= Solution
The <deformation gradient> is the <derivative> of the <deformation map> with respect to the <reference configuration> coordinates, at fixed time:
$$
\boxed{F_{iA}(X,t)=\frac{\partial x_i}{\partial X_A}(X,t),\qquad dx=F\,dX.}
$$
Its <determinant> $J=\det F$ is the local ratio of current to reference <volume>: $dv=J\,dV$ for an orientation-preserving <deformation in continuum mechanics>. Thus <elastic incompressibility> is the pointwise constraint $\boxed{J=1}$, rather than a requirement that each individual <principal stretch> be one.
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