Incidence-coordinate conormal kernel and canonical relation

Since $F$ is a submersion, its differential has rank $k$. The tangent space of its zero level is $T_{(g,x)}Z=\ker DF(g,x)$. In finite-dimensional linear algebra, the annihilator of the kernel of a rank-$k$ linear map is the image of its transpose: inclusion follows from ${\tau }^{T}DFv=0$ for $v\in \ker DF$, and both spaces have dimension $k$. Hence every conormal is uniquely $D{F}^T\tau =(D_g{F}^T\tau ,D_x{F}^T\tau )$.

By [F1], the incidence functional is integration of a test function against a smooth density on $Z$. Complete $F$ to submersion coordinates $(s,t)$ with $t=F(g,x)$, using [F3]. In these coordinates its action is ${\int }_{t=0}\mathrm{\Phi }(s,0)c(s)\mathrm{\mid }ds\mathrm{\mid }$ for a smooth coefficient $c$. Rewriting the ambient density as $\mathrm{\mid }dgdx\mathrm{\mid }$ and applying the ordinary change-of-variables rule gives $b(g,x)\delta (F(g,x))\mathrm{\mid }dgdx\mathrm{\mid }$ for a smooth $b$; the Jacobian factor is nonzero because the coordinate map is a diffeomorphism.

The cotangent relation uses the conormal $(\eta ,-\xi )$ by [F1]. Substituting the formula from step 1.1 gives $\eta =D_g{F}^T\tau$ and $-\xi =D_x{F}^T\tau$, so $\xi =-D_x{F}^T\tau$. Since $D{F}^T$ is injective, $\tau \mathrm{\ne }0$ gives a nonzero conormal. This proves the displayed relation, with every sign fixed by the kernel convention.