Let $M$ be a smooth manifold, let $E\to M$ be a smooth complex vector bundle of rank $r$, let $\nabla^0$ and $\nabla^1$ be smooth connections on $E$, and define

\begin{align*} A:=\nabla^1-\nabla^0\in \Omega^1(M;\operatorname{End}(E)). \end{align*}

For $t\in [0,1]$, let

\begin{align*} \nabla^t:=\nabla^0+tA \end{align*}

be the affine path of connections, and let

\begin{align*} F_{\nabla^t}\in \Omega^2(M;\operatorname{End}(E)) \end{align*}

be its curvature. Let $k\ge 1$, and let $P:\mathfrak{gl}(r,\mathbb C)\to \mathbb C$ be a [homogeneous polynomial](/page/Homogeneous%20Polynomial) of degree $k$ invariant under the adjoint action of $GL(r,\mathbb C)$. Write the same symbol $P$ for its associated symmetric $k$-linear polarization, extended to $\operatorname{End}(E)$-valued differential forms by the standard Chern-Weil wedge-composition convention. Then

\begin{align*} P(F_{\nabla^1})-P(F_{\nabla^0})=d\left(k\int_{[0,1]} P(A,F_{\nabla^t},\dots,F_{\nabla^t})\,d\mathcal L^1(t)\right). \end{align*}