Let $M$ be a closed oriented smooth manifold of dimension $n$, and let $\pi:E\to M$ be an oriented smooth real vector bundle of rank $n$. Let $z:M\to E$ denote the zero section, and write $0_p:=z(p)\in E_p$ for the zero vector in the fiber over $p\in M$. Let $s:M\to E$ be a smooth section transverse to the zero section submanifold $z(M)\subset E$, and define the zero set by $Z(s):=\{p\in M:s(p)=0_p\}$. For each zero $p\in Z(s)$, let $\nabla s_p:T_pM\to E_p$ denote the induced vertical derivative of $s$ at $p$, equivalently the derivative of the local representative of $s$ in any oriented local trivialization of $E$. Define $\operatorname{ind}_p(s)=+1$ if $\nabla s_p$ preserves orientation and $\operatorname{ind}_p(s)=-1$ if $\nabla s_p$ reverses orientation. Then

\begin{align*} \langle e(E),[M]\rangle=\sum_{p\in Z(s)}\operatorname{ind}_p(s). \end{align*}