Question

We have discussed the fact that a conformal transformation magnifies and rotates an infinitesimal geometrical figure. We showed that \(|d w / d z|\) is the magnification factor. Show that the angle of \(d w / d z\) is the rotation angle. Hint: \(\quad\) Consider the rotation and magnification of an arc \(d z=d x+i d y\) (of length \(d s\) and angle arc tan \(d y / d x)\) which is required to obtain the image of \(d z,\) namely dw.

Short answer

The angle of \( \frac{dw}{dz} \) is the rotation angle in the conformal transformation.

Step-by-step solution

- Understanding the Conformal Transformation

A conformal transformation is a function that preserves angles locally. This means that it can magnify and rotate small figures (infinitesimal geometrical figures) without altering their shape.

- Representation of Infinitesimal Arc

Consider an infinitesimal arc with a complex representation given by \( dz = dx + i dy \). The length of this arc is given by \( ds = \sqrt{dx^2 + dy^2} \), and the angle is \( \text{arctan}(dy/dx) \).

- Image of Infinitesimal Arc

To find the image of \( dz \) under the transformation, consider \( dw = \frac{dw}{dz} \, dz \) where \( \frac{dw}{dz} \) is some complex number. The magnification factor is given by \( |\frac{dw}{dz}| \).

- Deriving Rotation Angle

Let \( \frac{dw}{dz} = R e^{i\theta} \) where \( R = |\frac{dw}{dz}| \) is the magnification and \( \theta \) is the argument (angle) of \( \frac{dw}{dz} \). Therefore, \( dw = R e^{i\theta} \, dz \) rotates \( dz \) by the angle \( \theta \).

- Conclusion

The angle \( \theta \) of \( \frac{dw}{dz} \) is the rotation angle of the infinitesimal arc \( dz \). Hence, we have shown that the angle of \( \frac{dw}{dz} \) is indeed the rotation angle in the conformal transformation.

Key concepts

Magnification Factor

In a conformal transformation, small geometrical figures are magnified and potentially rotated. The magnification factor is a measure of how much the size of these figures is increased. Mathematically, it is denoted as \( |\frac{dw}{dz}| \).
This factor represents the absolute value (or modulus) of the derivative of the transformation function \( w(z) \) concerning \( z \).

Essentially, if you have a tiny distance represented by \( dz \) in the complex plane, the image of this distance under the transformation is scaled by \( |\frac{dw}{dz}| \).
Practical uses include:

• Determining how much a small area is stretched.
• Understanding local geometric changes in a figure.
  

Rotation Angle

The rotation angle in a conformal transformation indicates how much a small geometric figure is rotated. To find this angle, we can look at the argument (or phase) of \( \frac{dw}{dz} \).
Let's break this down:

• Consider a small arc represented by \( dz = dx + i dy \).
• The transformation's image is \( dw = \frac{dw}{dz} dz \) where \( \frac{dw}{dz} \) is a complex number.
• The complex number can be written as \( R e^{i\theta} \), where \( R \) is the magnification factor \( |\frac{dw}{dz}| \) and \( \theta \) is the rotation angle.

Therefore, when the arc \( dz \) is mapped to \( dw \), it is scaled by \( |\frac{dw}{dz}| \) and rotated by the angle \( \theta \).

Complex Representation

Complex representation provides a powerful way to deal with transformations in the complex plane. In our context, we use complex numbers to represent small distances and transformations.

For an infinitesimal arc, we can write it as:
\( dz = dx + i dy \)
where \( dx \) and \( dy \) are real changes along the x and y axis, respectively.
When we apply a transformation, this arc is mapped to another infinitesimal arc \( dw \).
This mapping can be written as \( dw = \frac{dw}{dz} dz \), where \( \frac{dw}{dz} \) is itself a complex number.
The magnitude \( |\frac{dw}{dz}| \) gives the magnification factor, and the argument \( \theta \) of \( \frac{dw}{dz} \) gives the rotation angle.

Using complex representation makes it easier to visualize and compute how small figures change under transformations, allowing for a clear understanding of both magnitude changes and rotational shifts.