Mathematical Methods in the Physical Sciences · Problem 9.15

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · conformal-mapping-directional-derivative

Problem

Compare the directional derivative

d\phi/ds

(Chapter 6, Section 6) at a point and in the direction given by

dz

in the

z

plane, and the directional derivative

d\phi/dS

in the direction in the

w

plane given by the image

dw

dz

. Hence show that the rate of change of

T

in a given direction in the

z

plane is proportional to the corresponding rate of change of

T

in the image direction in the

w

plane. (See Section 10, Example 2.) Show that the proportionality constant is

|dw/dz|

. Hint: See equations (9.6) and (9.7).