Understanding the Nonlinear Mapping of Analytic Functions

[chaoseverlasting]

Homework Statement

This is an example in Advanced Engineering Mathematics by Erwin Kreyszig p.675 which I don't understand. If you map $$w=z^2$$ using Cartesian Co-ordinates, w is defined as
$$w=u(x,y)+iv(x,y)$$, therefore, $$u=Re(z^2)=x^2-y^2$$ and $$v=Im(z^2)=2xy$$. The function is graphed using u and v as the axes, and a line x=c is graphed as a parabola as is the like y=k.

What I want to understand is, that is this so because the surface we were graphing these lines on (which was the xy plane) has been warped in such a manner as to define a new plane uv so that the projection of the lines x=c and y=c, on this uv plane turns out to be a parabola? Is that so?

 

[Dick]

w=z^2 is a nonlinear function. It's going to change lines in the z plane into curves. I'm not sure why this would surprise you. So, yes, it is so.

 

[chaoseverlasting]

Thank you. Is my explanation right? The second paragraph about the space being warped?

 

[Dick]

Well, yes, because the mapping of z->w is nonlinear, if that's what you mean by 'space being warped'. If you've shown they are parabolas then I think you are done.