= Hadamard jump condition
{c}
{title2=$Dy=A\text{ or }B,\quad y\text{ continuous on }N^\perp\implies A-B=a\otimes N$}
Suppose an affine deformation on each side of a plane with unit normal $N$ is continuous across it. Tangential derivatives agree, so $(A-B)t=0$ for every $t\perp N$. Decomposition of an arbitrary vector into its tangent component and $(x\cdot N)N$ then gives $(A-B)x=((A-B)N)(N\cdot x)$, proving the rank-one form. Conversely that form makes the two affine maps agree on the plane when their constants agree there.
Orientation preservation on both sides also requires $1-N\cdot A^{-1}a>0$: the <matrix determinant lemma> gives $\det B=\det A(1-N\cdot A^{-1}a)$. This inequality is additional geometric information supplied by the determinant signs, not by continuity alone.
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