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Complex functions, Laplace's equation and orthogonality of u & v curves

Deriving some interesting properties of complex analytic functions, starting from the Cauchy-Riemann equations. In particular, we show that the real and imaginary parts of a complex analytic function both obey Laplace's equation, and that curves of constant u are orthogonal to curves of constant v. These properties have some useful applications in Physics! The thumbnail shows contours of u and v for the function f(z) = z³. About me: I studied Physics at the University of Cambridge, then stayed on to get a PhD in Astronomy. During my PhD, I also spent four years teaching Physics undergraduates at the university. Now, I'm working as a private tutor, teaching Physics & Maths up to A Level standard. My website: https://benyelverton.com/ #physics #mathematics #complexnumbers #functions #orthogonal #contours #analyticfunction #function #laplaceequation #electrostatics #gravity #fluiddynamics #laplacian #differentialequation #gradient #differentiation #argandplane #cauchyriemannequations #complexanalysis #analysis #maths #math #science #education