Mathematical Methods in the Physical Sciences · Problem 11.35

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · cauchy-riemann-curvilinear-coordinates

Problem

Show that the Cauchy-Riemann equations [see (2.2) and Problem 2.46] in a general orthogonal curvilinear coordinate system [see Chapter 10, Sections 6 and 7] are

\dfrac{1}{h_1}\dfrac{\partial u}{\partial x_1} = \dfrac{1}{h_2}\dfrac{\partial v}{\partial x_2}, \quad \dfrac{1}{h_1}\dfrac{\partial v}{\partial x_1} = -\dfrac{1}{h_2}\dfrac{\partial u}{\partial x_2}

where, as in Chapter 10, Section 8, the variables are

x_1

x_2

and the scale factors are

h_1

h_2

. Hint: Consider the directional derivatives (Chapter 6, Section 6) in two perpendicular directions. (Compare Problem 2.46.) Also show that

u

and

v

satisfy Laplace's equation, Chapter 10, equation (9.10) (drop the

x_3

term and set

h_3 = 1

Related theory

Cauchy Riemann Equations

Browse the problem library