Suppose x = r cos θ, y = r sin θ, and f(z) = u(x, y) + iv(x, y). Show that (1)and  (2)Now...

Suppose x = r cos θ, y = r sin θ, and f(z) = u(x, y) + iv(x, y). Show that

 (1)

and  (2)

Now use (3) in the foregoing expressions for vr and vθ . By comparing your results with the expressions for ur and uθ , deduce the Cauchy-Riemann equations in polar coordinates given in (4).

 (3)

 (4)