(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Letf= (f_{1},f_{2}) be the mapping of R^{2}into R^{2}given by f_{1}=e^{x}cos(y), f_{2}=e^{x}sin(y).

(1) What is the range of f?

(2) Show that every point of R^{2}has a neighborhood in whichfis one-to-one.

(3) Show thatfis not one-to-one on R^{2}.

3. The attempt at a solution

(1) Since e^{x}is nonzero and cos and sin cannot be 0 at the same time, then the origin is not in the range, so the range is R^{2}\{0,0}.

(2) Calculating the Jacobian we get:

J = e^{2x}cos^{2}y + e^{2x}sin^{2}y= e^{2x}, which is nonzero, so every point has a neighborhood in which f is one-to-one.

(3) This is where I'm really stuck. I assume we need to find a point that is mapped by two different ordered pairs but I don't know how to go about finding those points.

Any help on #3 or comments on the previous parts is appreciated. Thanks.

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# Homework Help: One-to-One in R2

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