= Solution
The coordinate <partial derivatives> are $x_s=-x$, $y_s=-y$, $x_t=y$, and $y_t=-x$. Apply the <multivariable chain rule> to $v(s,t)=u(x(s,t),y(s,t))$:
$$
\boxed{v_s=-xu_x-yu_y,\qquad v_t=yu_x-xu_y.}
$$
All <partial derivatives> of $u$ are evaluated at $(e^{-s}\sin t,e^{-s}\cos t)$. This is a <logarithmic polar coordinate change>; its <Jacobian determinant> is $x^2+y^2=e^{-2s}>0$, so it is locally invertible away from the origin. A full angular turn repeats the same point, as usual for <polar coordinates>.
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