Understanding the Nonlinear Mapping of Analytic Functions

In summary, In Advanced Engineering Mathematics by Erwin Kreyszig p.675, there is an example of mapping w=z^2 using Cartesian Co-ordinates. 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. This is because the surface we were graphing on has been warped in such a manner to define a new plane uv, where the projection of the lines x=c and y=c turns out to be a parabola due to the nonlinear mapping of z->w.
  • #1
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 [tex]w=z^2[/tex] using Cartesian Co-ordinates, w is defined as
[tex]w=u(x,y)+iv(x,y)[/tex], therefore, [tex]u=Re(z^2)=x^2-y^2[/tex] and [tex]v=Im(z^2)=2xy[/tex]. 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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  • #2
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.
 
  • #3
Thank you. Is my explanation right? The second paragraph about the space being warped?
 
  • #4
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.
 

1. What is mapping analytic functions?

Mapping analytic functions is a mathematical technique used to represent complex relationships between variables. It involves creating a visual map or diagram to show how one variable affects another, and is often used in fields such as physics, economics, and engineering.

2. How is mapping analytic functions used in scientific research?

Mapping analytic functions can be used to study and understand complex systems or processes, such as climate patterns or biological interactions. It can also be used in data analysis to identify correlations and make predictions.

3. What are the benefits of using mapping analytic functions?

Mapping analytic functions can help simplify and visualize complex data, making it easier to understand and interpret. It also allows researchers to identify patterns and relationships that may not be apparent in a traditional numerical analysis.

4. Are there any limitations to mapping analytic functions?

While mapping analytic functions can be a powerful tool, it is not always the most accurate or precise method for analyzing data. It relies on assumptions and simplifications, and may not capture all aspects of a complex system.

5. What skills are required to use mapping analytic functions?

To use mapping analytic functions effectively, one should have a strong understanding of mathematics and data analysis techniques. Familiarity with software programs such as MATLAB or Python can also be helpful in creating and interpreting visual maps.

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