Finding the Inverse of a Transformation: Solving for u and v

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Homework Help Overview

The discussion revolves around finding the inverse of a transformation defined by the equations (x, y) = T(u,v) = (uv, v^u). Participants are exploring the relationships between the variables and the implications of the transformation's injectivity.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the challenges of finding u in terms of x and y, with one noting the involvement of a transcendental function. Others suggest alternative transformations and provide examples to illustrate the concept of inverses.

Discussion Status

The discussion includes various attempts to manipulate the transformation equations, with some participants providing specific examples and others expressing skepticism about the feasibility of finding a solution. There is no explicit consensus, but several lines of reasoning are being explored.

Contextual Notes

One participant mentions difficulty accessing LaTeX, which may affect the clarity of mathematical expressions. Additionally, the nature of the functions involved raises questions about the complexity of finding inverses in this context.

Castilla
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By some reason I can't access to Latex...

Let be (x, y) = T(u,v) = (uv, v^u)

So x = x(u,v) = uv
and y = y(u,v) = v^u.

I see that T is an injective function, but I can't find u = u(x, y). Can you help me?

Thanks.
 
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This involves a transcendental function. I'd try Mathematica see what it'd come up with, but I am not hopeful.
 
Well let's drop that function, I would thank if you could provide any example of a transformation with its inverse.
 
Suppose x=uv, y=vu^2. Then v=x/u. Substitute into y=vu^2 to get y=xu; solve for u=y/x.
 
OK, so u = u(x,y) = y/x and v = v(x,y) = (x^2)/y.

Thanks, Enuma!
 

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