= Solution
Differentiate the first-order <chain rule> again, including derivatives of the coordinate coefficients. The <second derivatives under a two-variable coordinate change> are
$$
\boxed{f_{xx}=f_{uu}u_x^2+2f_{uv}u_xv_x+f_{vv}v_x^2+f_u\,u_{xx}+f_v\,v_{xx},}
$$
and
$$
\boxed{f_{yy}=f_{uu}u_y^2+2f_{uv}u_yv_y+f_{vv}v_y^2+f_u\,u_{yy}+f_v\,v_{yy}.}
$$
The last two terms account for derivatives of the coordinate functions; they vanish for an <affine map>. Smoothness justifies equality of the two mixed partial derivatives.
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