Let $M$ is a $2n$-dim smooth Riemannian manifold, $\eta$ is a vector field on $M$, $p\in M$ is a isolated zero of $\eta$. Then we can define a map from $\partial B_{p}(\varepsilon)$ to $S^{2n-1}(1)$ by $\eta$: $$\eta_{p}=\frac{\eta}{|\eta|}: \partial B_{p}(\varepsilon)\rightarrow S^{2n-1}(1)$$
Here $B_{p}(\varepsilon)$ is a ball about of $p$ with radius $\varepsilon$.
Let $f_1,\cdots,f_{2n-1}$ be an orthonormal basis of $T(\partial B_{p}(\varepsilon))$, let $f^{*}_1,\cdots,f^{*}_{2n-1}$ be the metric dual basis.
We can find that
$$\eta^{*}\wedge(\nabla^{TM}_{f_1}\eta)^{*}\wedge(\nabla^{TM}_{f_{2n-1}}\eta)^{*}$$
$$= (d{\rm vol}_{g^{TM}})\int^{B}\eta^{*}\wedge (\nabla^{TM}_{f_1}\eta)^{*}\wedge\cdots\wedge (\nabla^{TM}_{f_{2n-1}}\eta)^{*}$$
then
$$f^{\ast}_{1}\wedge\cdots\wedge f^{\ast}_{2n-1}\int^{B}\eta^{\ast}\wedge (\nabla^{TM}_{f_1}\eta)^{\ast}\wedge\cdots\wedge (\nabla^{TM}_{f_{2n-1}}\eta)^{\ast}=\eta^{\ast}_{p}\omega ,(1) $$
here $\omega$ be the volume form on $S^{2n-1}(1)$.
I find this formula (1) in the article “$\eta$-invariants and the Poincaré-Hopf index formula”. But I don't know how to get it. Is there some way to compute it?