The classical Wu formula claims that if $M$ is a smooth closed $n$-manifold with fundamental class $z\in H_n(M;\mathbb{Z}_2)$, then the total Stiefel-Whitney class $w(M)$ is equal to $Sq(v)$, where $v=\sum v_i\in H^*(M;\mathbb{Z}_2)$ is the unique cohomology class such that $$\langle v\cup x,z\rangle=\langle Sq(x),z\rangle$$ for all $x\in H^*(M;\mathbb{Z}_2)$. Thus, for $k\ge0$, $v_k\cup x=Sq^k(x)$ for all $x\in H^{n-k}(M;\mathbb{Z}_2)$, and $$w_k(M)=\sum_{i+j=k}Sq^i(v_j).$$ Here the Poincare duality guarantees the existence and uniqueness of $v$.

My question: if $M$ is a smooth compact $n$-manifold with boundary, is there a similar Wu formula? In this case, there is a fundamental class $z\in H_n(M,\partial M;\mathbb{Z}_2)$ and the relative Poincare duality claims that capping with $z$ yields duality isomorphisms $$D:H^p(M,\partial M;\mathbb{Z}_2)\to H_{n-p}(M;\mathbb{Z}_2)$$ and $$D:H^p(M;\mathbb{Z}_2)\to H_{n-p}(M,\partial M;\mathbb{Z}_2).$$

Thank you!