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- Homework Statement
- Prove, using the definition, that ##\textbf{u}=\textbf{u}(\textbf{x})## is a bijection from the strip ##D=-\pi/2<x_1<\pi/2## in the ##x_1x_2##-plane onto the entire ##u_1,u_2##-plane.

- Relevant Equations
- ##u_1 = \tan{(x_1)}+x_2##

##u_2 = x_2^3##

How would one tackle this using the definition? (i.e. for some function ff that f(x)=f(y)⟹x=yf(x)=f(y)⟹x=y implies an injection and y=f(x)y=f(x) for all yy in the codomain of ff for a surjection, provided such x∈Dx∈D exist.)

One can solve the system of equations for x1x1 and x2x2 and that shows that u=u(x)u=u(x) has an inverse x(u)x(u) and that u(x(u))=xu(x(u))=x. This would only show that uu is surjective, correct?

One can solve the system of equations for x1x1 and x2x2 and that shows that u=u(x)u=u(x) has an inverse x(u)x(u) and that u(x(u))=xu(x(u))=x. This would only show that uu is surjective, correct?

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