Solving Complex Exponentials: Deduce & Explain Relation w/ Vector Diagram

[kevi555]

Hi,

If z=Ae^{i\theta}, deduce that dz = iz d\theta, and explain the relation in a vector diagram.

I know that z = x + iy but I don't know if that's going to help. Any hints or tips would be appreciated! Thanks!

 

[Philosophaie]

The derivative of z:

dz=i*A*e^(i*Θ)*dΘ

You have to ask yourself what are i,j: are they directions (r=xi+yj) or are they complex numbers ( x=a+i*b y=c+j*d).


Secondly, graphing on a polar plot one direction is for the real (x-axis) and the other for the imaginary (y-axis).

 

[kevi555]

Please note, I made an error in my first post...it is now correct.

 

[Philosophaie]

This may help:

z=A*cosΘ+i*A*sinΘ

 

[kevi555]

Alright, I can picture the Acos\theta and Asin\theta on the horizontal and vertical axes, respectively. I don't understand how the z is still in the derivative equation?

 

[Philosophaie]

dz is the change in z. i in a vector diagram is equal to 90deg, right. So when you take the derivative of a vector you shift 90deg.

 

[buffordboy23]

Alright, I can picture the Acos\theta and Asin\theta on the horizontal and vertical axes, respectively. I don't understand how the z is still in the derivative equation?

Philosophaie gave you a big hint. Look at the equation in his first post real carefully. What is "z" equal to? You already told us. =)