Discussion
Let \(f : \mathbb{R}^2 \to \mathbb{R}^2\) be defined as
\[ f(x, y) = (e^x \cos y,\ e^x \sin y). \]
Which one of the following is TRUE?
Answer: C
The true statement is f is locally invertible.
The Jacobian determinant is
\[ \det \begin{pmatrix} e^x \cos y & -e^x \sin y \\ e^x \sin y & e^x \cos y \end{pmatrix} = e^{2x} \gt 0 \]
at every point, so by the inverse function theorem f has a smooth local inverse around each point.
It is not one-to-one, because f is \(2\pi\)-periodic in y: \(f(x, y) = f(x, y + 2\pi)\). So it cannot be invertible on the whole plane (it also never takes the value (0,0)). The Jacobian is positive, not negative. This map is the complex exponential \(e^{x+iy}\) written in real coordinates.
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