Exercise 9.17

Question

Exercise 17: Let $\v f=(f_1,f_2)$ be the mapping of $\R2$ into $\R2$ given by
    $$
        f_1(x,y)=e^x\cos y,\qquad f_2(x,y)=e^x\sin y.
    $$
    (a) What is the range of $\v f$?

(b) Show that the Jacobian of $\v f$ is not zero at any point of $\R2$. Thus every point of $\R2$ has a neighborhood in which $\v f$ is one-to-one. Nevertheless, $\v f$ is not one-to-one on $\R2$.

(c) Put $\v a=(0,\pi/3)$, $\v b=\v f(\v a)$, let $\v g$ be the continuous inverse of $\v f$, defined in a neighborhood of $\v b$, such that $\v g(\v b)=\v a$. Find an explicit formula for
    $\v g$, compute $\v f'(\v a)$ and $\v g'(\v b)$, and verity the formula (52).

(d) What are the images under $\v f$ of lines parallel to the coordinate axes?

analambanomenos · Accepted Answer

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(d) Fix $y=y_0$. Then $\v f$ maps $(x,y_0)$, $x\in\mathbf R$, to the ray from (but not including) the origin through the point $(\cos y_0,\sin y_0)$ on the unit circle. The negative values of $x$
    map to points on the ray inside the circle, and the positive values of $x$ map to points outside the unit circle.

Fix $x=x_0$. Then $\v f$ maps $(x_0,y)$, $v\in\mathbf R$, to the circle of radius $e^x$. Note that $\v f(x_0,y)=\v f(x_0,y+2n\pi)$ for all integers $n$.

(a) The results of (d) show that the range of $\v f$ is $\R2$ minus the origin.

(b) Since
    \begin{align*}
        D_1f_1(x,y) = \phantom{-}e^x\cos y &\qquad D_1f_2(x,y) = e^x\sin y \
        D_2f_1(x,y) = -e^x\sin y &\qquad D_1f_2(x,y) = e^x\cos y
    \end{align*}
    the Jacobian of $\v f$ at $(x,y)$ is $e^{2x}(\cos^2y+\sin^2y)=e^{2x}
e 0$ for all $x$. From (d) we have $\v f(x,y)=\v f(x,y+2n\pi)$ for all integers $n$, so $\v f$ is not one-to-one on $\R2$.

(c) Let $w=f_1(x,y)=e^x\cos y$, $z=f_2(x,y)=e^x\sin y$. Then, in a neighborhood of $\v b=(1/2,\sqrt{3}/2)$ where $w
e 0$, we have $w^2+z^2=e^{2x}$, $z/w=	an y$, so
    \begin{gather*}
        \v g(w,z)=\big(\log\sqrt{w^2+z^2},\arctan(z/w)\big) \
        \v f'(x,y)=
        \begin{pmatrix}
            \phantom{-}e^x\cos y & e^x\sin y \
            -e^x\sin y & e^x\cos y
        \end{pmatrix} \
        \v g'(w,z)=
        \begin{pmatrix}
            w/(w^2+z^2) & -z/(w^2+z^2) \
            z/(w^2+z^2) & \phantom{-}w/(w^2+z^2)
        \end{pmatrix} \
    \v f'(\v a)\circ\v g'(\v b)=
        \begin{pmatrix}
            \phantom{-}1/2 & \sqrt{3}/2 \
            -\sqrt{3}/2 & 1/2
        \end{pmatrix}
        \begin{pmatrix}
            1/2 & -\sqrt{3}/2 \
            \sqrt{3}/2 & \phantom{-}1/2
        \end{pmatrix}
    =
        \begin{pmatrix}
            1 & 0 \
            0 & 1 
        \end{pmatrix}
    \end{gather*}

Exercise 9.17

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Exercise 9.17

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