Understanding the Nonlinear Mapping of Analytic Functions

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Homework Help Overview

The discussion revolves around the mapping of complex functions, specifically the nonlinear mapping of the function w=z^2 using Cartesian coordinates. The original poster seeks to understand the implications of this mapping on the graphical representation of lines in the xy plane and their transformation into parabolas in the uv plane.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to clarify whether the transformation of lines x=c and y=k into parabolas in the uv plane is due to the warping of the xy plane caused by the nonlinear mapping. Some participants affirm this notion, discussing the nature of nonlinear functions and their effects on geometric representations.

Discussion Status

The discussion is progressing with participants confirming the original poster's understanding of the warping concept related to nonlinear mappings. There is an exchange of ideas regarding the implications of this mapping, but no explicit consensus has been reached on the broader implications or interpretations.

Contextual Notes

Participants are navigating the complexities of nonlinear mappings and their graphical interpretations, with an emphasis on understanding the transformation of geometric shapes in different planes. The original poster's inquiry is framed within the context of a specific example from a textbook, which may impose certain constraints on the discussion.

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

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