= Spatial derivative control in perturbations of the identity map
To deduce $Du_t=I+tDF+o(t)$ from $u_t(x)=x+tF(x)+o(t)$, the remainder must be small in a topology controlling its spatial derivatives, such as $C^1$ locally. Small displacement alone is insufficient: $u_t(x,y,z)=(x+t^2\sin(x/t^2),y,z)$ differs uniformly from the identity by $o(t)$, but has <Jacobian determinant> $2$ at $x=0$ for nonzero $t$. A smooth <flow map> generated by a smooth <vector field> has the required derivative control.
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