I'm studying Bruce Driver's notes Curved Wiener Space Analysis, and I have a question.
We assume $M$ is a Riemannian manifold embedded in $\mathbb R^N$. The Riemannian metric is the classical inner product.
$P(m)$ is the projection map from $\mathbb R^N$ to the tangent plane at $m$. $Q(m)$ is the projection map into the normal plane at $m$.
$\Sigma$ is an $M$ valued semi-martingale. He uses $\delta\Sigma$ for Stratonovich integral and $d\Sigma$ for Itô integral.
At page 55, theorem 5.11 he gives the proof of $d\Sigma=P(\Sigma)\delta \Sigma$, i.e. $\Sigma_t-\Sigma_0=\int_0^tP(\Sigma_s)\delta\Sigma_s$.
At page 58, in the proof of Lemma 5.15, he says by the theorem 5.11 we have $$\int_0^\cdot F'(\Sigma)P(\Sigma)\delta\Sigma=\int_0^\cdot F'(\Sigma)\delta\Sigma$$ where $F:\mathbb R^N\to \mathbb R$ is a smooth function.
I can't get how he reach that conclusion. I think I'm missing a very easy point here. Why can we say $\int_0^\cdot F'(\Sigma)P(\Sigma)\delta\Sigma=\int_0^\cdot F'(\Sigma)\delta\Sigma$ from $d\Sigma=P(\Sigma)\delta \Sigma$.