Q15P - Mathematical Methods in Physical Sciences

Question

Compare the directional derivative $\raisebox{1ex}{$\mathbf{d}\mathbf{\varphi }$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}\mathbf{s}$}\right.$  (Chapter 6, Section 6) at a point and in the direction given by dz  in the z  plane, and the directional derivative $\raisebox{1ex}{$\mathbf{d}\mathbf{\varphi }$}\!\left/ \!\raisebox{-1ex}{$\mathbf{d}\mathbf{s}$}\right.$   in the direction in the w  plane given by the image dw  of 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 $\left|\raisebox{1ex}{$dw$}\!\left/ \!\raisebox{-1ex}{$dz$}\right.\right|$ . Hint: See equations (9.6) and (9.7).

Step-by-Step Solution

Answer

The rate of change of T  in a given direction in  z plane is proportional to the corresponding rate of change of T  in the image direction in the z  plane.

Step 1: Determine the Proportionality constant.

In a proportional connection, the constant of proportionality is the ratio that connects two given numbers.

Let w = f(z) = u + iv  be any analytical function.

Step 2: Show that the rate of change of T in a given direction in z plane is proportional to the corresponding rate of change of T in the image direction in the w plane.

Then, calculate:

 $$\begin{gathered} dz=dx+idy \\ dw=du+idv \\ {\left|dz\right|}^2=d{x}^2+d{y}^2 \\ {\left|dw\right|}^2=d{u}^2+d{v}^2 \end{gathered}$$

The square of the arc length element in the (x,y)  plane is as follows:

 $$\begin{gathered} d{s}^2=d{x}^2+d{y}^2 \\ ={\left|dz\right|}^2 \\ ={\left|\frac{dz}{dw}\right|}^2{\left|dw\right|}^2 \\ ={\left|\frac{dz}{dw}\right|}^{2}d{S}^2 \end{gathered}$$

Now, calculate further as follows:

 $\begin{array}{rcl}\frac{d{S}^2}{d{s}^2}& =& {\left|\frac{dz}{dw}\right|}^2\\ \frac{dS}{ds}& =& \left|\frac{dz}{dw}\right|\\ \frac{\frac{dT}{ds}}{{\displaystyle \frac{dT}{ds}}}& =& \left|\frac{dz}{dw}\right|\\ \frac{dT}{ds}& =& \left|\frac{dz}{dw}\right|\frac{dT}{ds}\\ & & \end{array}$

Therefore, the rate of change of  T in a given direction in  z plane is proportional to the corresponding rate of change of  T in the image direction in the   plane, and the proportionality constant is $\begin{array}{rcl}& & \frac{dw}{dz}\\ & & \end{array}$ .