}, which are the equipotentials and streamlines for the fluid flow V(x,y)=F′(z)‾\mathbf{V}(x,y) = \overline{F'(z)}. Prove that the two families of curves are orthogonal. Hint: Suppose that (x0,y0)(x_0, y_0) is a point common to the two curves ϕ(x,y)=c1\phi(x,y) = c_1 and ψ(x,y)=c2\psi(x,y) = c_2. Use the gradients of ϕ\phi and ψ\psi to show that the normals to the curves are perpendicular.
Intuitive explanation

The key insight is that the Cauchy–Riemann equations create a hidden perpendicularity: the gradients of ϕ\phi and ψ\psi are related by a 90° rotation. Since ∇ϕ\nabla\phi is normal to the equipotential curve and ∇ψ\nabla\psi is normal to the streamline, showing these gradients are perpendicular proves the curves themselves are orthogonal. Use the Cauchy–Riemann relations ϕx=ψy\phi_x = \psi_y and ϕy=−ψx\phi_y = -\psi_x to compute their dot product and verify it vanishes.

Intuitive explanation

The key insight is that the Cauchy–Riemann equations create a hidden perpendicularity: the gradients of ϕ\phi and ψ\psi are related by a 90° rotation. Since ∇ϕ\nabla\phi is normal to the equipotential curve and ∇ψ\nabla\psi is normal to the streamline, showing these gradients are perpendicular proves the curves themselves are orthogonal. Use the Cauchy–Riemann relations ϕx=ψy\phi_x = \psi_y and ϕy=−ψx\phi_y = -\psi_x to compute their dot product and verify it vanishes.

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